Fabien Candelier

Bernhard Mehlig

Jacques Magnaudet

Time-dependent lift and drag on a rigid body in a viscous steady linear flow

Keywords:

We compute the leading-order inertial corrections to the instantaneous force acting on a rigid body moving with a time-dependent slip velocity in a linear flow field, assuming that the square root of the Reynolds number based on the fluid-velocity gradient is much larger than the Reynolds number based on the slip velocity between the body and the fluid. As a first step towards applications to dilute sheared suspensions and turbulent particle-laden flows, we seek a formulation allowing this force to be determined for an arbitrarily shaped body moving in a general linear flow. We express the equations governing the flow disturbance in a non-orthogonal coordinate system moving with the undisturbed flow and solve the problem using matched asymptotic expansions. The use of the co-moving coordinates enables the leading-order inertial corrections to the force to be obtained at any time in an arbitrary linear flow field. We then specialize this approach to compute the time-dependent force components for a sphere moving in three canonical flows: solid-body rotation, planar elongation, and uniform shear. We discuss the behaviour and physical origin of the different force components in the short-time and quasi-steady limits. Last, we illustrate the influence of time-dependent and quasi-steady inertial effects by examining the sedimentation of prolate and oblate spheroids in a pure shear flow.

Introduction

The pioneering experiments by Segré & Silberberg (1962a,b) revealed that small neutrally buoyant particles in a Poiseuille flow experience a lateral migration that concentrates them in an annulus with a well-defined radius. Almost simultaneously, [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] proved that Stokes-type solutions for the disturbance induced by a fore-aft symmetric body immersed in an arbitrary flow field cannot result in any lift force. These two discoveries, combined with the recent success of the matched asymptotic expansions technique (hereinafter abbreviated as MAE) in low-Reynolds-number flows ( Kaplun & L agerstrom 1 957;P roudman & P earson 1957), strongly stimulated the theoretical study of shear-induced inertial effects acting on rigid spherical or spheroidal bodies. In a seminal paper based on the use of an improved version of the MAE approach [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF], [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF] showed that a sphere translating along the streamlines of a simple shear flow w ith a non-zero slip velocity experiences a force that acts to move it perpendicular to the streamlines. His work, in which inertial effects due to the shear are assumed to dominate over those due to the slip between the undisturbed flow a nd t he s phere, m otivated a stream of theoretical and experimental studies during the next 50 years (see [START_REF] Stone | Philip Saffman and viscous flow theory[END_REF] for an overview covering the first 3 5 y ears). I n p articular, H arper & C hang (1968) worked out the case of a sphere translating in an arbitrary direction in a simple shear flow, d etermining i nertial c orrections t o t he d rag i n e very d irection a s w ell a s the components of the lift force. [START_REF] Mclaughlin | Inertial migration of a small sphere in linear shear flows[END_REF] extended Saffman's approach to physical situations in which slip effects are of similar magnitude to or even larger than those of the shear (see also [START_REF] Asmolov | Dynamics of a spherical particle in a laminar boundary layer[END_REF]. These results revealed that inertial effects due to the slip velocity (i.e. Oseen-like effects) decrease the magnitude of the lift force compared to Saffman's prediction and may even change its sign when the shear becomes weak enough.

All aforementioned results only hold for solid spheres and quasi-steady conditions. They were generalized to spherical drops and bubbles by [START_REF] Legendre | A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow[END_REF], who showed in particular that the lift force on a spherical bubble with a vanishingly small viscosity is (2/3) 2 that on a solid sphere. Saffman's analysis was also extended to the case of a sinusoidally time-varying slip velocity by [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF], who determined that the component of the lift force in phase with the slip decreases monotonically as the frequency of the excitation increases, whereas the out-of-phase component (which vanishes in the quasi-steady limit) goes through a maximum when this frequency is of the same order as the shear rate. An approximate transformation of these results to the time domain was achieved by [START_REF] Candelier | Time-dependent lift force acting on a particle moving arbitrarily in a pure shear flow, at small Reynolds number[END_REF], thus yielding the lift force for arbitrary time-dependent sphere translations along the streamlines of the shear flow.

Besides the unidirectional shear flow, two other configurations involving bidirectional planar linear flows w ere a lso c onsidered u sing t he M AE a pproach, s till under Saffman's assumptions. One of them corresponds to the case of a purely elongational flow. I t w as w orked o ut b y D rew ( 1978), w ho s howed t hat a s phere t ranslating i n a direction that is not aligned with one of the principal axes of the strain experiences a lift force, and hence tends to be deflected t owards o ne o f t hese a xes. T he second of these bidirectional configurations i s t he o ne w here t he s phere i s i mmersed i n a solid-body rotation flow, w hich a ctually i nvolves t wo c ontrasting s ubcases. I n t he first of them, considered by [START_REF] Herron | On the sedimentation of a sphere in a centrifuge[END_REF], the sphere is entrained by the flow ( as i n a c entrifuge) a nd t he s lip v elocity i s m ost o ften i n t he r adial direction, due to the density difference between the body and the fluid. H ence t he ' lift' f orce is circumferential and causes the sphere to lead the flow o r t o l ag b ehind i t, depending on the sign of the density difference. In contrast, in the second subcase, the slip is oriented along the streamlines of the rotating flow a nd i s s uch t hat t he s phere appears fixed i n t he l aboratory. T his i s t he c onfiguration ob tained by re leasing bu bbles or rigid particles in a rotating cylinder with a horizontal axis [START_REF] Van Nierop | Drag and lift forces on bubbles in a rotating flow[END_REF][START_REF] Bluemink | Drag and lift forces on particles in a rotating flow[END_REF][START_REF] Sauma-Pérez | An experimental study of the motion of a light sphere in a rotating viscous fluid[END_REF]. This situation was considered by [START_REF] Gotoh | Brownian motion in a rotating flow[END_REF]. The connection between these two subcases, which yield different values for the inertial contributions to the hydrodynamic force, was clarified by [START_REF] Candelier | Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers[END_REF], who established the general form of this force encapsulating the two situations. More specifically, he showed that this force involves in general two distinct 'history' terms accounting for the fact that in the first (respectively, second) subcase the sphere velocity is time-dependent when evaluated in the laboratory (respectively, rotating) frame.

In the above review, we did not discuss the influence of nearby walls on the lift force. Near-wall effects change the velocity profile of the undisturbed flow, induce slip between the body and the fluid even in the case of neutrally buoyant particles, and modify the disturbance flow resulting from the presence of the latter. Obviously they are of primary importance to understand migration phenomena in pipe and channel flows, as well as in rotating containers. These problems involve regular or singular perturbations, depending on whether the wall-particle separation distance is smaller or larger than the distance at which inertial effects start to modify the structure of the disturbance. Several forms of Lorentz's reciprocal theorem and the MAE approach have been used to compute the lift force. We refer the reader to papers by [START_REF] Leal | Particle motions in a viscous fluid[END_REF], [START_REF] Hogg | The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows[END_REF] and [START_REF] Magnaudet | Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow[END_REF] for reviews of the corresponding studies.

Coming back to unbounded linear flows, another series of studies employed the so-called 'induced-force' method (hereinafter abbreviated as IF) as an alternative to the MAE approach, based on the formulation developed by [START_REF] Mazur | A generalization of Faxén's theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow[END_REF] to extend Faxén's formulae to a sphere undergoing an arbitrary time-dependent motion in an inhomogeneous flow. In this method, an extra force is added to the Navier-Stokes equation to ensure that the slip velocity vanishes everywhere within the body, rendering the modified equation valid in the entire domain, both in the fluid and the body. This approach was first applied by [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF] to find the frequency-dependent inertial corrections to the force experienced by a sphere translating in a planar or an axisymmetric purely elongational flow. Pérez-Madrid, Rubi & [START_REF] Pérez-Madrid | Motion of a sphere through a fluid in stationary homogeneous flow[END_REF] then obtained the quasi-steady form of the friction tensor for the three canonical planar flow configurations discussed above. While their result agreed with that of [START_REF] Herron | On the sedimentation of a sphere in a centrifuge[END_REF] in a solid-body rotation flow, the components of the resistance tensor obtained in the case of a pure shear flow differed from those determined by [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF]. In particular the component corresponding to the Saffman's lift force was found to be approximately 2.3 times larger than predicted by the MAE approach [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF][START_REF] Saffman | The lift on a small sphere in a slow shear flow-corrigendum[END_REF]). This issue was reconsidered by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF], who identified that a non-algebraic term was unduly neglected by [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF] and [START_REF] Pérez-Madrid | Motion of a sphere through a fluid in stationary homogeneous flow[END_REF], leading to erroneous results in the quasi-steady limit (except in the solid-body rotation case where this term does not contribute to the final result). Having dealt with this term through a transformation described later, [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] could recover Saffman's prediction and conclude that the MAE and IF approaches yield identical results as expected. [START_REF] Miyazaki | Dependence of the friction tensor on the rotation of a frame of reference[END_REF] employed the same technique to clarify the connection between the quasi-steady results established by [START_REF] Herron | On the sedimentation of a sphere in a centrifuge[END_REF] and [START_REF] Gotoh | Brownian motion in a rotating flow[END_REF] in a solid-body rotation flow.

The above review indicates that the current knowledge regarding low-but-finite-Reynolds-number shear-induced lift forces acting on a sphere is quite satisfactory for both pure shear and solid-body rotation flows in the quasi-steady limit. Some results describing the time development of this force are also known, mostly in the short-time limit; we discuss them in § §4.1 and 5.1. The situation is more uncertain as far as the elongational flow is concerned, since only the quasi-steady expression of the inertial corrections derived by [START_REF] Drew | The force on a small sphere in slow viscous flow[END_REF] may potentially be correct in that case. Besides these three canonical configurations, no result has been established for more general linear flows resulting from an arbitrary combination of strain and rotation, neither in two nor in three dimensions. Once expressed in the proper eigenbasis, these flow fi elds de pend on tw o an d fiv e ind ependent par ameters, res pectively, in contrast with only one parameter (the shear, rotation, or strain rate) in the above three configurations. I n s uch s ituations, i nertial c orrections c annot b e o btained b y linearly superposing expressions available for solid-body rotation and planar elongation, since the governing equation for the flow d isturbance i s n onlinear. T his i s o bviously a major limitation on the route towards accurate predictions of particle motion in turbulent flows, s ince t he l ocal v elocity g radient t ensor o f t he c arrying fl ow ge nerally results from the superposition of time-dependent strain and rotation components.

The chief technical difficulty i n t his c lass o f p roblems i s t hat, owing t o t he presence of a space-dependent term (the one that was overlooked by [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF] and [START_REF] Pérez-Madrid | Motion of a sphere through a fluid in stationary homogeneous flow[END_REF]), the unsteady disturbance in Fourier space is governed by a set of coupled partial differential equations. This makes it particularly difficult to obtain the solution. For the solid-body rotation, this difficulty i s e asily overcome by using a rotating reference frame, since the space-dependent term disappears in this frame [START_REF] Herron | On the sedimentation of a sphere in a centrifuge[END_REF][START_REF] Miyazaki | Dependence of the friction tensor on the rotation of a frame of reference[END_REF][START_REF] Candelier | Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers[END_REF]. Based on this observation, it seems natural to seek a generic coordinate transformation that removes this term whatever the carrying flow. T his i s t he b ackbone o f t he p resent w ork. M ore precisely, we express the unsteady disturbance problem in a system of moving non-orthogonal coordinates that follow the undisturbed flow. I n F ourier s pace, t he d isturbance i s then determined by a set of ordinary differential equations in these co-moving coordinates, making the problem much easier to solve. Solving these equations and transforming back to the laboratory frame yields the desired inertial corrections irrespective of the nature of the linear carrying flow. T his t echnique i s s imilar i n e ssence t o t he approach used in the rapid distortion theory (RDT), pioneered by [START_REF] Batchelor | The effect of rapid distortion of a fluid in turbulent motion[END_REF] to determine how a turbulent velocity fluctuation i s d istorted b y a s trong non-uniform mean flow. I n t he p articular c ontext o f t he fl ow pa st a ri gid bo dy, th is id ea was also used by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF], extending a technique developed by [START_REF] Onuki | Critical phenomena of classical fluids under flow. I: mean field approximation[END_REF] for a scalar field, b ut, c ompared t o o ur a pproach, t hey employed it differently, namely by considering time-dependent wavenumbers in the Fourier transform of the disturbance equation. These connections are discussed in more detail at the end of § 3.1.

