domain: phys math

text:
Pavel Kuzhir

Maxime Raboisson-Michel

Jordy Queiros Campos

Gregory Verger- Dubois

Andrey Yu Zubarev
email: a.j.zubarev@urfu.ru

Unified mathematical model of the kinetics of nanoparticle phase condensation in magnetic fields

Keywords: Magnetic nanoparticles, non-equilibrium phase transition, magnetic field

In this paper, we aim to present a unified mathematical modelling and description of the kinetics of magnetic nanoparticles phase condensation (conducting to the appearance of bulk elongated aggregates) under homogeneous permanent or alternating magnetic field. For such case, the aggregate growth rate usually takes the form 𝑑𝑉/𝑑𝑡 = 𝐺(𝑉)∆(𝑡), with V and t being the aggregate's volume and time, respectively, ∆(𝑡)the supersaturation of the nanoparticle suspension, and with the function 𝐺(𝑉) depending on the precise configuration of the applied field. The Liouville equation for the aggregate size distribution function is solved by the method of characteristics. The solution is obtained in parametric form for an arbitrary function 𝐺(𝑉), providing a general framework for any type of the applied magnetic field. In the particular case of low-frequency rotating magnetic field (𝐺(𝑉)~𝑉 2/3 ), an explicit expression of the distribution function is obtained., while the dimensionless average aggregate volume 〈𝑉〉 is found by the method of moments allowing a complete decoupling of the system of equations for the statistical moments 〈𝑉 𝑛 〉 of the distribution function. Numerical examples are provided for the cases of permanent and low-or medium frequency rotating fields. It is shown that in all cases, the average volume 〈𝑉〉 only slightly depends on the relative width of the initial size distribution. Nevertheless, at any times, 𝑡 > 0, the size distribution shows a significant spreading around the average value 〈𝑉〉 which increases progressively with time and achieves a final plateau at long times. This model can be helpful for several biomedical or environmental applications of magnetic nanoparticles, in which the nanoparticle suspension undergoes a field-induced phase condensation.

I. Introduction

Magnetic nanoparticles are gaining a growing interest as a versatile nanomaterial for biomedical and environmental applications [START_REF] Tartaj | Synthesis, properties and biomedical applications of magnetic nanoparticles[END_REF][START_REF] Pratt | Environmental applications of magnetic nanoparticles[END_REF]. In many of these applications, magnetic nanoparticles are subjected to aggregation induced by the applied external magnetic field due to dipole-dipole interactions. Such aggregation is often treated as a condensation phase transition or phase separation where initially homogeneous colloid separates into a concentrated phase (usually present in form of needle-like aggregates) and a dilute phase representing a homogeneous suspending medium for aggregates [START_REF] Socoliuc | Magnetically induced phase condensation in an aqueous dispersion of magnetic nanogels[END_REF]. The typical aggregation timescale has often to be compared to the characteristic timescales of the given application. Kinetics of field-induced aggregation becomes therefore very important for a successful realization of the application. Since various magnetic field configurations are used in different applications (permanent, oscillatory, rotating magnetic fields), it is very useful to find a unified description of the kinetics of nanoparticle aggregation independent of the field configuration and applicable to both permanent and alternating magnetic fields. To the best of our knowledge, such generalized theory of kinetics of field-induced aggregation (or phase condensation) does not exist. Theoretical [START_REF] Tsebers | Thermodynamic stability of magnetofluids[END_REF][START_REF] Sano | Theory of Agglomeration of Ferromagnetic Particles in Magnetic Fluids[END_REF][6] and computer [START_REF] Satoh | Three Dimensional Monte Carlo Siulations of Thick Chai-like Clusters Composed of Ferromagnetic Fine Particles[END_REF][START_REF] Hynninen | Phase Diagram of Dipolar Hard and Soft Spheres: Manipulation of Colloidal Crystal Structures by an External Field[END_REF][START_REF] Hynninen | Phase behavior of dipolar hard and soft spheres[END_REF] models of the field-induced phase condensation and its kinetics have been mainly developed for the permanent magnetic field. However, the kinetics of aggregation has never been treated for the practically important case of rotating magnetic field, which makes the aggregates rotate [START_REF] Sandre | Assembly of microscopic highly magnetic droplets: Magnetic alignment versus viscous drag[END_REF][START_REF] Stikuts | Spontaneous order in ensembles of rotating magnetic droplets[END_REF]; this is expected to change dramatically the law of the aggregate growth.

In particular, in a recently proposed novel treatment of brain strokes and thrombosis, the magnetic nanoparticles are injected to the blocked vessel to activate recirculation of the blood flow near the blood clots. Its goal is to enhance the drug transport to the clot allowing its faster dissolution [START_REF] Creighton | Magnetic-based systems for treating occluded vessels[END_REF]. This recirculation is achieved by the application of rotating magnetic fields, which first induce aggregation of nanoparticles to micron-sized needle-like aggregates. Then the rotational motion of these aggregates induces recirculation flows in the blood vessel. When the applied field is switched off, the nanoparticle aggregates are destroyed by thermal motion and convective fluxes, and biocompatible nanoparticles are slowly eliminated from the human body by liver macrophages [START_REF] Ali | Synthesis, characterization, applications, and challenges of iron oxide nanoparticles[END_REF]. The same mechanism is involved in the elimination of magnetic nanoparticles applied as contrast agents in magnetic resonance imaging [START_REF] Ito | Medical application of functionalized magnetic nanoparticles[END_REF]. The timescale of field-induced aggregation should be much smaller than the typical time of the medical intervention. In this context, the kinetics of nanoparticle aggregation under rotating magnetic fields plays a decisive role in the success of brain stroke treatment.

This paper is focused on the development of the unified mathematical modelling of the kinetics of field-induced phase condensation in ferrfoluids under a spatially homogeneous permanent or alternating magnetic field of an arbitrary configuration (harmonic or non-harmonic oscillatory, rotating with circular or non-circular polarization, etc.). This description is given under two following limits: (a) very fast initial nucleation stage allowing the appearance of all the nuclei at very small times; (b) the aggregation timescale is large enough with respect to the characteristic period of the magnetic field. Under these limits, the unified modelling of the kinetics becomes possible since, in most of the relevant cases, the aggregate growth rate 𝑑𝑉/𝑑𝑡 can be presented as a product of some function 𝐺(𝑉) of the aggregate volume by another function Δ(𝑡) of time. The precise details of the aggregation mechanism are contained in the function 𝐺(𝑉), which depends on the magnetic field configuration, while the second function Δ(𝑡) stands for the suspension supersaturation. For definiteness, two distinct cases of the permanent (with 𝐺(𝑉) ∝ 𝑉 3/7 ) and rotating (with 𝐺(𝑉) ∝ 𝑉 2/3 ) magnetic fields will be considered. The distribution function of the aggregate volume, as well as evolution of the average volume with time, will be calculated for these two particular cases as for a general case of an arbitrary function 𝐺(𝑉).