A second major limitation of most results available in the literature is that they only apply to spherical bodies. However, most rigid particles, like aggregates of fibres, in flows o f g eophysical o r e ngineering r elevance a re n ot s pherical. T his g enerally makes the determination of the lift force an even more difficult p roblem. A n e xception is the case of particles that do not exhibit a fore-aft symmetry and move in a linear flow. S uch p articles d o e xperience a l ift f orce, e ven i n t he c reeping-flow li mit, as recognized by [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] and analysed for arbitrary body shapes by Brenner (1964b). For this reason, the various components of the force and torque acting on particles with 'simple' asymmetric shapes suspended in a linear shear flow c ould be determined using standard techniques; for example, [START_REF] Nir | On the creeping motion of two arbitrary-sized touching spheres in a linear shear field[END_REF] considered an aggregate made of two arbitrary-sized touching spheres, or [START_REF] Singh | Rigid ring-shaped particles that align in simple shear flow[END_REF] looked at a ring-like particle with asymmetric inner and outer edges. The difficulty i ncreases s ubstantially a s s oon a s i nertia e ffects h ave t o b e c onsidered. The lift force acting on a fibre s ettling i n a w eak s hear fl ow wi th a se dimentation (slip) Reynolds number of O(1) was computed by [START_REF] Shin | Structure and dynamics of dilute suspensions of finite-Reynolds-number settling fibers[END_REF], who showed that the corresponding lift force is made of two contributions, one resulting from the coupling of shear and translational inertial terms, the other being related to the classical sedimentation-driven drift of an inclined fibre u nder S tokes conditions.

In the strong shear flow case, i.e. under assumptions similar to those considered by [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF], the first result for the inertial lift force acting on a non-spherical body was obtained by [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF], who, by using the inertial corrections acting on a sphere translating in an arbitrary direction in a linear shear flow, found a way to evaluate these corrections for a spheroid with an arbitrary aspect ratio. However, their approach still assumes the velocity disturbance to be steady, whereas spheroids immersed in a shear flow are known to tumble periodically in the zero-Reynoldsnumber limit [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]. Although a quasi-steady state has to be reached after the body has been in the flow for a long enough time, the possible long-term influence of the body rotation on the inertial correction to the force remains unknown in this approach.

The leading-order inertial corrections to the loads experienced by a rigid body depend linearly on the force and torque acting on it in the zero-Reynolds-number limit, and this force and torque depend linearly on the relative velocity and rotation rate between the body and fluid through resistance tensors entirely determined by the body geometry [START_REF] Brenner | The Stokes resistance of an arbitrary particle[END_REF](Brenner , 1964a,b;,b;[START_REF] Happel | Low Reynolds Number Hydrodynamics[END_REF]. How does the shape of the body enter the MAE and IF approaches? In the former, inertial corrections are entirely governed by the uniform component of the residual disturbance in the 'outer' domain [START_REF] Kaplun | Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers[END_REF][START_REF] Proudman | Expansions at small Reynolds numbers for the flow past a sphere and circular cylinder[END_REF]. Moreover, as first recognized by [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF], the presence of the body enters only through a point-source term in the 'outer' problem. Hence, as far as the resistance tensors are known in the zero-Reynolds-number limit, it is straightforward to extend results derived for spherical bodies to arbitrarily shaped bodies. In contrast, it is not obvious to infer how the IF method can be generalized to such bodies, and to the best of our knowledge this has not been achieved yet. Indeed this method requires the slip between the body and fluid to vanish everywhere within the inner domain bounded by the actual body surface. Hence resistance tensors are not involved explicitly, making the prescription of the 'induced force' non-trivial as soon as a non-spherical shape is considered. This is why the MAE technique appears to be much more suitable for bodies of general shape.

The primary goal of our work is to determine, using the MAE technique, how the instantaneous force and torque on an arbitrarily shaped body are affected by small inertia effects in a general quasi-steady linear flow. The present paper describes our approach in its full generality, but the applications presented hereinafter only concern the usual three canonical planar flows, and in addition the sedimentation of spheroidal particles in a shear flow. In that respect, the present contribution only represents a first step towards the prediction of finite inertial effects acting on particles immersed in a turbulent flow.

The paper is organized as follows. Section 2 outlines the formulation of the problem, stating in particular the asymptotic conditions under which the solution is sought, establishing the corresponding disturbance flow problem, and deriving the form of unsteady force and torque corrections in terms of the solution of this problem. Section 3 describes how to solve the disturbance problem by introducing a moving non-orthogonal coordinate system. This reduces the initial problem to a set of ordinary differential equations in Fourier space, and provides the general structure of the corresponding solution, first in Fourier space, then in the physical space in the form of a tensorial convolution kernel. The technical steps leading to the solution, which involve in particular the use of Magnus expansions, are detailed in appendix A. In § 4, we compute explicitly the kernel in the solid-body rotation and planar elongational flows, and examine its various components at both short and long where m p is the mass of the body and g denotes gravity. Similarly, the torque τ with respect to the body centroid is

τ = S p r × Σ ∞ • ds + S p r × Σ • ds =τ , where r = x -x p .
(2.

3)

The integrals are over the body surface S p , ds is the outward normal surface element, and Σ ∞ and Σ are the stress tensors associated with the undisturbed flow and the disturbance, respectively. The last terms in the right-hand side of (2.2) and (2.3) define the disturbance force, f , and the disturbance torque, τ , respectively. Since the undisturbed flow is known, the integrals involving Σ ∞ in (2.2) and in (2.3) are readily evaluated by using Stokes' theorem together with the fact that the undisturbed flow is a solution of the Navier-Stokes equation (e.g. [START_REF] Gatignol | The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow[END_REF]). To compute the disturbance force and torque, the stress tensor of the disturbance flow must be determined, requiring in principle solving the Navier-Stokes equations for the disturbance velocity w(r, t) = U(r + x p , t) -U ∞ (r + x p , t). In the following, we assume that this solution is known in the quasi-steady creeping-flow limit where no inertia effects are considered. In this limit, the disturbance force and torque on an arbitrarily shaped body may be expressed in the form [START_REF] Kim | Microhydrodynamics: Principles and Selected Applications[END_REF])

f (0) (t) τ (0) (t) = -μ M 1 (t) M 2 (t) M T 2 (t) M 3 (t) • ẋp -U ∞ ω p -Ω ∞ -μ N 1 (t) : S ∞ N 2 (t) : S ∞ . (2.4)
Here μ is the dynamic viscosity of the fluid, S ∞ ≡ (A + A T )/2 is the symmetric part of the strain-rate tensor A (throughout the paper, T denotes the transpose), and Ω ∞ ≡ (1/2)∇ × U ∞ is half the vorticity of the undisturbed flow. The M i are the second-order resistance tensors, while the N i are third-order tensors. The components of M i and of N i in the principal basis of the body are assumed to be known. In the laboratory frame, these components depend on time, since they depend on the instantaneous orientation of the body.

Dimensionless parameters and variables

In this study, one of the main objectives is to predict how fluid inertia modifies the slip velocity of the body. For this reason, translational velocities are normalized with the typical order of magnitude of the body's slip velocity, u c , in the creeping-flow limit, so that U = u c U . Distances are normalized by a characteristic body length, a, so that r = ar . Consequently pressures and forces are normalized in the form p = (μu c /a)p and f = μu c af , respectively. Components of A are normalized by a characteristic velocity gradient, s, defined as s = √ (A : A)/2 in the bidirectional flows considered in § 4, and as s = √ A : A in the one-directional shear flow on which § 5 focuses. The two different normalizations ensure that s is the magnitude of the velocity gradients in both cases. In the absence of a condition on the overall torque, the body rotation rate may be arbitrary, so that it provides an independent time scale, ω -1 p . Last, it is necessary to introduce the characteristic time τ c over which the relative translational and rotational velocities may vary at the body surface, so that the dimensionless time t is t = t/τ c . With these definitions, t he p roblem i s g overned b y f our d imensionless numbers, namely

Re s = a 2 s ν , Re p = au c ν , Re ω = a 2 ω p ν and Sl = 1 sτ c , (2.5a-d)
where Re s , Re p and Re ω are the shear, slip and rotation Reynolds numbers, respectively, Sl is the Strouhal number characterizing the unsteadiness of the problem and ν = μ/ρ f is the kinematic viscosity. In the remainder of the paper, the primes are dropped but it must be understood that all equations and quantities are dimensionless. In these variables, the equations governing the disturbance flow, expressed in a frame translating with the body, read

Re s Sl∂ t w| r + Re s [A • w + (A • r) • ∇w] + Re p [-u s • ∇w + w • ∇w] = -∇p + ∇ 2 w, (2.6)
subject to the incompressibility condition

∇ • w = 0, (2.7)
and to the boundary conditions

w → 0 for |r| → ∞ and w = u s + Re ω Re p ω p × r - Re s Re p (Ω ∞ × r + S ∞ • r) for r ∈ S p .
(2.8a,b) In (2.6) and (2.8), the gradients are spatial derivatives with respect to r = xx p , the time derivative is evaluated at fixed r, and we have introduced the slip velocity u s = ẋp -U ∞ (x p ). We consider the problem in the Saffman limit where all three Reynolds numbers are small, and Re s and Re p satisfy the ordering condition Re p √ Re s .

(2.9)

We further assume that Sl 1 Re s , (2.10) which allows unsteadiness to be large, but not 'too' large. In particular, τ c must in principle be much larger than the viscous time a 2 /ν. Conditions (2.9) and (2.10) significantly simplify the problem, which can then be treated via a perturbation approach. Close to the body, the disturbance velocity is then well approximated by the quasi-steady Stokes solution w ∼ 1/r. Thanks to this scaling form, it is readily found that beyond the so-called Saffman length defined as (2.11) terms proportional to Re p in (2.6) become negligibly small compared to the other terms in the disturbance equation. The part of the inner solution that stems from the last two terms in the right-hand side of (2.8) consists of a combination of rotlets and stresslets which decay as 1/r 2 [START_REF] Lamb | Hydrodynamics[END_REF]. Hence, at distances of the order of s , the corresponding contributions are respectively of O(Re ω Re 1/2 s Re -1 p ) and O(Re 3/2 s Re -1 p ) compared to that due to the slip. Although these ratios may be large, i.e. the corresponding part of the solution may dominate the disturbance in the outer region up to a certain distance from the body, the slip-induced disturbance is always dominant for |r| → ∞ provided Re p is non-zero, since the strongest singularity in this problem results from the slip-induced contribution. Moreover, we shall show later that the part of the outer solution related to the rotlet and stresslet contributions does not provide any correction to the force and torque at the leading order considered here, for symmetry reasons. As a consequence, with the above set of assumptions, i.e. (2.9), (2.10) plus the restriction that Re p , Re s and Re ω are all small and Re p is non-zero, the disturbance problem in the far field reduces to

|r| ∼ 1 √ Re s ≡ s ,
Re s Sl ∂ t w| r + Re s (A • w + (A • r) • ∇w) = -∇p + ∇ 2 w,
(2.12) ∇ • w = 0, (2.13) w → 0 for |r| → ∞ and w = u s for r ∈ S p .

(2.14a,b) 2.3. Asymptotic solution As discussed in the introduction, the MAE approach has been extensively employed to determine how small inertia effects alter the force acting on a small rigid or deformable body since the pioneering studies of [START_REF] Kaplun | Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers[END_REF] and [START_REF] Proudman | Expansions at small Reynolds numbers for the flow past a sphere and circular cylinder[END_REF]. The standard method used in the presence of a non-uniform background flow was devised by [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF] and [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF]. Specificities of this method are that the body is replaced by a point force (through a Dirac-delta function with appropriate strength and direction) in the far-field equation of the disturbance, and that the matching is performed in Fourier space. In all studies reviewed in § 1, with the exception of the work by [START_REF] Mclaughlin | Inertial migration of a small sphere in linear shear flows[END_REF], expansions were performed with respect to the small parameter

= -1 s = Re 1/2 s . (2.15)
This is also the key small parameter here. However, most studies to date based on the MAE approach considered either the quasi-steady approximation Sl → 0 or harmonic variations of the slip velocity [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF]. Here, in contrast, unsteady effects corresponding to arbitrary evolutions are considered up to O( 2-n ) with 0 n < 2, together with O( 2 ) shear-induced inertial effects. As will become apparent soon, the additional difficulty resulting from such unsteady effects is that, in Fourier space, the governing equation for the velocity disturbance becomes a partial differential equation with respect to time and space, which greatly complicates the determination of its solution. This is why, to the best of our knowledge, only [START_REF] Candelier | Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers[END_REF] considered such arbitrary evolutions in the specific case of a solid-body rotation flow, where this additional difficulty is overcome by a simple change of reference frame.

As discussed above, under conditions (2.9), the overlap between the inner and outer regions of the perturbation problem arises at distances from the body of the order of the Saffman length, s = -1 .

In the inner region, |r| 1/ , the disturbance is sought in the form of a regular expansion in powers of , namely

w in = w (0) in + w (1) in + • • • and p in = p (0) in + p (1) in + • • •. (2.16a,b)
The inner solution, w in , satisfies the required boundary condition at the body surface but fails in the outer region |r| 1 and r ∼ O(1/ ) since it tends to produce inertial contributions of larger magnitude than the viscous term [START_REF] Oseen | Über die Stoke'sche Formel und über die verwandte Aufgabe in der Hydrodynamik[END_REF][START_REF] Van Dyke | Perturbation Methods in Fluid Mechanics[END_REF].

Therefore, inertial and viscous terms must both be considered in the outer region. Following [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF], the effect of the inner boundary condition is replaced by a point force, f (0) , weighted by the three-dimensional delta function, δ(r). Hence, assuming provisionally Sl = 1 (which amounts to scale time with s -1 ) the disturbance flow i s g overned by

2 [∂ t w| r + A • w + (A • r) • ∇w] = -∇p + ∇ 2 w + f (0) δ(r),
(2.17) subject to the divergence-free condition (2.13), and to the vanishing condition w → 0 for |r| → ∞. To leading order, f (0) is just the leading-order force exerted by the body on the fluid, i.e. the opposite of the Stokes force acting on the body in the linear flow as given by (2.4). Thus we write

f (0) (t) ≡ -f (0) (t) = M 1 (t) • u s (t) + M 2 (t) • ω s (t) + N 1 (t) : S ∞ . (2.18)
As mentioned in § 2.1, the components of M 1 , M 2 and N 1 generally depend upon time.

The situation greatly simplifies for a sphere, for which M 1 = 6π1 and M 2 = N 1 = 0, where 1 is the identity (Kronecker) tensor. Equation (2.17) was written under the assumption Sl = O(1). However, following (2.10), we stress again that the solutions derived throughout this paper are valid even for larger Strouhal numbers, Sl = O( -n ) with n 0, provided that n < 2.

Fourier transforming (2.17) yields

2 [∂ t ŵ| k + A • ŵ -k • A • ∇ ŵ] = -ikp -k 2 ŵ + f (0) , ( 2.19)
where i 2 = -1, k 2 = k • k, ∇ denotes the gradient with respect to k, and the direct and inverse Fourier transforms are respectively defined as

ŵ = w(r, t) exp(-ik • r) d 3 r and w = 1 8π 3 ŵ(k, t) exp(ik • r) d 3 k. (2.20a,b)
In (2.19), the pressure term can be eliminated thanks to the continuity equation which, in Fourier space, reduces to k • ŵ = 0. In this way one obtains

-ip = 2 2 (A • ŵ) • k k 2 - f (0) • k k 2 (2.21) and therefore (2.19) becomes 2 ∂ t ŵ| k + A • ŵ -k • A • ∇ ŵ -2 (A • ŵ) • k k 2 k + k 2 ŵ = 1 - kk k 2 • f (0) . (2.22)
At this stage it is worth noting that if one introduces the stretched wavevector k = -1 k, the entire left-hand side of (2.22) becomes of O( 2 ŵ) once rewritten in stretched coordinates. In contrast, since 0) in the right-hand side of (2.22), so that the overall balance requires the leading-order term in ŵ to be of O( ). This is where the magnitude of the force correction due to inertial effects may be inferred: in the low-Reynolds-number regime, an O( ) uniform upstream velocity yields a net force with a similar magnitude on the body. This is why the inertial force is expected to be of O( ) in the present case, provided that the outer flow correction ŵ contains a continuous component.

δ(r) exp(-ik • r) d 3 r = 1, one has δ(r / ) exp(-ik • r ) d 3 r = 3 , with r = r. This implies that f (0) is replaced with 3 f (
In what follows, rather than solving (2.22) in stretched coordinates, we directly perform a series expansion of the solution in Fourier space. This expansion may be carried out in terms of generalized functions of k [START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF] with respect to , i.e.

ŵ(k, , t) = T (0) (k, t) + T (1) (k, t) + O( 2 ), (2.23)
where by definition,

T (0) (k, t) = lim →0 ŵ(k, , t) and T (1) (k, t) = lim →0 ŵ(k, , t) - T (0) (k, t) . (2.24a,b)
The first term, T (0) (k, t), is the Fourier transform of the solution of the outer problem for = 0 (i.e. the Stokeslet solution). When transformed back into the physical domain, this term reads

T (0) (r, t) = 1 8π 1 |r| + rr |r| 3 • f (0) (t).
(2.25)

Making use of the homogeneity properties of the outer solution, we see that the next term in the expansion has the form

T (1) (k, t) = lim →0 1 3 [ ŵ(k , t) - T (0) (k , t)].
(2.26)

Evaluating this limit in the sense of generalized functions yields

T (1) (k, t) = δ(k) [ ŵ(k, t)| =1 - T (0) (k, t)] d 3 k, (2.27)
which, once transformed back into real space becomes a time-dependent constant, independent of r, namely

T (1) (t) = 1 8π 3 [ ŵ(k, t)| =1 - T (0) (k, t)] d 3 k.
(2.28)

The contribution ŵ(k, t)| =1 in (2.28) is the solution of the outer problem (2.22) for = 1. Since the function T (1) (t) is r-independent, it defines a uniform velocity in the far field, to which the sought force correction is linearly proportional. Equations (2.25) and (2.28) provide the boundary conditions to be satisfied by the inner solution for |r| ∼ -1 .

The inner and outer solutions must match at |r| ∼ -1 . Hence the lowest-order term in the inner expansion, w (0) in , corresponds to the solution of Stokes equation satisfying

w (0) in = u s for r ∈ S p and lim |r|→∞ w (0) in ∼ T (0) (r, t). (2.29a,b)
The second term in the inner expansion, w (1) in , is also a solution of the Stokes equation, but now with boundary conditions w (1) in = 0 for r ∈ S p and lim

|r|→∞ w (1) in ∼ T (1) (t). (2.30a,b)
Determining w (1) in is thus equivalent to solving the Stokes flow problem about the body kept fixed in a uniform stream. Hence, determining inertial corrections to the force is equivalent to determining the corrected uniform flow at infinity, so that one readily concludes that the (dimensionless) disturbance force in (2.2) reads to order

f = -M 1 (t) • u s -M 2 (t) • ω s -N 1 (t) : S ∞ + M 1 (t) • 1 8π 3 R 3 [ ŵ(k, t)| =1 - T (0) (k, t)] d 3 k , (2.31) while the disturbance torque in (2.3) reads τ = -M T 2 (t) • u s -M 3 (t) • ω s -N 2 (t) : S ∞ + M T 2 (t) • 1 8π 3 R 3 [ ŵ(k, t)| =1 - T (0) (k, t)] d 3 k . (2.32)
These are the desired expressions for the unsteady force and torque acting upon an arbitrarily shaped rigid body moving in a general linear flow. As discussed earlier, the results (2.31) and (2.32) were obtained without considering the part of the outer solution that stems from the last two terms in the right-hand side of (2.8). The reason for this is that these two terms cannot contribute to T (1) (t). This is seen by noting that (2.22) remains unchanged if k is replaced by -k, which implies that the outer solution ŵ(k, t) must be an even function of k. As mentioned above, the neglected terms in (2.8) induce rotlets and stresslets in the inner solution. As [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF] showed, such contributions result in additional source terms in the form of force dipoles, say D • ∇δ, in the outer problem (2.19). Such terms are odd functions of k, and so the linearity of (2.22) implies that the corresponding solution must also be odd in k. Consequently, this part of the solution vanishes upon integration over k and cannot contribute to (2.28). Therefore, only the part of the outer solution corresponding to a point force contributes to the force (2.31) and torque (2.32) at O( ). The body rotation and variations of the background velocity at the body surface can of course induce higher-order corrections to the force and torque at O(Re ω ) and O(Re s ), as shown by [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF] in the pure shear case. The chief difficulty in finding the outer solution ŵ involved in (2.31)-(2.32) results from the advective term

(A • r) • ∇w (2.33)
on the left-hand side of (2.17). Although this term is linear, it is inhomogeneous in the sense that it explicitly depends on the r vector, yielding partial derivatives with respect to k in (2.22). This is what renders the determination of the solution technically difficult, even in the quasi-steady approximation, except under particular circumstances. One of these is the case where all spatial derivatives in (2.22) are with respect to the same component of k, as in Saffman's original problem. As explained in the introduction, the key idea here is to obtain a general solution to the disturbance problem by removing the inhomogeneous advective term (2.33) with the aid of a non-orthogonal coordinate system that moves and deforms with the undisturbed flow, so as to reduce this problem to a set of ordinary differential equations with respect to time that are much more easily solved.

Solution strategy

3.1. Time-dependent non-orthogonal coordinates Although (2.17) is written in a reference frame translating with the position of the body centroid, x p (t), the unit vectors are those of the laboratory frame. Components r i of the vector r in the corresponding basis with unit vectors e i define a rectilinear orthogonal coordinate system. Alternatively, one may introduce a new system with coordinates R i such that

r i = F i j R j . (3.1)
Here the F i j (t) are the time-dependent components of a transformation matrix, F(t), and summation is implied on repeated indices. We assume det(F) = 0, so that the relation (3.1) between r i and R i can be inverted. Since (3.1) is linear and the F i j do not depend on the variable r, the R i coordinate system remains rectilinear. In contrast, these coordinates are generally non-orthogonal. We introduce the time-dependent basis, E i , associated with these coordinates as

E i (t) = F(t) • e i . (3.2)
In what follows we adopt the convention that components of vectors and tensors expressed in the E i and e i bases are denoted with uppercase and lowercase letters, respectively. For instance, we write w = w i (r j , t)e i = W i (R j , t)E i . All vectors are expressed in contravariant form, and their components are thus denoted with upper indices. Because they contract with such vectors, several second-order tensors need to be expressed in covariant or mixed form, thus involving lower indices; e.g. F i j in (3.1). In appendix A we show that the partial time derivatives of any vector at fixed r i and fixed R j components are related through

∂w ∂t r i = ∂(W i (R j , t)E i ) ∂t R j -v • ∇w with v ≡ ∂r ∂t R j . (3.3)
The goal is now to express (2.17) with respect to coordinates R j in such a way that the v-term cancels the inhomogeneous advective contribution (A • r) • ∇w. This is achieved by setting

v = A • r. (3.4) It then follows from (3.1) that F(t) must satisfy dF dt = A • F with F(0) = 1. (3.5)
The initial condition ensures that r i = R i at t = 0. In continuum mechanics, F is the deformation-gradient tensor mapping an infinitesimal vector dx corresponding to the initial configuration onto another infinitesimal vector dX corresponding to the deformed configuration; F T • F is referred to as the Cauchy-Green tensor [START_REF] Truesdell | The Non-Linear Field Theories of Mechanics[END_REF][START_REF] Eringen | Mechanics of Continua[END_REF]. It provides the square of the local change in distances due to deformation, since dX

• dX = dx • (F T • F) • dx.
In fluid mechanics, F arises in characterizing the time-dependent orientations of rod-like particles advected in turbulence [START_REF] Wilkinson | Fingerprints of random flows?[END_REF][START_REF] Voth | Anisotropic particles in turbulence[END_REF]; in this case, F maps the initial rod orientation onto the final one.

As A is time-independent, the solution to (3.5) is merely

F = Exp(At). (3.6)
Now it remains to determine how the derivatives in (2.17) transform. Noting that the transformation (3.1) depends linearly on the R i , and using the fact that dE i /dt = A • E i , it is readily found that (with = 1), the ith component of (2.17) transforms into

∂W i ∂t R i + 2A i j W j = -R ij (t) ∂P ∂R j + R jk (t) ∂ 2 W i ∂R j ∂R k + F (0)i (t)δ(R i E i ), (3.7)
where R ij (t) = δ k (F -1 ) i (F -1 ) j k , δ k denoting the Kronecker symbol. The R ij are the components of the inverse of the metric tensor with components g ij = E i • E j [START_REF] Aris | Vectors, Tensors and the Basic Equations of Fluid Mechanics[END_REF]. The term 2A i j W j on the left-hand side of (3.7) may be thought of as a generalization of the Coriolis acceleration. Obviously, this term vanishes if the co-moving reference frame only translates with respect to the laboratory frame, in which case

R ij = δ ij .
As mentioned in the introduction, (3.7) presents much similarities with the equation governing the linear (i.e. short-time) evolution of a turbulent fluctuation subject to a strong mean shear or strain, as described by the rapid distortion theory (RDT). This approach was pioneered by [START_REF] Batchelor | The effect of rapid distortion of a fluid in turbulent motion[END_REF] and is described in detail by [START_REF] Townsend | The Structure of Turbulent Shear Flow[END_REF]. In RDT, the linearized Navier-Stokes equation are solved in Fourier space by using moving Lagrangian coordinates (see for instance equation (8.23) in [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]). The same approach was used by [START_REF] Batchelor | Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution[END_REF] to solve the heat equation past a particle suspended in a steady linear flow field, to evaluate the influence of advective effects on the heat/mass transfer rate. In the present problem, (3.7) differs from the RDT governing equation due to the additional forcing term F (0)i δ accounting for the presence of the body in the linear flow. Owing to this additional term, the governing equation for the outer disturbance in the Fourier domain is an inhomogeneous ordinary differential equation, whereas it reduces to a homogeneous differential equation in RDT. The additional inhomogeneous term makes the problem significantly more difficult. In particular, as will be seen in the next section, the solution of the outer equation in Fourier space takes the form of a convolution product. One consequence is that, in contrast with RDT, the state of the disturbance flow at a given time cannot be determined directly from its state in the recent past, since it depends on the whole history of the flow.