II. Liouville equation solution for arbitrary function 𝐺(𝑉)

In this section, we propose a mathematical model of the evolution of the dense aggregates, consisting of many magnetic nanoparticles, under a homogeneous either permanent or homogeneous alternating magnetic field. The intensity of these magnetic fields may be an arbitrary function of time, under the restriction that the period of the lowest harmonics of the magnetic field is much smaller than the aggregation timescale. In many experiments, straight elongated aggregates with a shape assimilated to prolate ellipsoids have been reported under permanent [START_REF] Socoliuc | Magnetically induced phase condensation in an aqueous dispersion of magnetic nanogels[END_REF], rotating [START_REF] Sandre | Assembly of microscopic highly magnetic droplets: Magnetic alignment versus viscous drag[END_REF] or oscillating [START_REF] Socoliuc | Magnetically induced phase condensation in an aqueous dispersion of magnetic nanogels[END_REF] magnetic fields, at least in a low-frequency range below a critical frequency at which the straight elongated shape becomes unstable [START_REF] Bacri | Behavior of a magnetic fluid microdrop in a rotating magnetic field[END_REF]. Thus, we consider dense elongated aggregates that, depending on the magnetic field configuration, can either be stationary or perform an angular motion (rotation as an example) under the action of this field. The aggregates are illustrated in Fig. 1. All real magnetic colloids (ferrofluids) are polydisperse systems, very often with a wide distribution over particles sizes. As a rule they include small particles, not able to aggregate under the magnetic forces, as well as relatively large particles with the energy of magnetic interaction large enough to provoke their aggregation in external magnetic field. Here we suppose that the aggregates appear and grow due to absorbtion of the large particles.

To prevent irreversible coagulation of the particles under the colloidal attractive forces, they are covered with special surfactant or ionic layers. However, these layers do not always completely screen the colloidal attractive forces and the particles, at the stage of the ferrofluid synthesis, form some so-called primary agglomerates (sometimes called flocculi), consisting of several or several tens of nanoparticles. That is why, the onset of the ferrofluid aggregation is determined by many factorsby the size distribution of the particles, the presence of the primary agglomerates, etc. The biggest particles or the biggest primary agglomerates can serve as centres of the heterogeneous nucleation of the main part of the particles. The rate of the primary nuclei appearance can be very high; the characteristic time of the nucleation is usually below the ability of detection of the process [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF]. Therefore, the most probable scenario of the kinetics of the phase separation is the following. Once the external magnetic field is applied, for practically negligible time, the nuclei of the dense phase appear on some condensation centres. After that, due to the magnetic interparticle interaction, the nuclei absorb the free particles from the main part of the ferrofluid and grow; supersaturation of the system decreases. These growing nuclei will be hereinafter called aggregates. At the final stage of the phase condensation, the aggregates amalgamate because of magnetic interaction between them.

In this work, we will consider the main second part of the phase condensation, when the aggregates grow due to capture of free nanoparticles from the surrounding dilute phase. New nuclei no longer appear during this stage, while the coalescence of aggregates under attractive dipolar interaction is still negligible [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF], such that the number of aggregates N per unit volume will be considered to be constant with time. Because of the size polydispersity of the nucleation centres, the aggregates are expected to have unequal size. It is convenient to describe the aggregate size distribution at a given moment of time t by a distribution function (probability density) 𝐹(𝑉, 𝑡) over the aggregate volumes V. After the very fast nucleation stage, the aggregate size usually achieve a few microns. The effect of thermal fluctuations on the rate of volume growth is expected to be negligible for such supercritical aggregates. That is why, as commonly admitted in the theory of phase transitions, the distribution function 𝐹(𝑉, 𝑡) of the aggregate size obeys the classical Liouville equation free of any diffusive terms [START_REF] Landau | Statistical Physics[END_REF]:

𝜕𝐹 𝜕𝑡 + 𝜕 𝜕𝑉 (𝑉 ̇𝐹) = 0, (1)
with the normalization condition

∫ 𝐹(𝑉, 𝑡)𝑑𝑉 = 1 ∞ 0
(2) and the initial condition 𝐹(𝑉, 0) = 𝐹 0 (𝑉)

relevant for the considered case of constant number density N of aggregates with the initial size distribution 𝐹 0 (𝑉) in the issue of very short heterogeneous nucleation stage. In the present work, this initial distribution is supposed to be known, while in practice it can either be measured in experiments or modelled through a detailed consideration of the heterogeneous nucleation stage.

The aggregate growth rate 𝑑𝑉/𝑑𝑡 depends on both the aggregate volume V and time t. However, in most physically relevant cases, separation of variables takes place and the function 𝑑𝑉/𝑑𝑡 takes the following general form:

𝑉 ̇(𝑉, 𝑡) = 𝐺(𝑉)∆(𝑡), 𝑉 ̇≡ 𝑑𝑉 𝑑𝑡 (4)
where Δ(𝑡) = 𝜑(𝑡) -𝜑 0 is the ferrofluid supersaturation, 𝜑(𝑡) is the mean volume concentration of the free particles in the inter-aggregate space (in the dilute phase) and 𝜑 0 is the concentration of the nanoparticles in the dilute phase when the equilibrium between the dilute and concentrated (inside the aggregate) phases is achieved. The function 𝐺(𝑉) depends on the intensity, frequency and shape of the temporal dependency of the magnetic field. For example, in the case of the permanent magnetic field, 𝐺(𝑉) ∝ 𝑉 3/7 [START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF], while in the case of the circularly polarized rotating field, 𝐺(𝑉) ∝ 𝑉 2/3 at low frequency and can show an intermediate behaviour between 𝑉 3/7 and 𝑉 2/3 at higher frequencies, as shown in the Appendix.

The size distribution must also respect the volume conservation condition that can be written in terms of the supersaturation as follows [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF][START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF]:

∆(𝑡) = ∆ 0 -𝑁𝜑 𝑖 〈𝑉〉, (5a) 〈𝑉〉 = ∫ 𝐹(𝑉, 𝑡)𝑉𝑑𝑉 ∞ 0 , ( 5b
)
where 𝜑 𝑖 is the volume fraction of nanoparticles inside the aggregates and 〈𝑉〉 is the average aggregate volume at a given time t.