General solution of the disturbance problem (2.17)

Since (3.7) no longer contains an inhomogeneous advective term, it can be solved in Fourier space. This yields the general solution of the disturbance problem in the time-dependent basis, E i . In a second step, this solution must be re-expressed in the Cartesian basis, yielding formally the desired general solution of the disturbance problem (2.17) as

ŵ(k, t) - T (0) (k, t) = - t 0 e -t τ K 2 (t-τ ) dτ Ĝ • df (0) (τ ) dτ dτ - t 0 K 2 (t -τ )e -t τ K 2 (t-τ ) dτ [ Ĝ -Y 2 (t, τ ) • Ĝ2 (t -τ ) • F(t -τ )] • f (0) (τ ) dτ . (3.8)
Details of the derivation are given in appendix A. Here Ĝ is the Fourier transform of the Green tensor associated with the Stokes equation, namely

Ĝ = 1 k 2 1 - kk k 2 , (3.9)
such that

T (0) = Ĝ • f (0) . The kernel K 2 is directly related to the Cauchy-Green tensor through K 2 (ξ ) = k • F(ξ ) • F T (ξ ) • k, (3.10) and Ĝ2 (ξ ) = 1 K 2 (ξ ) 1 - F(ξ ) • F T (ξ ) • kk K 2 (ξ ) , (3.11)
where ξ = t -τ denotes the time lag. Finally, the second-order tensor Y 2 (t, τ) is defined as

Y 2 (t, τ ) = Y(t) • Y -1 (τ ), (3.12)
where Y(t) is the solution of the fundamental differential equation

dY(τ ) dτ = -2K 2 (t -τ ) Ĝ2 (t -τ ) • A • Y(τ ) with Y(0) = 1. (3.
f = -M 1 (t) • u s -M 2 (t) • ω s -N 1 (t) : S ∞ -M 1 (t) • t 0 K(t -τ ) • d dτ f (0) dτ , (3.14) and τ = -M T 2 (t) • u s -M 3 (t) • ω s -N 2 (t) : S ∞ -M T 2 (t) • t 0 K(t -τ ) • d dτ f (0) dτ , (3.15)
where the kernel K(t) may be split in the form

K(t) = K h (t) + t 0 K i (ξ ) dξ, (3.16) with K h (ξ ) = 1 8π 3 e -t τ K 2 (t-τ ) dτ Ĝ d 3 k, (3.17a) K i (ξ ) = 1 8π 3 K 2 (ξ )e -t τ K 2 (t-τ )dτ [ Ĝ -Y 2 (t, τ ) • Ĝ2 (ξ ) • F(ξ )] d 3 k. (3.17b)
Equations (3.14) and (3.15) are the main results of this paper. They provide an explicit expression for the disturbance loads acting on a rigid body with an arbitrary shape moving in a general linear flow in the Saffman limit. Similar to the Basset-Boussinesq force acting on a sphere having a time-dependent motion with respect to the fluid (e.g. [START_REF] Landau | Fluid Mechanics[END_REF], the instantaneous force acting on a body translating and rotating arbitrarily in a steady linear flow field takes the form of a convolution integral. It is important to keep in mind that, although the present theory captures the dominant, O( ), time-dependent effects, it does not predict higher-order contributions to these effects. In particular, it does not capture the so-called added-mass (or inducedinertia) force. This force is proportional to the product of the mass of fluid displaced by the body, which scales as ρ f a 3 , and the body acceleration, which is proportional to u c /τ c (see (2.5)). Indeed, once normalized by the steady Stokes drag, this force is of O( 2 Sl), which makes it negligibly small compared to all contributions in (3.14) for O(1)-Strouhal numbers.

Remarkably, the kernel K(t) defined b y ( 3.16) d oes n ot d epend o n t he b ody shape, the influence o f w hich i s e ntirely a ccounted f or b y t he r esistance t ensors M i a nd N i appearing in the expression for f (0) (see (2.18)), and as prefactors of the convolution integral in (3.14) and (3.15). That the kernel is independent of the body shape is readily understood by keeping in mind that, at leading order in , the body is seen as a point force by the far-field fl ow. Th anks to th is cr ucial pr operty, th e ke rnel ma y be determined once and for all and the method can then be applied to any body shape, provided that the resistance tensors are known. It is important to note that this state of affairs drastically differs from the problem of the leading-order inertial corrections to the rotational dynamics of neutrally buoyant non-spherical bodies immersed in shear flows. S uch c orrections w ere fi rst de rived by Su bramanian & Ko ch (2 005) fo r rod-like bodies, then by Einarsson et al. (2015b) and [START_REF] Dabade | The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow[END_REF] for spheroids with arbitrary aspect ratios. In all cases, the corrections were obtained using a regular perturbation expansion in powers of in which the first non-zero correction to the torque was found to occur at O( 2 ). That this correction is provided by a regular expansion indicates that it is driven by the near-field fl ow, an d thus depends directly upon the body shape, in contrast to the O( )-correction derived here. This suggests that, regarding the translational dynamics, higher-order corrections to present results, similar to the so-called second-order Saffman's lift force [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF][START_REF] Mclaughlin | Inertial migration of a small sphere in linear shear flows[END_REF], may incur direct dependencies on the body shape as well.

As a last point, we stress that results (3.14)-(3.17b) are distinct from the expression of the unsteady force and torque acting on an arbitrarily shaped body derived by [START_REF] Gavze | The accelerated motion of a rigid bodies in non-steady Stokes flow[END_REF]. His work is concerned with an entirely different limit of the problem, where unsteady contributions are of the same order as the quasi-steady Stokes drag (this limit is obtained by setting Re s → 0, Re p → 0 and Re s Sl = O(1) in (2.6)). Under such conditions, the problem is equivalent to the time-dependent Stokes equation solved by [START_REF] Boussinesq | Sur la résistance qu'oppose un fluide indéfini au repos sans pesanteur au mouvement varié d'une sphère solide qu'il mouille sur toute sa surface quand les vitesses restent bien continues et assez faibles pour que leurs carrés et produits soient négligeables[END_REF] and Basset (1888) past a sphere. However, when generalized to bodies of arbitrary shape, the unsteady part of the solution (which yields added-mass and 'history' effects) is found to involve two supplementary shape-dependent tensors similar to the 'grand-resistance' tensor in (2.4). Here in contrast, as inertial effects are assumed to provide only small corrections to the quasi-steady Stokes drag, they are connected to the body shape exactly in the same way as the primary drag, namely through the resistance tensors M 1 and M 2 .

The kernel in the two canonical bidirectional linear flows

To prove the versatility of the approach derived in the previous section, we first specialize it to the canonical cases of a solid-body rotation flow and a planar elongational flow, respectively. In the former case, transforming the disturbance flow equations in the co-moving coordinate system merely corresponds to performing a change of reference frame, which makes this situation a compulsory test case.

Solid-body rotation

As reviewed in the introduction, this configuration is well documented in the literature. Interestingly, determining the inertial drag correction experienced by a sphere translating with a constant velocity along the axis of a solid-body rotation flow was the question that motivated [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF] to design the MAE approach in the way that later became standard in the class of problems considered here; his predictions were checked experimentally by [START_REF] Maxworthy | An experimental determination of the slow motion of a sphere in a rotating, viscous fluid[END_REF].

In the laboratory frame, the base flow reads

U ∞ (x) = A • x, with A = ⎛ ⎝ 0 -1 0 1 0 0 0 0 0 ⎞ ⎠ . (4.1)
The matrix exponentiation (3.6) provides the deformation-gradient tensor F in the form

F(t) = ⎛ ⎝ cos t -sin t 0 sin t cos t 0 0 0 1 ⎞ ⎠ . (4.2) Using (4.2), (3.10) yields K 2 (ξ ) = k 2 , hence t τ K 2 (t -τ ) dτ = k 2 ξ , so that the fundamental problem (3.13) reduces to dY(τ ) dτ = 2 k 2 ⎛ ⎝ k 1 k 2 (k 2 2 + k 2 3 ) 0 -(k 2 1 + k 2 3 ) -k 1 k 2 0 k 2 k 3 -k 1 k 3 0 ⎞ ⎠ • Y(τ ) Y(0) = 1. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (4.3)
As the right-hand side of (4.3) depends upon the time lag only, integration can be achieved again through a matrix exponentiation. The solution (4.3) yields the secondorder tensor Y 2 defined in (3.12) in the form

Y 2 (ξ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k 1 k 2 k 3 k sin Z + cos Z k 2 2 + k 2 3 k 3 k sin Z 0 - k 2 1 + k 2 3 k 3 k sin Z - k 1 k 2 k 3 k sin Z + cos Z 0 k 2 k sin Z - k 1 k 3 cos Z + k 1 k 3 - k 1 k sin Z - k 2 k 3 cos Z + k 2 k 3 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 4.4)
where Z = 2k 3 ξ/k. Then the two kernels K h and K i involved in (3.17a) and (3.17b) are obtained in the form

6πK h (ξ ) = 1 √ πξ , 6πK i (ξ ) = ⎛ ⎝ I 1 (ξ ) I 2 (ξ ) 0 -I 2 (ξ ) I 1 (ξ ) 0 0 0 I 3 (ξ ) ⎞ ⎠ , (4.5a,b)
where

I 1 (ξ ) = 1 16
3 sin ξ cos 2 ξ -6ξ 2 sin ξ -3ξ cos ξ + 8ξ 3 √ πξ 9/2 , (4.6)

I 2 (ξ ) = - 3 16 cos ξ -cos 3 ξ -2ξ 2 cos ξ + ξ sin ξ √ πξ 9/2 , (4.7) I 3 (ξ ) = - 1 8 -4ξ 3 + 3 sin ξ cos ξ -6ξ cos 2 ξ + 3ξ √ πξ 9/2 . (4.8)
After integrating by parts, these kernels are found to be identical to those obtained by [START_REF] Candelier | Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers[END_REF]. Expanding (4.5) for short times and setting τ = 0 yields K(t) in the form

6πK(t) = 1 √ π ⎛ ⎜ ⎝ t -1/2 + 1 10 t 3/2 -1 75 t 5/2 0 1 75 t 5/2 t -1/2 + 1 10 t 3/2 0 0 0 t -1/2 + 2 15 t 3/2 ⎞ ⎟ ⎠ + • • •. (4.9)
The contribution to the hydrodynamic force associated with the t -1/2 diagonal terms (which result from the kernel K h in (4.5)) corresponds the usual Basset-Boussinesq 'history' force [START_REF] Landau | Fluid Mechanics[END_REF]. Inertial corrections due to the background linear flow result from the kernel K i . They are seen to grow as t 3/2 on the diagonal, faster than off-diagonal corrections corresponding to a lift force, which grow as t 5/2 . Integrating K i over time, the quasi-steady kernel corresponding to the long-time limit

t → ∞ is found to be 6πK = 6π ∞ 0 K i (ξ ) dξ = ⎛ ⎜ ⎝ 3 √ 2(19+9 √ 3) 280 -3 √ 2(19-9 √ 3) 280 0 3 √ 2(19-9 √ 3) 280 3 √ 2(19+9 √ 3) 280 0 0 0 4 7 ⎞ ⎟ ⎠ . (4.10)
This is the result obtained independently by [START_REF] Gotoh | Brownian motion in a rotating flow[END_REF] and [START_REF] Miyazaki | Dependence of the friction tensor on the rotation of a frame of reference[END_REF] through the MAE and IF approaches, respectively. The [K] 3 3 component was determined much earlier by [START_REF] Childress | The slow motion of a sphere in a rotating, viscous fluid[END_REF]. Inertial effects due to the solid-body rotation are seen to increase the drag whatever the direction of the slip velocity, with a slightly larger prefactor when the body moves along the rotation axis (6π[K] 3 3 ≈ 0.571) than within the plane of the flow (6π[K] 1 1 = 6π[K] 2 2 ≈ 0.542). Inertial effects also induce a small non-zero lift component (6π[K] 2 1 = -6π[K] 1 2 ≈ 0.052) which is centrifugal if the sphere is at rest in the laboratory frame. More generally, this lift component is centrifugal (respectively centripetal) if the sphere translates in such a way that it lags behind (respectively leads) the fluid. Note that the same situation was considered by [START_REF] Drew | The force on a small sphere in slow viscous flow[END_REF]; however his calculation erroneously predicted the lift component to be zero.

Figure 2 shows how K(t) reaches the above steady state; according to (3.14), the inertial corrections to the force directly follow this evolution if the slip velocity is set abruptly to a non-zero constant value at time t = 0. The diagonal components are seen to reach levels close to their steady-state value in approximately two time units. In contrast, it takes approximately twice as long for the off-diagonal component, [K] 1 2 , to reach a quasi-converged level. This is due to the different growth rates identified in (4.9). In all cases, damped oscillations with a period T o = 2π corresponding to the imposed rotation rate (1/2)|Ω ∞ | = 1 take place subsequently.

Planar elongational flow

As a second example, we consider the purely extensional planar flow defined by 3 ; grey dashed line: t -1/2 short-time behaviour resulting from the contribution K h (t) in (4.5); black dash-dotted line: 6π|[K]| 1 2 ; grey dash-dotted line: short-time expansion 6π

U ∞ (x) = A • x, with A = ⎛ ⎝ 1 0 0 0 -1 0 0 0 0 ⎞ ⎠ . ( 4
[K] 1 2 ∼ (1/75 √ π)t 5/2 .
This configuration was first considered by [START_REF] Drew | The force on a small sphere in slow viscous flow[END_REF] in the steady-state limit, using the MAE approach, then by Pérez-Madrid et al. (1990) using the IF method. However, as discussed in § 1, the results of Pérez-Madrid et al. are not correct, since they neglected the inhomogeneous term arising from (2.33). The same warning applies to the conclusions of [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF], who also examined the corresponding time-dependent situation.

The deformation-gradient tensor corresponding to (4.11) is

F(t) = ⎛ ⎝ e t 0 0 0 e -t 0 0 0 1 ⎞ ⎠ .
(4.12)

It follows from (3.10) that

K 2 (ξ ) = k 2 1 e 2ξ + k 2 2 e -2ξ + k 2 3 , (4.13)
and thus

t τ K 2 (t -τ ) dτ = 1 2 k 2 1 (e 2ξ -1) + 1 2 k 2 2 (1 -e -2ξ ) + k 2 3 ξ. (4.14)
Equations (3.11)-(3.13) show that the fundamental problem to solve is

dY(τ ) dτ = 2 k 2 1 e 2(t-τ ) + k 2 2 e -2(t-τ ) + k 2 3 ⎛ ⎜ ⎝ -(k 2 2 e -2(t-τ ) + k 2 3 ) -e 2(t-τ ) k 1 k 2 0 e -2(t-τ ) k 1 k 2 (k 2 1 e 2(t-τ ) + k 2 3 ) 0 k 1 k 3 -k 2 k 3 0 ⎞ ⎟ ⎠ • Y(τ ), Y(0) = 1. ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (4.15)
The solution of this differential equation yields the second-order tensor Y 2 defined in (3.12) in the form The kernel K is readily obtained after inserting (4.14) and (4.16) into (3.17a) and (3.17b) and performing integrations. At short times, the non-zero components of K may be obtained in the form of a regular expansion in t. Truncating this expansion to O(t 3/2 ) terms, we find

Y 2 (ξ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k 2 1 + e -2ξ (k 2 2 + k 2 3 ) k 2 k 1 k 2 (1 -e 2ξ ) k 2 0 k 1 k 2 (1 -e -2ξ ) k 2 k 2 2 + e 2ξ (k 2 1 + k 2 3 ) k 2 0 k 1 k 3 (1 -e -2ξ ) k 2 k 2 k 3 (1 -e 2ξ ) k 2 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 4
6πK(t) = 1 √ π ⎛ ⎜ ⎝ t -1/2 + 7 10 t 1/2 -1 105 t 3/2 0 0 0 t -1/2 -7 10 t 1/2 -1 105 t 3/2 0 0 0 t -1/2 + 11 210 t 3/2 ⎞ ⎟ ⎠ + • • •.
(4.17) Again, the leading-order term in this expansion evolves as t -1/2 , a behaviour characterizing the response of the hydrodynamic force to an impulsive velocity change [START_REF] Landau | Fluid Mechanics[END_REF]. The two t 1/2 -contributions in (4.17) have the same magnitude and are in agreement with the high-frequency behaviour determined by [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF].

To determine the steady-state limit of K, we evaluated the k-integrals involved in (3.17a) and (3.17b) numerically. The result is shown in figure 3. After the three components separate for t ≈ 0. 1,[K] 1 1 and [K] 3 3 gradually reach their steady-state value for t = O(1). In contrast, the component [K] 2 2 (t) corresponding to the compressional direction of the strain sharply decreases and becomes negative for t ≈ 1.4. At longer times, its absolute value increases and becomes of O(1). So far, despite various attempts to stretch the integrand in the vicinity of k = 0 (which yields the dominant contribution to the steady-state components of K), we have been unable to compute [K] 2 2 (t) beyond t ≈ 32, where we find (using Maple ) 6π[K] 2 2 ≈ -1.48. At the present stage, considering that the tendency for the absolute value of [K] 2 2 to increase goes on at later times, the steady-state kernel is expected to be 6πK = 6π

∞ 0 K i (ξ ) dξ ⎛ ⎝ 0.901 0 0 0 6π[K] 2 2 < -1.48 0 0 0 0 .420 ⎞ ⎠ . (4.18)
Only the diagonal components are non-zero in (4.17) and (4.18), and they all differ in magnitude. Consequently, if the body does not move along one of the principal directions of the strain, it experiences a transverse or lift force. For instance, suppose a sphere moves ahead of the fluid with a unit slip velocity along the first bisector of the (e 1 , e 2 ) plane. Then, according to (4.17), it experiences a growing transverse force f

T (t) ≡ -18π 2 ([K] 1 1 -[K] 2
2 )(e 1e 2 ) = (21/10)(2π) 1/2 t 1/2 (e 2e 1 ) at short times, which eventually becomes

f T = 3π(0.901 -6π[K] 2 2 ) ((e 2 -e 1 )/ √
2) in the steady-state limit. As [K] 2 2 is expected to be negative, this transverse force tends to deviate the sphere towards the compressional e 2 -axis at both short and long times. This is qualitatively consistent with the conclusion of [START_REF] Drew | The force on a small sphere in slow viscous flow[END_REF], who, in present notations, found f T = 3.012π ((e 2e 1 )/ √ 2). However this prefactor is uncertain because Drew's result for the kernel component corresponding to the e 1 -extensional direction is 6π[K] 1 1 = 0.602, instead of 0.901 in (4.18).

The kernel in a linear shear flow

We now consider the more widely studied case of a linear shear flow in which A takes the form

A = ⎛ ⎝ 0 0 1 0 0 0 0 0 0 ⎞ ⎠ .
(5.1)

In that case, the unit vector e 1 points in the flow direction, e 3 points in the shear direction, and e 2 is aligned with the direction of the undisturbed vorticity, such that ∇ × U ∞ = -e 2 . Exponentiating A, (3.6) implies

F(t) = ⎛ ⎝ 1 0 t 0 1 0 0 0 1 ⎞ ⎠ . (5.2)
Inserting this result into (3.10) yields

K 2 (ξ ) = k 2 + 2k 1 k 3 ξ + k 2 1 ξ 2 , so that t τ K 2 (t -τ ) dτ = k 2 ξ + k 1 k 3 ξ 2 + 1 3 k 2 1 ξ 3 . (5.3) Then (3.13) takes the form dY(τ ) dτ = 2 k 2 + 2k 1 k 3 (t -τ ) + k 2 1 (t -τ ) 2 ⎛ ⎝ 0 0 -[k 2 2 + k 2 3 + k 3 k 1 (t -τ )] 0 0 k 1 k 2 0 0 [k 1 k 3 + k 2 1 (t -τ )] ⎞ ⎠ • Y(τ ),
(5.4) with the initial condition Y(0) = 1. The solution of (5.4) yields the second-order tensor Y 2 defined in (3.12) in the form

Y 2 (ξ ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 - (k 2 + 2k 3 k 1 ξ + k 2 1 ξ 2 )k 2 2 S(k 1 , k 2 , k 3 , ξ) (k1 2 + k2 2 ) k 2 1 (k 2 1 + k 2 2 ) - ξ(k 3 k 3 1 ξ + k 2 2 k 2 + 2k 2 1 k 2 3 ) (k 2 1 + k 2 2 )k 2 0 1 (k 2 + 2k 3 k 1 ξ + k 2 1 ξ 2 )k 2 k 1 S(k 1 , k 2 , k 3 , ξ) (k1 2 + k2 2 ) k 2 1 (k 2 1 + k 2 2 ) + ξ(k 2 1 + k 2 2 -k 2 3 -k 3 k 1 ξ)k 2 k 1 (k 2 1 + k 2 2 )k 2 0 0 k 2 + 2k 3 k 1 ξ + k 2 1 ξ 2 k 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
(5.5) 10 4 10 3 10 2 t 10 1 10 0 10 -1 10 -1 10 0 FIGURE 4. Time variation of the components of the kernel K in a linear shear flow. Black lines correspond to the diagonal components (i.e. the inertial corrections to the drag force), with 6π[K] 1 1 (solid line), 6π[K] 2 2 (dash-dotted line) and 6π[K] 3 3 (dashed line). Dark grey lines correspond to the off-diagonal components, with 6π[K] 1 3 (dashed line) and 6π[K] 3 1 (solid line); the latter is the time-dependent counterpart of the Saffman lift force. Circles correspond to the inverse Fourier transform of the results obtained in the frequency domain by [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF]. Pale grey lines correspond to the t -1/2 -Basset-Boussinesq kernel (dashed line), and to the off-diagonal components of the kernel derived by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] in the short-time limit (dash-dotted line).

where the function S(k 1 , k 2 , k 3 , ξ) is given by

S(k 1 , k 2 , k 3 , ξ) = tan -1 k 1 (k 3 + k 1 ξ) k 2 1 (k 2 1 + k 2 2 ) -tan -1 k 1 k 3 k 2 1 (k 2 1 + k 2 2 )
.

(5.6)

The kernel K(t) is obtained after inserting the above expressions into (3.16)-(3.17) and performing the required integrations. Figure 4 shows the corresponding result for each non-zero component of K(t), integrations having again been performed with Maple . At short times, each component exhibits a power-law form, albeit with a different exponent for the diagonal and off-diagonal components. Also shown are numerical data (circles) for the [K] 3 1 component, which corresponds to the Saffman lift force in the limit t → ∞ in the case of a sphere, i.e. with f (0) = 6πe 1 . To obtain these data, we numerically performed the inverse Fourier transform of the frequency-dependent expression derived by [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF], as described by [START_REF] Candelier | Time-dependent lift force acting on a particle moving arbitrarily in a pure shear flow, at small Reynolds number[END_REF]. [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF] could compute the full frequency dependence for this specific kernel component because in that specific case, the partial differential equation (2.22) simplifies to an ordinary differential equation. We did not succeed in simplifying the kernel components (3.17a) and (3.17b) for arbitrary values of t. This is why we illustrate the physical mechanisms at play and compare present findings with available results [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF][START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] in the two limit cases t 1 and t 1.

Short-time limit

In the former limit, similar to the case of the purely elongational flow, the non-zero components of K may be obtained in the form of a regular expansion with respect to t.

Keeping only the first two terms in each infinite series, K reduces to

6πK(t) = 1 √ π ⎛ ⎜ ⎝ t -1/2 + 1 70 t 3/2 0 7 20 t 1/2 + 1 800 t 5/2 0 t -1/2 + 13 280 t 3/2 0 7 20 t 1/2 -53 5600 t 5/2 0 t -1/2 + 13 420 t 3/2 ⎞ ⎟ ⎠ + • • •.
(5.7)

Not surprisingly, the leading-order behaviour of the diagonal terms is again found to behave as t -1/2 . As is well known, the corresponding contributions, which yield the classical Basset-Boussinesq 'history' force, result from vorticity diffusion across the boundary layer that develops around the body after the flow is abruptly started. These effects only involve the inner solution corresponding to |r| -1 , as discussed in § 2.3, since vorticity stays concentrated in the neighbourhood of the body for t 1. The initial t -1/2 decrease of the drag components may then be readily understood by equating the rate of work of the drag force to the viscous dissipation throughout the fluid. Since the boundary layer thickness grows as t 1/2 , velocity gradients within it decay as t -1/2 , making the local dissipation rate decrease as t -1 . Therefore the integral of this dissipation throughout the boundary layer volume decreases as t -1/2 and the drag force changes accordingly.

Figure 4 shows that the diagonal components of K (black lines) start to depart from the t -1/2 behaviour after a few time units, which typically corresponds to the time it takes for the vorticity to reach the Saffman distance, s = -1 . The next-order terms are the signature of inertial effects resulting from the increasing role of vorticity advection at distances from the body larger than s . The t 3/2 terms involved in the diagonal components differ from one component to the other, due to the anisotropy of the base flow.