Equation (5a) allows us to determine the maximal average volume of the aggregates at the end of the aggregate growth stage when the supersaturation tends to zero: 〈𝑉〉 𝑚𝑎𝑥 = ∆ 0 /(𝑁𝜑 𝑖 ). It is thus reasonable to scale the aggregate volume by the maximal average volume and introduce the dimensionless aggregate volume as 𝑣 = 𝑉/〈𝑉〉 𝑚𝑎𝑥 . Furthermore, the function 𝐺(𝑉) has a dimension of the aggregate growth rate (m 3 /s) and can be presented as 𝐺(𝑉) ≡ 𝜅〈𝑉〉 𝑚𝑎𝑥 𝑔(𝑣), with  being some kinetic constant having a dimension of inverse time (s -1 ), and 𝑔(𝑣)a function of the dimensionless volume v that can in principle depend on 〈𝑉〉 𝑚𝑎𝑥 . With this in mind, let us introduce the following dimensionless quantities: 𝑓 = 〈𝑉〉 𝑚𝑎𝑥 𝐹the dimensionless distribution function; 𝜏 = Δ 0 𝜅𝑡the dimensionless time and 𝜔 = Δ/Δ 0the reduced supersaturation. The initial value problem (1) -( 5) for the aggregate size distribution can now be written in the following dimensionless form:

𝜕𝑓 𝜕𝜏 + 𝜔(𝜏) 𝜕 𝜕𝑣 (𝑔(𝑣)𝑓) = 0, (6a)
∫ 𝑓(𝑣, 𝜏)𝑑𝑣 = 1 ∞ 0 , (6b) 𝑓(𝑣, 0) = 𝑓 0 (𝑣), (6c) 𝜔(𝜏) = 1 -〈𝑣〉 = 1 -∫ 𝑓(𝑣, 𝜏)𝑣𝑑𝑣 ∞ 0 . ( 6d
)
Introducing new functions:

𝑈(𝑣, 𝜏) = 𝑔(𝑣)𝑓(𝑣, 𝜏), (7a)
𝜃(𝜏) = ∫ 𝜔(𝜏′) 𝜏 0 𝑑𝜏′, (7b)
𝑦(𝑣) = ∫ 𝑑𝑣′ 𝑔(𝑣′) 𝑣 0 , ( 7c
)
we come to the problem with the initial condition as follows:

𝜕𝑈 𝜕𝛩 + 𝜕𝑈 𝜕𝑦 = 0, 𝑈(𝑦, 0) = 𝑔(𝑣(𝑦))𝑓 0 (𝑣(𝑦)), (8)
where 𝑣(𝑦) is the inverse function of 𝑦(𝑣) defined in (7c).

By using the methods of characteristics, one can get the solution of the initial value problem (8) as

𝑈(𝑦, 𝜃) = 𝑔(𝑣(𝑦 -𝜃))𝑓 0 (𝑣(𝑦 -𝜃))𝐻(𝑦 -𝜃), (9)
where 𝐻(𝑥) is the Heaviside step function, introduced to avoid negative values of the argument (dimensionless volume) of the initial size distribution 𝑓 0 (𝑣). Let us denote 𝑣(𝑦 -𝜃) = 𝑣 * (𝑦, 𝜃).

Since 𝑣(𝑦) is the inverse function of 𝑦(𝑣) than 𝑣(𝑦 -𝜃) is expected to be an inverse function of 𝑦(𝑉) -𝜃(𝜏). This implies 𝑦(𝑣 * ) = 𝑦(𝑣) -𝜃(𝜏). Using the definition of y [Eq. (7c)], we get:

𝜃 = ∫ 𝑑𝑣′ 𝑔(𝑣 ′ ) 𝑣 𝑣 * → 𝑣 * = function(𝜃, 𝑣). (10)
With this new notation, the distribution function takes the following form in coordinates (v, ):

𝑓(𝑣, 𝜃) = 1 𝑔(𝑣) 𝑈(𝑣, 𝜃) = 1 𝑔(𝑣)
𝑔(𝑣 * (𝑣, 𝜃))𝑓 0 (𝑣 * (𝑣, 𝜃))𝐻(𝑣 * (𝑣, 𝜃)), [START_REF] Stikuts | Spontaneous order in ensembles of rotating magnetic droplets[END_REF] where 𝑣 * (𝑣, 𝜃) is the solution of Eq. [START_REF] Sandre | Assembly of microscopic highly magnetic droplets: Magnetic alignment versus viscous drag[END_REF]. From the solution [START_REF] Stikuts | Spontaneous order in ensembles of rotating magnetic droplets[END_REF], it follows that at a given equivalent time 𝜃, the dimensionless volume 𝑣 has a minimal non-zero value 𝑣 𝑚𝑖𝑛 defined by the condition 𝑣 * (𝑣 𝑚𝑖𝑛 , 𝜃) = 0. Substituting this expression to Eq. ( 10), we get the following transcendental equation allowing finding the minimal volume as a function of 𝜃:

𝜃 = ∫ 𝑑𝑣′ 𝑔(𝑣 ′ ) 𝑣 𝑚𝑖𝑛 0 → 𝑣 𝑚𝑖𝑛 = function(𝜃). (12)
Before finding (), let us first find the average aggregate volume as a function of the unknown :

〈𝑣〉 = ∫ 𝑓(𝑣, 𝜃)𝑣𝑑𝑣 ∞ 0 = ∫ 𝑔(𝑣 * (𝑣,𝜃))𝑓 0 (𝑣 * (𝑣,𝜃)) 𝑔(𝑣) 𝑣𝑑𝑣 ∞ 𝑣 min (𝜃) = function(𝜃). ( 13
)
Then we use the volume conservation condition (6d), 𝜔(𝜏) = 𝑑𝜃/𝑑𝜏 = 1 -〈𝑣〉, whose solution is trivial:

𝜏(𝜃) = ∫ 𝑑𝜃′ 1-〈𝑣〉(𝜃′) 𝜃 0 . ( 14
)
Thus, the average volume and the distribution function can be found in the function of time in parametric form by variation of the parameter . The main steps of such parametric calculation are summarized as follows:

1) first, we find the minimal volume 𝑣 𝑚𝑖𝑛 (𝜃) by solving the transcendental equation ( 12); 2) then, we find 𝑣 * (𝑣 𝑚𝑖𝑛 , 𝜃) by solving the transcendental equation [START_REF] Sandre | Assembly of microscopic highly magnetic droplets: Magnetic alignment versus viscous drag[END_REF];

3) then we find, the average volume 〈𝑣〉(𝜃) and time 𝜏(𝜃) by Eqs. ( 13), ( 14); 4) finally, we plot the distribution function 𝑓(𝑣, 𝜃) as a function of 𝑣 at different moments of time 𝜏(𝜃).