Let us now consider the off-diagonal components in (5.7), depicted by dark grey lines in figure 4. Only those corresponding to a slip velocity lying in the plane of the shear are non-zero. The component [K] 3 1 corresponds to a force directed along the shear (hence at right angle from the streamlines) when the slip velocity is aligned with the undisturbed flow. This is the component that yields the Saffman lift force in the long-term limit. The component [K] 3 1 corresponds to a force aligned with the streamlines when the slip velocity lies in the direction of the shear; it was first computed by [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF]. Both lift components cannot exist in the creeping-flow limit, owing to reversibility [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF], and are therefore due to fluid inertia effects. As already observed in the elongational flow, these two off-diagonal components are equal in the short-time limit. They are identical to the leading-order inertial corrections of the diagonal components in (4.17) and agree with the high-frequency asymptote of the mobility tensor, U(ω) = (M 1 ) -1 (ω), computed by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF], which in dimensional variables is written in every linear flow (equation (5.10) in their paper) as

U(ω) ∼ - -iωa 2 ν 1/2 1 + 7 40 A + A T iω , (5.8)
where ω denotes the radian frequency. The short-time contribution in [K] 3 1 is also identical to that determined by [START_REF] Asmolov | The inertial lift on an oscillating sphere in a linear shear flow[END_REF].

The t 1/2 short-time evolution of these off-diagonal components may be understood by considering how nonlinear effects modify the vorticity disturbance ω = ∇ × w, especially how they tilt the upstream vorticity ∇ × U ∞ oriented along the e 2 -direction to generate a non-zero vorticity component oriented along the e 1 -direction in the wake. This streamwise vorticity component is known to be the key ingredient yielding a non-zero lift force on a three-dimensional body (Lighthill 1956). According to (2.12), the vortex stretching/tilting term is written as Re s {A • ω + (∇ × (A • r)) • ∇w}, so that its streamwise component is Re s {ω 3 -(e 2 • ∇)w 1 }. At short time, ω 3 and (e 2 • ∇)w 1 decay as t -1/2 within the boundary layer, owing to the t 1/2 thickening of the latter, and so does the vortex stretching/tilting term. To balance this decay, the time rate of change of ω 1 (and the diffusion term ∇ 2 ω 1 ) must decay at the same rate, which results in a t 1/2 -growth of ω 1 . Since ω 1 = (e 2 • ∇)w 3 -(e 3 • ∇)w 2 , the growth of ω 1 induces inertial corrections to the transverse velocity components w 2 and w 3 that also grow as t 1/2 . This in turn results in a similar growth of the transverse pressure gradient, which yields the observed t 1/2 -growth of the non-diagonal components of the force on the body. That these two components are identical for t 1 may readily be understood by considering successively a sphere translating along the e 1 -and e 3 -directions, with the same slip velocity. The corresponding two inner solutions are identical (up to a switch in the dependency with respect to the x 1 and x 3 coordinates), and so is the vorticity distribution about the sphere. Hence the initial stretching and tilting of the vorticity in the wake have the same magnitude in both configurations, a nd s o d o t he two components of the lift force. Figure 4 indicates that these two components separate beyond a time of the order of a few units, as already observed for the diagonal terms. The reason for this separation is discussed in the next subsection.

Long-time limit

To determine the steady-state limit of K, the k-integrals in (3.17a)-(3.17b) were evaluated numerically up to t = 10 000, yielding with a four-digit accuracy 6πK ⎛ ⎝ 0.0737 0 0.9436 0 0.5766 0 0.3425 0 0.3269 ⎞ ⎠ .

(5.9)

These values are in almost perfect agreement with those obtained by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] (their equation (5.27)), the largest deviation being 0.3 %. The reason for the tiny differences left between the two sets of coefficients most likely results from truncation errors associated with numerical integration. Note that, despite some similarities, the integrals that appear in the calculation of [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] and those involved in (3.17a)-(3.17b) are different, and we do not know how to transform them into each other. Note also that the values of the non-zero components in the first two rows of (5.9) differ significantly from those determined by [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF], presumably because of the limited accuracy that they could reach in the numerical integration procedure.

In figure 4, the convergence of the kernel components to their steady-state value is seen to be slow, especially for the [K] 1 3 component. The dimensional time it takes to reach the steady state is of the order of 10 s -1 for [K] 3 1 , but is typically two orders of magnitude larger for [K] 1 3 . For that component, we found that the asymptotic value is approached in a power-law fashion, namely (5.10) where, according to Maple , C 1 3 ≈ 1.252.

6π[K] 1 3 (t) ∼ 6π[K] 1 3 -C 1 3 t -1/2 ,
The slow convergence of the lift components towards their steady-state value has direct consequences on the migration of particles in turbulent flows. In particular, if one plans to examine lateral migration phenomena using a point-particle approach, it is clear from figure 4 that such features may be grossly overestimated if the steadystate values [K] 1 3 and [K] 3 1 are used instead of the instantaneous values, unless in the specific situation where the particle stays in a given vortex (i.e. experiences a given shear rate) during a dimensional time much larger than s -1 .

As pointed out by [START_REF] Hogg | The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows[END_REF], the physical mechanism that produces the lift force corresponding to [K] 3 1 may be understood by considering the fluid displaced laterally by the body as it translates along the streamlines of the base flow. In the wake, assuming a positive shear rate, this displaced fluid moves faster (respectively slower) with respect to the body at a given x 3 > 0 (respectively x 3 < 0). At large enough distances from the body, nonlinear advective processes associated with the last two terms in the left-hand side of (2.17) dominate and this asymmetry results in a lateral pressure gradient directed towards negative x 3 ; hence a lift force directed towards positive x 3 . The mechanism responsible for the lift component associated with

[K] 1
3 is more subtle because in that case the body translates across the streamlines of the shear flow, and the shear forces the wake to bend. Suppose that the body moves in the direction of increasing velocity, i.e. towards positive x 3 , and stands at the position where the undisturbed velocity vanishes. Then the fluid contained in the wake experiences a negative transverse velocity that increases with the downstream distance to the body, resulting in a bending of the wake axis towards negative x 1 . Because of this bending, within a section of the wake perpendicular to its axis, the magnitude of the transverse velocity provided by the undisturbed flow increases with x 1 . Then, repeating the above argument leads to the conclusion that, at large enough distances from the body, a transverse pressure gradient directed towards x 1 < 0 takes place within each cross-section of the wake, resulting in a lateral force on the body directed towards x 1 > 0. As there is no reason for the transverse pressure gradient to be identical in the two situations, it is no surprise that [K] 3 1 = [K] 1 3 .

6. Influence of small inertia effects on the sedimentation of non-spherical particles in a linear flow

It is known that inertia effects make a crucial contribution to the nature of the motion of small, neutrally buoyant non-spherical particles immersed in a shear flow. In particular, [START_REF] Feng | The unsteady motion of solid bodies in creeping flows[END_REF] showed numerically that effects of unsteadiness, be they due to the body or the fluid inertia, tend to suppress the periodic oscillations predicted by the quasi-steady approximation. Influence of the body inertia in the case where the particle stands close to a wall also induces dramatic changes because, combined with the wall-particle hydrodynamic interaction, it induces a drift of the particle towards the wall [START_REF] Gavze | Motion of inertial spheroidal particles in a shear flow near a solid wall with special application to aerosol transport in microgravity[END_REF]. Small-but-finite fluid inertia effects are known to affect the hydrodynamic torque and angular motion in such a way that the marginal stability of the Jeffery orbits of spheroidal particles is broken [START_REF] Subramanian | Inertial effects on fibre motion in simple shear flow[END_REF][START_REF] Candelier | The role of inertia for the rotation of a nearly spherical particle in a general linear flow[END_REF]Einarsson et al. 2015a;[START_REF] Rosen | Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers[END_REF][START_REF] Dabade | The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow[END_REF]); these conclusions were recently extended to an arbitrary linear flow field [START_REF] Marath | The inertial orientation dynamics of anisotropic particles in planar linear flows[END_REF]. Unsteady fluid inertia effects have also been shown to make a significant contribution to the body-shape dependence of the stability exponents of the Jeffery orbits (Einarsson et al. 2015b). Most of the above results were obtained by deriving an approximate angular equation of motion for the particle orientation by using a regular first-order perturbation approach with respect to Re s . In the case where the particle and fluid d ensities are different, the particle does not exactly follow the flow, s o t hat a n on-negligible slip takes place and the hydrodynamic force is modified b y fl uid in ertia ef fects at order Re s 1/2 . The question is then that of the influence o f t he c orresponding contributions to the force on the particle path. Addressing this issue requires the translational problem to be solved, which is more challenging than the angular problem, because the corresponding perturbation is singular, as discussed in § 2.3. In this section, we make use of the developments and results provided earlier in the paper to consider this question, first f or a rbitrarily s haped p articles s edimenting i n a g eneral l inear flow, then in more detail for spheroids immersed in a linear shear flow.

General results at O( )

In what follows, we implicitly assume that the body density, ρ p , is of the same order as that of the fluid, ρ f , s o t hat t he l ow-Reynolds-number c onditions ( 2.9) m ay be satisfied w ithin a s ignificant ra nge of flu id vis cosities and bod y siz es. Sti ll assuming Sl = 1, the body motion is governed by the force and torque balances

f (0) τ (0) + f (1) τ (1) + ⎡ ⎣ V p ρ p ρ f -1 g 0 ⎤ ⎦ = O( 2 ) O( 2 ) . (6.1)
Here V p is the non-dimensional volume of the body and g denotes the gravitational acceleration normalized by a(sν) -1 . The force and torque f (0) and τ (0) are those corresponding to the Stokes limit (2.4), whereas f (1) and τ (1) are those due to leadingorder fluid inertia effects. Terms of O( 2) in the right-hand side of (6.1) comprise various inertial contributions, especially those due to the body acceleration, ẍp , which include added-mass effects, and those due to the local acceleration of the undisturbed flow, DU ∞ /Dt = A • (A • r) (keeping in mind that only steady undisturbed flows are considered in this work). Since the present theory is valid only up to O( ), it is consistent to neglect such inertial contributions. However, during the initial transient following the introduction of the particle in the flow, its acceleration may be large enough for 2 ẍp to be of O( ). Hence, results based on (6.1) are not expected to be valid during this initial transient, the duration of which is of the order of the viscous time scale, a 2 /ν. According to (3.14) and (3.15), one has

f (1) τ (1) = - M 1 (t) M 2 (t) M T 2 (t) M 3 (t) • ⎡ ⎣ t 0 K(t -τ ) • d dτ f (0) dτ 0 ⎤ ⎦ . (6.2)
Assuming that the slip velocity between the body and fluid is zero at t = 0, the point force in (6.2) has the form

f (0) (t) = -H(t)f (0) , (6.3)
where H(t) denotes the Heaviside function. In keeping with the approximations used throughout the paper, we solve (6.1) through an expansion in the small parameter , seeking the expansion in the form ẋp = ẋ(0)

p + ẋ(1) p + O( 2 ), ω p = ω (0) p + ω (1) p + O( 2 ).
To leading order, one has

f (0) (ẋ (0) p , ω (0) p ) τ (0) (ẋ (0) p , ω (0) p ) = ⎡ ⎣ V p 1 - ρ p ρ f g 0 ⎤ ⎦ , (6.4) so that ẋ(0) p ω (0) p = U ∞ Ω ∞ - M 1 (t) M 2 (t) M T 2 (t) M 3 (t) -1 • ⎡ ⎢ ⎣ N 1 (t) : S ∞ + V p 1 - ρ p ρ f g N 2 (t) : S ∞ ⎤ ⎥ ⎦ . (6.5)
Equation ( 6.5) describes the gravity-driven settling of the body in the Stokes limit. No external force acts at O( ), so that

f (0) (ẋ (1) p , ω (1) p ) τ (0) (ẋ (1) p , ω (1) p ) = - f (1) (ẋ (0) p , ω (0) p ) τ (1) (ẋ (0) p , ω (0) p ) , (6.6) which yields ẋ(1) p ω (1) p = - ⎡ ⎣ t 0 K(t -τ ) • d dτ (f (0) (ẋ (0) p (τ ), ω (0) p (τ ))) dτ 0 ⎤ ⎦ .
(6.7)

Using (6.4) and ( 6.3) and noting that Ḣ(t) = δ(t), one is finally left with

ẋ(1) p ω (1) p = ⎡ ⎣ V p 1 - ρ p ρ f K(t) • g 0 ⎤ ⎦ . (6.8)
Hence, only the translational velocity of the body is altered by inertia effects at order Re 1/2 s , irrespective of the body geometry. In the long-term limit, the kernel K(t) tends towards its steady-state value, K. Gathering (6.5) and (6.8) yields in that limit

ẋp ω p = U ∞ Ω ∞ - M 1 (t) M 2 (t) M T 2 (t) M 3 (t) -1 • ⎡ ⎢ ⎣ N 1 (t) : S ∞ + V p 1 - ρ p ρ f g N 2 (t) : S ∞ ⎤ ⎥ ⎦ + ⎡ ⎣ V p 1 - ρ p ρ f K • g 0 ⎤ ⎦ . (6.9)
Given the structure of the kernel, K is independent of the shape, initial orientation and possible rotation of the body. Hence, according to (6.9), the long-term translational and angular velocities of the body depend on time only through the variations of the tensors M i and N i . In other terms, the long-term time dependence of the body dynamics is similar to that corresponding to Stokes conditions. That no additional time dependence is introduced by the O( ) inertial effects is due to the fact that the external body force and the carrying flow do not depend upon time. The remaining question is that of the time it takes for the body to reach such a quasi-steady state. We shall come back to this in the next subsection.