This procedure can be applied to an arbitrary function g(v) depending on the magnetic field configuration. In the next two sections, we focus our attention on a specific form of the function 𝑔(𝑣) = 𝑣 2/3 , relevant for rotating magnetic field and allowing more concise analytical solution for 𝑓(𝑣, 𝜃) and 𝜏(𝜃) (Sec. III) and finding the average volume 〈𝑣〉 by the method of moments without solving the Liouville equation (Sec. IV).

III. Resolution of the Liouville equation for the particular case g(v)=v 2/3

The particular case 𝑔(𝑣) = 𝑣 2/3 corresponds to the circularly polarized rotating magnetic field (see Appendix) and is important for the application of magnetic nanoparticles in magnetically assisted brain stroke treatment, as pointed out in Introduction. This case allows rather simple expressions for all unknown quantities, such as 𝑣 𝑚𝑖𝑛 , 𝑣 * , 𝜃 and 𝑓(𝑣, 𝜃). Straightforward application of the calculation steps described at the end of Sec. II gives the following results:

𝑣 𝑚𝑖𝑛 = ( 𝜃 3 ) 3 , ( 15a
)
𝑣 * = (𝑣 1/3 - 𝜃 3 ) 3 , ( 15b
) 〈𝑣〉 = ∫ 𝑣 * 2/3 𝑓 0 (𝑣 * )𝑣 1/3 𝑑𝑣 ∞ 𝑣 min (𝜃) = 3 ∫ 𝜉 2 𝑓 0 (𝜉 3 ) (𝜉 + 𝜃 3 ) 3 𝑑𝜉 ∞ 0 = 𝐴 + 𝐵𝜃 + 𝐶𝜃 2 + 𝐷𝜃 3 (15c) 𝜏(𝜃) = ∫ 𝑑𝜃′ 1-[𝐴+𝐵𝜃′+𝐶𝜃′ 2 +𝐷𝜃′ 3 ] 𝜃 0 , ( 15d
)
𝑓(𝑣, 𝜃) = 𝑣 -2/3 (𝑣 1/3 - 𝜃 3 ) 2 𝑓 0 ((𝑣 1/3 - 𝜃 3 ) 3 ) 𝐻 (𝑣 1/3 - 𝜃 3 ), (15e)
𝐴 = 3 ∫ 𝜉 5 𝑓 0 (𝜉 3 )𝑑𝜉 ∞ 0

; 𝐵 = 3 ∫ 𝜉 4 𝑓 0 (𝜉 3 )𝑑𝜉 ∞ 0

; 𝐶 = ∫ 𝜉 3 𝑓 0 (𝜉 3 )𝑑𝜉 ∞ 0

; 𝐷 = 1 9 ∫ 𝜉 2 𝑓 0 (𝜉 3 )𝑑𝜉 ∞ 0 , (15f)
where we have made a change of variables 𝜉 ≡ 𝑣 * 1/3 = 𝑣 1/3 -𝜃/3 when calculating the integral in (15c). The integral in (15d) can be evaluated numerically, while the numerical values of the coefficients A, B, C and D in (15f) will depend on the shape of the initial size distribution 𝑓 0 (𝑣). Numerical examples will be analyzed in Sec. V.

The analysis shows that in the particular case of the Dirac delta initial distribution of the aggregate size 𝑓 0 (𝑣) = 𝛿(𝑣 -𝑣 0 ), i.e. all aggregates of the same initial volume 𝑣 0 , the size distribution remains monodisperse at any moment of time 𝜏 or 𝜃:

𝑓(𝑣, 𝜃) = 𝛿(𝑣 -𝑣 1 ), (16)
with the aggregate volume 𝑣 1 (the same for all aggregates) and the dimensionless time 𝜏 given by

𝑣 1 = (𝑣 0 1/3 + 𝜃 3 ) 3 ; 𝜏(𝜃) = ∫ 𝑑𝜃′ 1-(𝑣 0 1/3 +𝜃 ′ /3) 3 𝜃 0 , (17)
or, eliminating  from ( 17), we get the following explicit expression for the dimensionless time as function of the aggregate volume in this particular case of monodisperse size distribution:

𝜏(𝑣 1 ) = ∫ 𝑑𝑣 𝑣 2 3(1-𝑣) 𝑣 1 𝑣 0 = 𝜓(𝑣 1 ) -𝜓(𝑣 0 ); 𝜓(𝑥) = 3 1/2 atan 1+2𝑥 1/3 3 1/2 + 1 2 ln 1+𝑥 1/3 +𝑥 2/3 (1-𝑥 1/3 ) 2 (18)
The case of monodisperse aggregate size will be compared to the case of the polydisperse size distribution (with some initial distribution 𝑓 0 (𝑣) ≠ 𝛿(𝑣 -𝑣 0 )) in Sec. V.

IV.

Method of moments for determination of the average volume in particular case g(v)=v 2/3 .

In this section, we show that the special case g(v)=v 2/3 admits a complete decoupling of the moments

〈𝑣 𝑛 〉 = ∫ 𝑣 𝑛 𝑓(𝑣)𝑑𝑣 ∞ 0 , 𝑛 = 1 3 , 2 3
, 1 of the distribution function 𝑓(𝑣, 𝜏) without any closure approximation. Thus, we do not need to solve the Liouville equation (6a) for the distribution function to find the average volume 〈𝑣〉. Such decoupling of moments is of particular interest for statistical physics and can generally provide faster computations as compared to the general method provided in Sec. III.

First, in the considered case, g(v)=v 2/3 , the Liouville equation (6a) written in dimensionless variables takes the following form:

𝜕𝑓 𝜕𝜏 + 𝜔(𝜏) 𝜕 𝜕𝑣 (𝑣 2/3 𝑓) = 0 (19)
Then, multiplying both parts of Eq. ( 19) by 𝑣 and integrating over 𝑣, one gets:

𝑑<𝑣> 𝑑𝜏 + 𝜔 ∫ 𝑣 𝜕 𝜕𝑣 (𝑣 2/3 𝑓) ∞ 0 𝑑𝑣 = 0.
Integrating by parts, taking into account that 𝑓 = 0 at 𝑣 = 0 , 𝑣 → ∞, we come to the equation:

𝑑〈𝑣〉 𝑑𝜏 = 𝜔 < 𝑣 2/3 >, (20)
Similarly, multiplying Eq. ( 19) by 𝑣 2/3 and by 𝑣 1/3 , and integrating over 𝑣, we get respectively:

𝑑〈𝑣 2/3 〉 𝑑𝜏 = 2 3 𝜔 < 𝑣 1/3 >, (21)
𝑑<𝑣 1/3 > 𝑑𝜏 = 1 3 𝜔 (22)
Along with the volume conservation equation (6d), implying 𝜔(𝜏) = 1 -〈𝑣〉, equations ( 20) -( 22) present a closed system of differential equations with respect to the moments < 𝑣 > , < 𝑣 2/3 >, < 𝑣 1/3 >. Eliminating the moments < 𝑣 2/3 > and < 𝑣 >, we arrive at a single differential equation as follows for the moment < 𝑣 1/3 >:

𝑑<𝑣 1/3 > 𝑑𝜏 = 1 3 [1 -(< 𝑣 1/3 > 3 + 3𝐶 1 < 𝑣 1/3 > +𝐶 2 )], (23)
where the integration constants C1 and C2 are found from the initial size distribution 𝑓 0 (𝑣) as follows:

𝐶 1 = 〈𝑣 2/3 〉 0 -(〈𝑣 1/3 〉 0 ) 2 , 𝐶 2 = 〈𝑣〉 0 + 2(〈𝑣 1/3 〉 0 ) 3 -3〈𝑣 2/3 〉 0 〈𝑣 1/3 〉 0 , (24)
with

〈𝑣 𝑛 〉 0 = ∫ 𝑣 𝑛 𝑓 0 (𝑣)𝑑𝑣 ∞ 0 , 𝑛 = 1 3 , 2 3 , 1.
The differential equation ( 23) allows expressing the moment < 𝑣 1/3 > as an inverse function of the dimensionless time 𝜏, while the system (20)-( 22) allows one to relate the average volume < 𝑣 > to < 𝑣 1/3 >. Thus, the time dependence of the average volume can be written in the following parametric form using < 𝑣 1/3 >≡ 𝑥 as a parameter:

< 𝑣 > (𝑥) = 𝑥 3 + 3𝐶 1 𝑥 + 𝐶 2 , (25a)
𝜏(𝑥) = 3 ∫ 𝑑𝑥′ 1-(𝑥′ 3 +3𝐶 1 𝑥′+𝐶 2 ) 𝑥 𝑥 0 , 𝑥 0 = 〈𝑣 1/3 〉 0 . (25b)
It can be easily checked, that in particular case of monodisperse initial size distribution, 𝑓 0 (𝑣) = 𝛿(𝑣 -𝑣 0 ), the constants 𝐶 1 = 𝐶 2 = 0, the size distribution remains monodisperse with time and given by Eq. ( 16), while the aggregate volume 𝑣 1 (the same for all aggregates) is given by the inverse function of time provided in Eq. [START_REF] Landau | Statistical Physics[END_REF].

V. Results

For the sake of the definiteness, we apply our models developed in Secs. II-IV to the physically relevant case of initial log-normal distribution of aggregate sizes:

𝑓 0 (𝑣) = 1 √2𝜋𝜎𝑣 𝑒𝑥𝑝 (- (ln( 𝑣 〈𝑣〉 0 )+ 𝜎 2 2 ) 2 2𝜎 2 ) 𝐻(𝑣), (26)
where 〈𝑣〉 0 is the initial dimensionless average aggregate volume and 𝜎 is a parameter characterizing the distribution width.

Let us first analyze the evolution of the dimensionless average aggregate volume 〈𝑣〉 with dimensionless time  for the three following cases:

(a) a permanent magnetic field with g(v)=v 3/7 (as shown in [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF][START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF])the case labelled as "P"; (b) low frequency circularly polarized rotating magnetic field with g(v)=v 2/3 (see Appendix)the case labelled as "R1"; (c) moderate frequency circularly polarized magnetic field with

𝑔(𝑣) = [𝑎𝑣 -2/3 + 𝑏𝑣 -3/7 ] -1 ,
where a and bdimensionless constants (see Appendix)the case labelled as "R2".

All cases are treated using the general methodology developed at the end of Sec. II, while it is checked that for the case R1 (g(v)=v 2/3 ), the calculations using this methodology (applied in Sec. III for this case) and the method of moments (developed in Sec. IV) coincide. The 〈𝑣〉 versus  dependency is plotted in Fig. 2 for the three cases with the initial log-normal size distribution [Eq. ( 26)] characterized by 〈𝑣〉 0 = 0.1 and 𝜎 = 1.5. The case R2 is calculated at a=b=1. As is seen from this figure, the average aggregate volume gradually increases with time from its initial value 〈𝑣〉 0 = 0.1 and asymptotically tends to unity at long times. Recalling that the dimensional volume 𝑉 has been normalized by the maximal average volume, 〈𝑉〉 𝑚𝑎𝑥 = ∆ 0 /(𝑁𝜑 𝑖 ), this means that at long times the average dimensional volume 𝑉 tends to its maximal value when the ferrofluid supersaturation  (or ) tends to zero. We have to bear in mind that the coalescence of aggregates has been neglected at this stage, and in fact, our preliminary experiments show that coalescence is strongly hindered by hydrodynamic repulsion between aggregates in the case of rotating magnetic field, which makes the aggregate turn synchronously with the field. Analyzing the difference between the three plotted curves in Fig. 2, one could presume that the permanent magnetic field (curve "P") provides a faster aggregation than the rotating one, while low-frequency rotating field (curve "R1") ensures faster aggregation than medium-frequency rotating field (curve "R2"). We have to bear in mind however that the kinetic constant  (introduced below Eq. ( 5)) can be essentially different for three considered cases implying different dimensional times 𝑡 = 𝜏/(Δ 0 𝜅) at the same dimensionless time . For example, in a permanent magnetic field, the kinetic constant is related to the inverse characteristic time of Brownian diffusion of a single nanoparticle. On the other hand, in a rotating field, the kinetic constant is expected to be a function of the field frequency while the convective transport of nanoparticles towards the aggregate is expected to be enhanced with increasing frequency leading to faster aggregation at a higher frequency. At this stage, we restrict our analysis to the dimensionless quantities and focus our attention to a general theoretical description of the solution of the kinetic problem. A deeper insight into the physical aspects related to precise dependence of the kinetic constant on the dimensional physical parameters (as magnetic field intensity, frequency, particle size, etc.) will be provided in future in comparison with experiments.

Let us now inspect the effect of the width  of the initial size distribution [Eq. ( 26)] on the average volume. The 〈𝑣〉 versus  dependency is plotted in Fig. 3 for the case R1 (g(v)=v 2/3 ) for 〈𝑣〉 0 = 0.1 and three different values of , with =0 corresponding to the Dirac delta initial distribution admitting the analytical solution Eq. ( 18). As is seen from this figure, the curves are shifted to the right with increasing the initial distribution width ( parameter), indicating that the aggregation gets slower. However, the difference with the monodisperse distribution (Dirac delta with =0) remains really small even at a very wide initial distribution with =2. Such a difference can be on the order of errors related to different model assumptions and it seems reasonable to use simple analytical result [Eq. ( 18)] (derived for monodisperse distribution) for the average aggregate volume even in the case of wide initial size distribution.