6.2. Sedimentation of prolate and oblate spheroids in a linear shear flow As an application of the above results, we now specialize them to the case of spheroids sedimenting in a linear shear flow, with the aim of examining how the sedimentation dynamics is affected by small inertia effects, especially before the quasi-steady state defined above is reached. As is well known, a spheroid generally rotates when immersed in a non-uniform flow, owing to the hydrodynamic torque acting on it. However, the resistance tensors M 2 and N 1 in (2.4) vanish for a spheroidal body, owing to its geometrical symmetries. This implies that there is no coupling between the angular and the translational dynamics of the body, which can therefore be treated separately. Although the body rotation is generally affected by effects of fluid inertia, this alteration only takes place at O( 2 ) [START_REF] Subramanian | Inertial effects on fibre motion in simple shear flow[END_REF]Einarsson et al. 2015b;Candelier, Einarsson & Mehlig 2016;[START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF]. Hence, in the O( ) approximation considered here, the angular velocity of the spheroid is that corresponding to the creeping-flow limit.

In that limit, a small spheroid in a shear flow is known to tumble periodically with an angular velocity obeying [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]) (6.10) where Λ = (λ 2 -1)(λ 2 + 1) -1 is a shape parameter that depends on the body aspect ratio, λ, which is the ratio of the body length along the symmetry axis to that of its equatorial diameter. The normalizing length, a, considered so far is taken to be the half-length of the major semiaxis, while b is the half-length of the minor semiaxis, so that λ = a/b (respectively b/a) for a prolate (respectively an oblate) spheroid. The kinematic equation (d/dt)n = ω p × n, governing the evolution of the orientation of the body symmetry axis (see figure 1), has an infinite number of marginally stable periodic solutions, commonly known as Jeffery orbits. Here we assume that the symmetry vector n tumbles within the (e 1 , e 3 ) plane where the shear flow takes place (this is the orbit expected to produce the largest unsteadiness). With this choice, the angular velocity of the spheroid is related to the angle θ(t) made by the spheroid's axis with the streamlines of the shear flow through ω p = θ(t)e 2 , and θ(t) obeys (6.11) This ordinary differential equation has a periodic solution [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF], characterized by a period T J = (2π)/ √ 1 -Λ 2 . As a result, n rotates within the (e 1 , e 3 ) plane according to n(t) = e 1 cos θ(t) + e 3 sin θ(t). (6.12)

ω p = Ω ∞ + Λn × (S ∞ • n),
θ(t) = 1 2 + Λ 1 2 -cos(θ (t)) 2 .
This periodic angular motion acts as an unsteady disturbance for the translational problem. We assume θ(0) = 0, i.e. the symmetry axis of the spheroid is initially aligned with the streamlines of the base flow. Following (2.4), one has (6.13) where u s (t) = ẋp (t) -U ∞ (x p (t)) denotes the instantaneous slip between the body and fluid. The resistance tensor is known to be diagonal in the principal axes of the spheroid [START_REF] Kim | Microhydrodynamics: Principles and Selected Applications[END_REF], so that (6.14) where the components M and M ⊥ depend on the aspect ratio, λ. For a prolate spheroid (λ > 1) they are (6.15) whereas for an oblate spheroid (λ < 1) one has

f (0) = M 1 (t) • u s (t),
M 1 = M nn + M ⊥ (1 -nn),
M = 8 3λ 6π -2λ λ 2 -1 + 2λ 2 -1 (λ 2 -1) 3/2 ln λ + √ λ 2 -1 λ - √ λ 2 -1 , M ⊥ = 8 3λ 6π λ λ 2 -1 + 2λ 2 -3 (λ 2 -1) 3/2 ln λ + √ λ 2 -1 , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
M = 8 3 6π 2λ 1 -λ 2 + 2(1 -2λ 2 ) (1 -λ 2 ) 3/2 tan -1 √ 1 -λ 2 λ , M ⊥ = 8 3 6π - λ 1 -λ 2 - (2λ 2 -3) (1 -λ 2 ) 3/2 sin -1 √ 1 -λ 2 . ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (6.16)
Due to the body rotation, the orientation vector n(t) depends upon time and so do the components of M 1 in the laboratory frame.

To reveal the influence of small inertia effects on the body dynamics, we numerically integrated (6.1)-( 6.2) with the following dimensional parameters: a = 1 mm, ν = 10 -4 m 2 s -1 , s = 10 s -1 and ρ p /ρ f = 1.5. These parameters imply ≈ 0.32, so that predictions provided by the asymptotic approach are expected to be at least qualitatively valid. We consider four distinct spheroid aspect ratios, namely λ = 1/10, 1/2, 2 and 10. Gravity is set in the form g = -ge 3 , so that spheroids are settling along the shear. Integration of (6.1)-(6.2) is achieved using a method inspired from [START_REF] Daitche | Advection of inertial particles in the presence of the history force: higher order numerical schemes[END_REF], with the history integral in (6.2) evaluated using an implicit scheme. The results of this integration are displayed in figures 5-8. As we continued to assume Sl = 1, equation (2.5) implies τ c = s -1 , so that t = 1 corresponds to the time required for the disturbance to diffuse over a distance of O(Re -1/2 s ), i.e. to reach the outer region of the perturbation problem defined in § 2. This diffusion time is much shorter than the Jeffery's period, T J , which is larger than 2π whatever λ. Initially, the particle acceleration is very large, typically of O( -1 ), so that terms proportional to 2 ẍp which were neglected in (6.1) are of the same order as the O( )-terms computed in the present theory. We checked that in all cases the components of ẍp become less than -1 within a time period shorter than 0.1. Consequently, results displayed in figures 5-8 are accurate beyond the short initial transient corresponding to t 0.1, and the discussion below disregards this very first transient.

Let us first comment on the predictions in which inertial corrections are ignored. Clearly, predictions of the horizontal slip component, u s • e 1 , obtained by considering only the Stokes quasi-steady drag (dashed lines) and those in which the Basset-Boussinesq 'history' force is also taken into account (grey lines) are indiscernible, indicating that the relative acceleration between the body and the fluid does not play any role on that component. In contrast the two predictions for the vertical slip component, u s • e 3 , differ significantly all along the body path, especially when the aspect ratio is of order unity (figures 5 and 7). It is worth noting that the time-averaged horizontal slip, u s1 , is not strictly zero, as may especially be inferred from the above two figures. Although surprising at first glance, the small positive value of u s1 may be understood by noting that the time rate-of-change of the horizontal slip component, (ẋ p -U ∞ ) • e 1 , involves the time derivative of the fluid velocity along the body path, U∞ = (ẋ p • e 3 )e 1 . Hence, as the body settles (ẋ p • e 3 < 0), it 'sees' a decreasing horizontal background velocity, which makes the horizontal slip drift towards positive values, resulting in a slightly positive u s1 .

Let us now turn to the influence of inertial corrections. In all four cases, it is seen that these corrections result in a large positive shift of the horizontal slip. In particular, this slip component is now positive for all times, i.e. the spheroid drifts in the e 1 -direction, for a spheroid with λ = 2 (figure 5a). The same feature is observed with the oblate spheroid corresponding to λ = 0.5 (figure 7a), except at very short times. The comparison of the left panels in figures 5-8 suggests that effects of time dependence in K(t) manifest themselves over a much larger number of tumbling periods for spheroids with O(1) aspect ratios. However, this is merely a consequence of the influence of λ on T J , which makes the tumbling period approximately four times shorter for λ = 2 or 1/2 compared to λ = 10 or 1/10. One may also note a slight reduction of the vertical slip velocity, which corresponds to the increase of the drag force associated with the positive diagonal component [K] 3 3 in (5.7) and (5.9). This slip component is found to converge towards the quasi-steady prediction (which is identical to that of [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF] for that component) within a few tumbling periods in all cases. The convergence of the horizontal slip component is much slower. This is no surprise since this component directly depends on [K] 1 3 (t), which was found to converge very slowly towards its steady-state value in figure 4. Moreover, as revealed by the same figure ,[K] 1 3 is the kernel component with the largest magnitude for t 5 (it is approximately three times larger than the Saffman's component [K] 3 1 at steady state), which makes the consequences of its slow convergence potentially large, as figures 5(a) and 6(a) indicate. Indeed, considering the steady-state expression (5.9) of the kernel (which, although based on the same set of assumptions as that of [START_REF] Harper | Maximum dissipation resulting from lift in a slow viscous shear flow[END_REF], has significantly different values of [K] 1 1 and [K] 1 3 for reasons explained in § 5.2) overestimates the horizontal slip by up to 50 % for both λ = 1/2 and λ = 2, even after several tumbling periods (see the last half-period in figures 5a and 7a). Thus during the very long transient required for the kernel to relax towards its quasi-steady value, estimating the O( ) inertial corrections to the force using K instead of K(t) yields O( ) errors. Consequently, only the prediction based on the instantaneous kernel is able to capture properly the evolution of the horizontal slip during that transient, the duration of which depends solely on the kernel properties, not on the shape of the particle nor on its rotation.

The above examples shed light on the importance of inertial corrections to the hydrodynamic force on the path of spheroids sedimenting in a shear flow. In particular, they show that, for values of of O(10 -1 ), the horizontal component of the slip velocity cannot realistically be predicted on the basis of the forces derived in the creeping-flow a pproximation, b e i t q uasi-steady o r f ully unsteady. Moreover they demonstrate that, owing to the very slow convergence of some components of the inertial kernel, large errors can be made in the prediction of this slip component during transients much longer than the tumbling period, if the quasi-steady approximation of the inertial corrections to the force is used in place of their time-dependent expression.

Summary and prospects

In this paper, we developed a generic methodology aimed at determining the leading-order inertial corrections to the instantaneous force and torque acting on an arbitrarily shaped rigid body moving with a time-dependent slip velocity in a quasi-steady linear flow field. We carried out the corresponding developments in the framework of the MAE approach, under the assumption that effects of the slip velocity between the body and the fluid are negligible compared to those due to the ambient velocity gradients. The key of the success was to express the flow disturbance in a non-orthogonal co-moving coordinate system that reduces the initial set of partial differential equations governing the disturbance problem in Fourier space to a set of ordinary differential equations that are much more easily solved whatever the nature of the background linear flow. The solution of this differential problem was obtained in the form of a closed convolution kernel, thanks to the use of Magnus expansions. The above idea is in essence similar to that used by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] in the framework of the IF formulation. However, it is somewhat hidden in their work, where it appears only through the use of time-dependent wavevectors during the step when the solution of the disturbance problem is sought in Fourier space. Because of these differences, the kernels provided by the two approaches exhibit a quite different mathematical structure, although they must yield identical predictions once integrated for any specific time variation of the slip velocity.

We proved the versatility of our approach by computing explicitly the kernel in the case of a body moving in a planar flow corresponding to a solid-body rotation, a planar elongation or a uniform shear. In the first case, as expected, we recovered the kernel derived by [START_REF] Candelier | Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers[END_REF] using a change of reference frame. In the uniform shear configuration, all kernel components agree very well with those computed by [START_REF] Miyazaki | Drag on a sphere in slow shear flow[END_REF] in both the short-and long-time limits, thus providing a stringent validation of the various steps involved in the present procedure. We actually computed the kernel for arbitrary times, which revealed in particular that some components require a much longer time than others to reach their steady-state value. Few results are available for the purely elongational case, which we also considered. Our results recover the short-time behaviour predicted by [START_REF] Bedeaux | Drag on a sphere moving slowly through a fluid in elongational flow[END_REF]. At longer time, we noticed an unexpected behaviour. While the kernel component corresponding to the extensional direction converges gradually towards its steady-state value (which differs from that predicted by [START_REF] Drew | The force on a small sphere in slow viscous flow[END_REF], the component corresponding to the compressional direction changes sign, implying that long-term inertial effects tend to decrease the drag force in that direction. Unfortunately, we have not yet computed the steady-state value of this component, due to technical difficulties encountered in the numerical integrations.

As shown in § 6.1, once the kernel is determined, the MAE approach allows the leading-order inertial force and torque corrections on a non-spherical body to be evaluated in a straightforward manner, provided the body's resistance tensors are known (the same remark applies to drops and bubbles with a prescribed shape, for which the appropriate kernel may be directly deduced from that of the corresponding solid body by applying the argument developed by [START_REF] Legendre | A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow[END_REF]). We illustrated this in § 6.2 by considering the sedimentation of spheroids, the rotational dynamics of which is unaffected by inertial effects at the order considered here. In contrast, we found that the horizontal component of their slip velocity is dramatically enhanced by these effects. Consequently, neither the Stokes approximation nor the refined approximation including the Basset-Boussinesq history force provides a reasonable prediction of this characteristic. Although closer to the actual evolution, the prediction based on the quasi-steady limit of the inertial corrections is also poorly accurate, especially for spheroids with moderate aspect ratios, owing to the aforementioned slow convergence of some of the kernel components.