Having found a slight effect of the initial distribution width, it is now interesting to inspect what happens with the size distribution with time. The distribution function f(v) of the dimensionless aggregate volume is plotted in Fig. 4 for the R1-case with 〈𝑣〉 0 = 0.1, =1.5 and three different dimensionless times, including the initial moment =0 with a given initial distribution f0(v) [Eq. ( 26)] and the moment of time =8.4 at which the average aggregate volume is equal to 〈𝑣〉 = 0.999 (thus approaches its maximal value 〈𝑣〉 𝑚𝑎𝑥 = 1). It is seen from Fig. 4 that the size distribution displaces along the abscissa axis and spreads over this axis during time although the average volume remains close to the one of the monodisperse distribution. One can, therefore, conclude that the distribution globally keeps its initial shape and nearly equally spreads around its average value roughly given by the monodisperse approximation [Eq. [START_REF] Landau | Statistical Physics[END_REF]].

For the practical biomedical applications, the whole size distribution (and not only the average aggregate volume) could appear to be an important parameter. For example in brain stroke treatments, the largest aggregates can achieve the size of the vessels and the smallest aggregates can be too small for effective manipulation by external fields. It was, therefore, important to learn about relatively large width of the aggregate size distribution (Fig. 4) even in the case when the average aggregate volume was nearly independent of the width of initial distribution (Fig. 3).

VI. Concluding remarks

In this paper, we present a mathematical model of nanoparticle phase condensation (or, equivalently, nanoparticle aggregation) under homogeneous permanent or homogeneous alternating magnetic field with arbitrary time dependence of the magnetic field intensity. The model is derived under basic assumptions of very fast nucleation kinetics but negligible aggregate coalescence (thus, the aggregate number density is constant over time) and for the magnetic field frequency range respecting two following conditions: (a) the aggregate growth is slow enough with respect to the field variation, and (b) the aggregates keep their straight elongated shape. Under these conditions, the aggregate growth rate takes the general form 𝑑𝑉/𝑑𝑡 = 𝐺(𝑉)∆(𝑡), with the function 𝐺(𝑉) depending on the precise configuration of the applied magnetic field. The ensemble of aggregates is described by their size distribution function 𝐹(𝑉, 𝑡), which is found by the solution of the Liouville equation ( 1) at an appropriate initial condition. The problem is solved by the method of characteristics, and the solution is obtained in parametric form for an arbitrary function 𝐺(𝑉), providing a general framework for any type of the applied magnetic field. In its dimensionless form, the solution is expressed through the integrals, whose integrands contain the dimensionless counterparts, 𝑔(𝑣) and 𝑓 0 (𝑣), of the function 𝐺(𝑉) and the initial size distribution 𝐹 0 (𝑉), respectively. In the particular case of low-frequency rotating magnetic field (𝑔(𝑣) = 𝑣 2/3 ), an explicit expression of the dimensionless distribution function 𝑓(𝑣, 𝜃) is obtained [Eq. (15e)], while the dimensionless average aggregate volume 〈𝑣〉 is easily found by the method of moments allowing a complete decoupling of the statistical moments 〈𝑣 𝑛 〉, 𝑛 = Numerical examples are provided for initial log-normal size distribution and the cases of a permanent magnetic field (𝑔(𝑣) = 𝑣 3/7 ), low-frequency rotating field (𝑔(𝑣) = 𝑣 2/3 ), and medium-frequency rotating field (with intermediate behaviour between 𝑣 3/7 and 𝑣 2/3 ). All these cases share the following common features: (a) at long times, the average volume asymptotically tends to its maximal value 〈𝑉〉 𝑚𝑎𝑥 = ∆ 0 /(𝑁𝜑 𝑖 ) (or, 〈𝑣〉 𝑚𝑎𝑥 = 1 in the dimensionless form) when the ferrofluid supersaturation ∆ (or 𝜔) tends to zero; (b) the average volume depends only slightly on the relative width 𝜎 of the initial size distribution, allowing one to evaluate 〈𝑣〉 for any 𝜎 (within the range 0 ≤ 𝜎 ≤ 2) using the explicit expression [Eq. [START_REF] Landau | Statistical Physics[END_REF]] derived for the Dirac delta distribution; (c) despite a negligible dependence of the average volume on 𝜎, the width of the size distribution at any moment 𝜏 > 0 strongly depends on 𝜎; the size distribution shifts along the abscissa axis with time and shows a significant spreading around the average value 〈𝑣〉, however, this spreading seems to be quasi-symmetric with respect to 〈𝑣〉, such that the average volume appears to be slightly affected by the width of the initial distribution. The three considered cases show some difference when the average dimensionless volume 〈𝑣〉 is plotted against the dimensionless time 𝜏, however, at the tendency can change if the analysis is conducted in dimensional quantities because the kinetic constant  (which intervenes into the normalization of the dimensional time t) depends on the physics behind each particular case (permanent or rotating magnetic field). A deeper insight into this problem will be presented in future in conjunction with the comparison with experiments.

From the practical perspective, it was important to learn about relatively large width of the aggregate size distribution (Fig. 4) even in the case when the average aggregate volume was nearly independent of the width of initial distribution (Fig. 3). In fact, the presence of too small or too large aggregates can be undesirable in biomedical or environmental applications of magnetic nanoparticles, for example, in brain stroke treatment. Thus, the mathematical model of aggregation kinetics presented in this work can provide useful feedback to a given application allowing optimization of operating parameters.

Appendix. The rate of the aggregate growth in a rotating field

In this Appendix, we derive a scaling law of the aggregate growth rate 𝑉 ̇ (the increase of its volume by unit time) as a function of the aggregate volume. At this stage, we are not looking for the exact expression for this rate, as long as it does not affect the mathematical structure of the kinetic equation (6a) considered in Sec. II and the complex physics of the aggregate growth only affect the kinetic constant  in the scaling law 𝑑𝑉/𝑑𝑡 = function(𝑉). We consider a dense aggregate, of a length L and diameter da consisting of many ferromagnetic nanoparticles, elongated by the applied rotating field and synchronously spinning with this field. The angular frequency of the rotating field is denoted by . These aggregates are illustrated in Fig. 1. We denote 𝜑 𝑖 = 𝑂(1) the volume fraction of nanoparticles inside the aggregates Kinetics of the aggregate growth is described by the following approximate balance equation:

𝜑 𝑖 𝑉 ̇= 𝐽, (A.1)
where J is the flux of the particles toward the aggregate.