As they stand, the kernels derived in this paper make it possible to compute the trajectory of a small isolated particle moving in an arbitrary direction in any of the three linear flow fields analysed in § § 4 and 5. The particle may have an arbitrary shape, provided the 'resistance' tensors involved in (2.4) are known, and effects of unsteadiness associated with its velocity variations may be large. These are the main two practical outcomes of this paper. For these predictions to apply, the asymptotic conditions discussed in § 2.2 must be satisfied. Besides the fact that all three Reynolds numbers of the problem have to be small, the most restrictive condition is most often (2.9), which requires the slip velocity to be very small, implying that the body-tofluid density ratio must be close to unity in most practical cases (see the discussion in [START_REF] Mclaughlin | Inertial migration of a small sphere in linear shear flows[END_REF] regarding the limitations of Saffman's assumptions).

Beyond these direct but quite specific applications, present results represent a first step towards a rational extension of the Basset-Boussinesq-Oseen approximation describing the unsteady motion of a small rigid particle to situations involving small-but-nonzero inertial effects due to the carrying flow. Such an extension is key to improving the determination of the forces and torques that govern the motion of particles in dilute sheared suspensions and in turbulent flows. To progress towards this objective, we now plan to extend present results in several directions. Our first goal remains to obtain the expression of the time-dependent kernel in a general steady linear flow characterized by an arbitrary traceless velocity gradient tensor, A. To maximize the usefulness of the outcome in terms of applications, we shall seek the components of that kernel in a general form involving explicitly the A i j components, in such a way that the inertial corrections can be straightforwardly computed in any linear flow once A is known. By proceeding in this manner, all results corresponding to the canonical flows considered here will be recovered as special cases. Analytical forms of the kernel will certainly be limited to the short-and long-times limits and we will have to develop approximate fits to provide expressions valid for arbitrary times.

Then, two central assumptions extensively used in the present work will have to be removed. First of all, we will have to consider that the deformation tensor, A, may be time-dependent. This is essential for the prediction of particle motion in turbulent flow, owing to the aforementioned slow convergence of several kernel components. Indeed, the turnover time of small-scale eddies, which have the largest velocity gradients, is too short to allow these kernel components to reach their steady-state value within a time interval during which the carrying flow m ay b e c onsidered frozen. For instance, only eddies larger than the Taylor microscale have a turnover time larger than the viscous time η k 2 /ν corresponding to particles with a characteristic size of the order of the Kolmogorov length scale, η k . Additional technical difficulties are expected with time-dependent flows b ecause ( 3.6) i s n o l onger v alid a nd a n ew term appears in (3.7). Moreover, when A depends upon time, the solution of (3.13) does not depend on the time lag t -τ only, so that the final e xpression o f t he inertial corrections no longer simplifies t o a c onvolution product.

Second, in the same spirit as the extension carried out by [START_REF] Asmolov | Dynamics of a spherical particle in a laminar boundary layer[END_REF] and [START_REF] Mclaughlin | Inertial migration of a small sphere in linear shear flows[END_REF] in a pure shear flow f or t he q uasi-steady S affman l ift f orce, we will have to go beyond Saffman's condition (2.9) in a general linear flow. F or t his, we must allow the two Reynolds numbers, Re s and Re p , to be of similar magnitude and examine how the kernel varies with the ratio sp = √ Re s /Re p comparing the Oseen length scale, o = Re p -1 , to the Saffman length scale, s . In this way we shall cover time-dependent situations in which dominant advective corrections are due to shearing effects (as in the present paper) as well as situations in which Oseen-like effects dominate. The limit sp → 0 corresponds to the time-dependent problem considered by [START_REF] Lovalenti | The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number[END_REF], who showed that advective effects drastically reduce the long-term magnitude of the 'history' force because a vorticity disturbance resulting from a change in the slip velocity is more efficiently r emoved f rom t he b ody's vicinity by these effects than by viscous diffusion once it has entered the Oseen region of the body-induced flow. F or i nstance ' history' e ffects d ecay a s t -2 a t l arge t imes in the case of a sudden start of the body [START_REF] Sano | Unsteady flow past a sphere at low Reynolds number[END_REF][START_REF] Lovalenti | The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number[END_REF], in contrast to the t -1/2 behaviour predicted by the Basset-Boussinesq kernel. Technically, considering finite v alues o f sp a mounts t o r eplacing t he t erm A • wˆi n ( 2.22) by {Asp -1 Re s -1/2 (ik • u s )1} • ŵ. Since u s is generally time-dependent, the extra difficulty is similar to that encountered with time-dependent velocity gradients.

The above extensions will involve a substantial amount of numerics, since solving the disturbance equation in Fourier space by hand or with the help of a symbolic computation software is only possible in very specific c ases. M ore p recisely, the solution may be obtained in this way at large k, but in most cases the ordinary differential equation needs to be solved numerically in the small-k range, which is the one that provides the leading contributions to the long-term kernel.

As a last extension, we wish to determine the second-order inertial corrections, at least in selected situations (e.g. spheroidal bodies, canonical flows), for s everal reasons. First, both the translational and the rotational dynamics are modified b y i nertial effects at O( 2 ), making this order of approximation relevant to obtain a nearly complete view of the influence o f s mall-but-finite in ertial ef fects on th e dy namics of particles in turbulent flows, a s w ell a s o n t he r heology o f s heared s uspensions. I t i s a lso an order of approximation where couplings between translation and rotation may happen, even for symmetric body shapes for which the coupling resistance tensor M 2 is zero. This is for instance the case of the O(Re ω ) lift force experienced by a spinning sphere translating in a fluid a t r est ( Rubinow & K eller 1 961), a nd a n O ( 2 ) e xpansion should capture this effect. Last, added-mass effects resulting from the differential acceleration between the body and the carrying flow a re a lso o f O ( 2 ). T his i s w hy t hese effects were not captured in the kernels computed in § § 4 and 5, unlike the O( ) 'history' effects. Expanding the solution of the disturbance problem up to O( 2 )-terms in linear flows w ould a llow u s t o c larify t he e xpression o f t he d ifferential acceleration involved in the added-mass force: although it is known that this contribution is proportional to the difference between the Lagrangian acceleration DU ∞ /Dt and the body acceleration dẋ p /dt in an inviscid flow [START_REF] Taylor | The forces on a body placed in a curved or converging stream of fluid[END_REF][START_REF] Auton | The force exerted on a body in inviscid unsteady non-uniform rotational flow[END_REF], the counterpart in the regime of low-but-finite Reynolds numbers is unknown. Clarifying this issue and gathering all inertial effects in a rational way up to O( 2 ) would represent a major extension of the Basset-Boussinesq-Oseen equation, even in the simplest case of a sphere, since only ad hoc extensions of this equation towards the inertial regime are available so far in non-uniform flows.

FIGURE 2 .

 2 FIGURE 2. Time variation of the components of the kernel K in a solid-body rotation flow. Solid line: 6π[K] 1 1 = 6π[K] 2 2 ; dashed line: 6π[K] 3 3 ; grey dashed line: t -1/2 short-time behaviour resulting from the contribution K h (t) in (4.5); black dash-dotted line: 6π|[K]| 1 2 ; grey dash-dotted line: short-time expansion 6π[K] 1 2 ∼ (1/75

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. Evolution of (a) the e 1 -component, and (b) the e 3 -component of the slip velocity, u s , of a prolate spheroid with aspect ratio λ = 2, as predicted using different approximations (time is normalized with the inverse of the shear rate, s -1 , and all velocities are normalized with the Stokes settling velocity of a sphere with radius a). Black line: present unsteady theory; dash-dotted line: present quasi-steady theory; dashed line: prediction based on the Stokes quasi-steady drag; grey line: prediction based on the sum of the Stokes quasi-steady drag and the Basset-Boussinesq force.

FIGURE 7 .

 7 FIGURE 7. Same as figure 5 for an oblate spheroid with aspect ratio λ = 1/2.

FIGURE 8 .

 8 FIGURE 8. Same as figure 5 for an oblate spheroid with aspect ratio λ = 1/10.

† Email addresses for correspondence: fabien.candelier@univ-amu.fr, jmagnaud@imft.fr

Acknowledgements B.M. was supported by Vetenskapsrådet (grant 2017-03865), Formas (grant no. 2014-585), and by the grant 'Bottlenecks for particle growth in turbulent aerosols' from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048.

Appendix A. Solving the disturbance equation in the co-moving reference frame A.1. Change of coordinates Equation (3.3) links the partial derivative of the velocity with respect to time evaluated at fixed r i with that evaluated at fixed R i . To derive it, we first compute the (total) time rate of change of the velocity while following the motion of a R i -coordinate position. Along the corresponding path, the velocity w may be written either in the Cartesian basis, e i , w = w i (r(R j , t), t)e i , (A 1)

or in the co-moving basis, E i ,

The time rate of change of w is then either dw dt = ∂(w i (r j (t), t)e i ) ∂t

in a Lagrangian-like approach. From (A 3) and (A 4), it is immediate to conclude that

which proves (3.3).

A.2. Solution in the co-moving basis Index notation was used throughout § 3 to avoid ambiguity in the derivation of (3.7). However, in order to solve effectively this equation, it is appropriate to switch to matrix notation. To do so, contravariant components of vectors, such as W i , are stored in column vectors, while covariant components, such as K i , are stored in row vectors. Components of the metric tensor, or of tensors F and A, are stored in matrix form. It must be pointed out that the mathematical objects we are dealing with in what follows are not necessarily tensors but may be simple matrices.

Using the above conventions and after the pressure has been eliminated with the help of the divergence-free condition, the Fourier transform of (3.7) may be cast in the form ∂ Ŵ ∂t

with

and

where, as defined in § 3, R is the inverse of the metric tensor g associated with the coordinate transformation. In order to solve (A 6), we first need to determine the solution of the homogeneous problem, which we denote by Ŵ(h) (t). The homogeneous problem takes the form of a linear system with non-constant coefficients. Its solution may be formally written under general conditions in terms of a Magnus expansion (see e.g. [START_REF] Blanes | The Magnus expansion and some of its applications[END_REF]. It then reads

where C is a parameter to be varied, and Exp(B(T)) is the exponential of the matrix B defined as

In (A 11), the matrix Ω H is given in the form of a sum, namely

The matrix Ω H represents the Magnus expansion of Ŵ(h) and the Ω (k) H that appear in (A 12) are defined as

) where the square brackets denote Lie brackets, so that for instance [H(t 1 ),

Applying the method of variation of parameters to this formal solution, and assuming that the disturbance velocity is zero at t = 0, we are led to

where

Equation (A 14) is the formal solution of the disturbance flow problem in Fourier space, expressed in the co-moving coordinate system. However, to determine the inertial correction to the force acting on the body, we need to subtract the Stokeslet solution T(0) (t) from (A 14). One way to achieve this is to perform an integration by parts of the latter. To this end, we first notice that

Using again the fact that the slip velocity is zero at t = 0, we obtain after a few manipulations

The components of the Green tensor expressed in the co-moving coordinates at any time, τ , may be written in the form

Then, making use of (A 8) and (A 18), (A 17) may be re-cast as

In (A 19), the first two terms in the second integral cancel each other but they have been introduced artificially on purpose. Attention must be paid to the fact the Greenlike matrix ĜC (t) involved in these two terms is evaluated at the current time, t, instead of τ . Integrating again by parts, making use of (A 16) and noting that

it may be shown that

Thanks to (A 21), (A 19) may then be written in the final form

A.3. Solution in the Cartesian basis Integrating (A 22) over the three-dimensional K-space yields the components of the disturbance force acting on the body in the E i -basis. However, the body slip velocity and acceleration are much more naturally expressed in the Cartesian e i -basis. This is why it is appropriate to re-write the disturbance solution in the Cartesian basis before integrating over the k-space. To this end, we may use the fact that

In addition, we note that covariant components in the co-moving basis are linked to their counterpart in the Cartesian basis through the relation

whereas contravariant components in the moving basis are linked to their counterpart in the Cartesian basis through

One may also notice that

Here it is important to point out that (A 26) is a component-to-component relation, not an intrinsic relation between tensors. Finally, at the current time t, one has

Writing the solution of the problem in the Cartesian basis then leads to

where we have used the fact that K 2 (K, τ ) = K 2 (k, t -τ ) (see (3.10)). Equation (A 28) may be further simplified by first writing the first integral on the right-hand side in the form

The second integral on the right-hand side of (A 29) can be integrated by parts as We finally obtain (3.8) by expressing the last term within parentheses in the integrand of (A 31) thanks to the two relations