The magnetophoretic flux arising due to magnetic attraction of individual nanoparticles to strongly aggregates has been found to be negligible in comparison to the diffusive flux [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF][START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF], while the typical ratio of the convective to the diffusive flux, at the distances comparable with the aggregate diameter da, is estimated through the Péclet number: 𝑃𝑒 = 𝐿Ω𝑑 𝑎 /𝐷, where 𝐷 is the Brownian diffusivity of a single nanoparticle. Estimates show that for individual nanoparticles of typical size 20 nm and the typical aggregates with a diameter 𝑑 𝑎 ~1 -10𝜇𝑚 and length 𝐿~100𝜇𝑚 suspended in water under the action of the field rotating with the frequency Ω~10𝑠 -1 , the strong inequality Pe>>1 holds. At such condition, the surface density of the diffusive flux on the aggregate surface can be evaluated using the boundary layer approach [START_REF] Levich | Physicochemical Hydrodynamics[END_REF]: 𝑗 𝐷 ~𝐷Δ/𝛿; 𝛿~𝑑 𝑎 𝑃𝑒 -1/3 . (A.2) Here Δ is the ferrofluid supersaturation (see Sec. II) and 𝛿 is the thickness of the diffusion boundary layer near the aggregate surface. According to the concept of the boundary layer approach, outside this layer, the convective flux dominates over the diffusion one; while the diffusion flux dominates inside the layer. Integrating the flux density [Eq. (A.2)] over the aggregate surface, one gets the following scaling for the particle flux (and consequently aggregate growth rate): 𝑉 ̇~𝐽𝐷𝐿𝑃𝑒 1/3 Δ, (A.

3) The aggregate length and diameter are related to its volume by the following scaling [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF][START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF]: 𝐿 ∝ 𝑉 3/7 and 𝑑 𝑎 ∝ 𝑉 2/7 (A.4)

Combining Eqs. (A.3), (A.4), with the definition of the Péclet number, we arrive at the following scaling relationship for the aggregate growth rate: 𝑉 ̇∝ 𝑉 2 3 Δ(𝑡), (A.5) where a dimensional multiplier before 𝑉 2/3 Δ depends on the field frequency  and the nanoparticle diffusivity D; however, derivation of the exact expression for this multiplier is out of the scope of the present work, as long as it does not affect the mathematical structure of the kinetic equation.

Notice that in reality, the relationship 𝑑𝑉/𝑑𝑡 = function(𝑉) can appear to be more complicated. In fact, the aggregate rotates in some volume having a disk-like shape and after one spin the aggregate can absorb most of the particles situated inside this disk-like cavity. In such situation, the free nanoparticles will diffuse from outside towards the cavity and the total flux will be affected by both convective transport of particles inside the cavity towards the aggregate and purely diffusive transport of particles towards the cavity. From obvious electric analogy, one can deduce that the total flux follows the rule of the series circuit with the inverse of the equivalent conductivity (the proportionality factor between J and ) being the sum of inverse conductivities of purely convective [Eq. (A.5)] and purely diffusive (𝐽𝑉 3/7 Δsee [START_REF] Ezzaier | Two-stage kinetics of field-induced aggregation of medium-sized magnetic nanoparticles[END_REF][START_REF] Yu | Kinetics of a magnetic fluid phase separation induced by an external magnetic field[END_REF]) transports. In such circumstances we get 𝑑𝑉/𝑑𝑡 ∝ (𝛼𝑉 -2 3 + 𝛽𝑉 - Notice that equation (A.6) is expected to apply at intermediate field frequencies, which remain small enough such that straight elongated shape of the aggregate is still stable, while Eq. (A.5) is expected to apply for relatively low frequencies when, after one aggregate spin, the nanoparticles have enough time to fill the cavity and compensate the "loss" of the "cavity particles" absorbed by the aggregate. The frequency has to be however high enough such that the aggregation timescale is always much slower than the rotation period and the high Péclet number limit is still sasisfied.

Fig. 1 .

 1 Fig.1. Sketch of the aggregates, which can either be stationary or perform an angular motion depending on the applied magnetic field. Dots around the aggregates stand for individual magnetic particles.

Fig. 2 .

 2 Fig. 2. Dimensionless average aggregate volume 〈𝑣〉 as function of dimensionless time  for three different functions g(v) and initial log-normal size distribution with 〈𝑣〉 0 = 0.1 and 𝜎 = 1.5. The case R2 is calculated for a=b=1.

Fig. 3 .

 3 Fig. 3. Dimensionless average aggregate volume 〈𝑣〉 as function of dimensionless time  for the R1-case and initial log-normal size distribution with 〈𝑣〉 0 = 0.1 and three different values 𝜎 of the distribution width.

Fig. 4 .

 4 Fig. 4. Aggregate size distribution for the R1-case with 〈𝑣〉 0 = 0.1, =1.5 and at the dimensionless times , covering the whole range of the average volume 〈𝑣〉 0 ≤ 〈𝑣〉 ≾ 1.

3 , 1 ,

 31 without the necessity to solve the Liouville equation.

  the dimensional constants  and  depending on exact expressions for the diffusive and convective fluxes. The term in parentheses in Eq. (A.6) is nothing but the function 𝐺(𝑉), intervening into the aggregate growth rate [Eq. (4)] with its dimensionless counterpart taking the form 𝑔(𝑣) = [𝑎𝑣 -2/3 + 𝑏𝑣 -3/7 ] -1 , where a and bdimensionless constants proportional to  and , respectively.

Acknowledgements

PK acknowledges the French 'Agence Nationale de la Recherche', Project Future Investments UCA JEDI, No. ANR-15-IDEX-01 (projects ImmunoMag and MagFilter) and the private company Axlepios Biomedicals for financial support, JQC acknowledges the financial support of UCA JEDI and Axlepios Biomedicals through the PhD fellowship. AZ thanks the Russian Science Foundation, project 20-12-00031, for the financial support.

Authors contribution.

PKthe main idea of this work, writing and revising the manuscript, obtaining the funding; MRM participation in the development of the general framework (Sec. II); JQCparticipation in the modelling of the special case (Sec. III and IV); GVDfunding of the studies and practical recommendations for given biomedical applications; AZdevelopment of the main part of the mathematical model.

Conflict of interests

Authors declare no potential conflict of interests ORCID Andrey Zubarev 0000-0001-5826-9852

Pavel Kuzhir 0000-0001-7089-6197


Ahmed Rida Sekkat
email: ahmed-rida.sekkat@univ-rouen.fr

Yohan Dupuis
email: yohan.dupuis@esigelec.fr

Pascal Vasseur
email: pascal.vasseur@univ-rouen.fr

Paul Honeine
email: paul.honeine@univ-rouen.fr

The OmniScape Dataset

Despite the utility and benefits of omnidirectional images in robotics and automotive applications, there are no datasets of omnidirectional images available with semantic segmentation, depth map, and dynamic properties. This is due to the time cost and human effort required to annotate ground truth images. This paper presents a framework for generating omnidirectional images using images that are acquired from a virtual environment. For this purpose, we demonstrate the relevance of the proposed framework on two well-known simulators: CARLA Simulator, which is an open-source simulator for autonomous driving research, and Grand Theft Auto V (GTA V), which is a very high quality video game. We explain in details the generated OmniScape dataset, which includes stereo fisheye and catadioptric images acquired from the two front sides of a motorcycle, including semantic segmentation, depth map, intrinsic parameters of the cameras and the dynamic parameters of the motorcycle. It is worth noting that the case of two-wheeled vehicles is more challenging than cars due to the specific dynamic of these vehicles.

I. INTRODUCTION

Perceiving and understanding the environment is an essential task for a mobile robot or an autonomous vehicle. One of the main issues for the development of these vehicles is the existence of datasets. Among the datasets of pinhole camera images dedicated to the development and study of autonomous vehicles, mention may be made of KITTI [START_REF] Fritsch | A new performance measure and evaluation benchmark for road detection algorithms[END_REF], Cityscape [START_REF] Cordts | The cityscapes dataset for semantic urban scene understanding[END_REF], Berkeley DeepDrive [START_REF] Yu | BDD100K: A diverse driving video database with scalable annotation tooling[END_REF], CamVid [START_REF] Brostow | Semantic object classes in video: A high-definition ground truth database[END_REF] and Mapillary Vistas Dataset [START_REF] Neuhold | The mapillary vistas dataset for semantic understanding of street scenes[END_REF]. Omnidirectional cameras can perceive the surrounding environment with a field of view that can reach 360°. For this reason, they are increasingly used in the field of intelligent vehicles, including fisheye cameras due to their compactness and inexpensive design. Several datasets contain fisheye images, such as CVRG [START_REF] Baris | Classification and tracking of traffic scene objects with hybrid camera systems[END_REF], LMS [START_REF] Eichenseer | A data set providing synthetic and realworld fisheye video sequences[END_REF], LaFiDa [START_REF] Urban | Lafida -a laserscanner multi-fisheye camera dataset[END_REF], SVMIS [START_REF] Caron | Spherical Visual Gyroscope for Autonomous Robots using the Mixture of Photometric Potentials[END_REF], "Go Stanford" [START_REF] Hirose | Gonet: A semi-supervised deep learning approach for traversability estimation[END_REF], GM-ATCI [START_REF] Levi | Tracking and motion cues for rear-view pedestrian detection[END_REF], and RTH Zurich multi-FoV synthetic datasets [START_REF] Zhang | Benefit of large field-of-view cameras for visual odometry[END_REF]. However, it is noted that there is a lack of road scenes omnidirectional images datasets embedded in a vehicle dedicated for computer vision applications. Recent work on semantic segmentation of fisheye images of road scenes had been performed on perspective images to which a distortion simulating the fisheye effect is applied [START_REF] Sweeney | Cnn-based fisheye image real-time semantic segmentation[END_REF], [START_REF] Deng | Cnn based semantic segmentation for urban traffic scenes using fisheye camera[END_REF], [START_REF] Deng | Restricted deformable convolution based road scene semantic segmentation using surround view cameras[END_REF]. Such deformation induces artefacts in the resulting images. There is a growing need to generate more reliable datasets of omnidirectional images, without the need to go through simple image rectification. Much recent work, especially in deep learning applied on spherical images, have been confirming the need of omnidirectional image databases [START_REF] Yin | Fisheyerecnet: A multi-context collaborative deep network for fisheye image rectification[END_REF], [START_REF] Su | Learning spherical convolution for fast features from 360°imagery[END_REF], [START_REF] Perraudin | Deepsphere: Efficient spherical convolutional neural network with healpix sampling for cosmological applications[END_REF], [START_REF] Coors | Spherenet: Learning spherical representations for detection and classification in omnidirectional images[END_REF], [START_REF] Boomsma | Spherical convolutions and their application in molecular modelling[END_REF], [START_REF] Esteves | Learning so(3) equivariant representations with spherical cnns[END_REF], [START_REF] Kondor | Clebsch-gordan nets: a fully fourier space spherical convolutional neural network[END_REF], [START_REF] Zhao | Distortionaware cnns for spherical images[END_REF].

In this paper, we propose a framework that can be applied to any simulator or virtual environment to generate omnidirectional images. We present the data acquired from a simulator and describe its use to generate omnidirectional images. We present in detail two well-known types of omnidirectional images, fisheye and catadioptric images. The models can be computed from a calibrated camera, using a large class of models proposed by Geyer and Daniilidis [START_REF] Geyer | A unifying theory for central panoramic systems and practical implications[END_REF], Barreto and Araujo [START_REF] Barreto | Issues on the geometry of central catadioptric image formation[END_REF], Mei and Rives [START_REF] Mei | Single view point omnidirectional camera calibration from planar grids[END_REF] or Scaramuzza et al. [START_REF] Scaramuzza | A toolbox for easily calibrating omnidirectional cameras[END_REF].

While the proposed framework is generic, we demonstrate its relevance in two famous simulators: CARLA Simulator and Grand Theft Auto V (GTA V). CARLA is an opensource simulator for urban autonomous driving. It gives the possibility to generate datasets with several ground truth modalities [START_REF] Dosovitskiy | CARLA: An open urban driving simulator[END_REF]. Grand Theft Auto V (GTA V) is a very high quality AAA video game. Both simulators provide an environment similar to real life, thanks to dynamic weather, seasons, regulated road traffic, traffic lights, signaling, pedestrians, different types of vehicles, ... It is worth noting that there is no support in these simulators or any other simulator for omnidirectional images, which makes the proposed framework of great interest for researchers working on omnidirectional images in robotics and automotive applications.

We show the relevance of this work by releasing the OmniScape dataset1 , which is a dataset of a motorised two-wheeler in the aforementioned simulators. OmniScape comprises stereo fisheye and stereo catadioptric images acquired from the two front sides of a motorcycle, with the corresponding depth maps, semantic segmentation, intrinsic parameters of the cameras and dynamic parameters of the motorcycle, such as velocity, angular velocity, acceleration, location and rotation. See Fig. 1. The OmniScape dataset will be progressively augmented with more omnidirectional data using the same principle with different vehicles, modalities and environments. We have chosen to provide data enquired from a motorcycle because motorcycles present challenging problems that were not addressed before. Indeed, the dynamics of a motorcycle are totally different from the dynamics of cars. As we know, a car is almost all the time parallel to the road. In addition to the distortion in spherical or omnidirectional images, motorcycles undergo rotations yaw, pitch and roll on the three axes, which make the task even harder, due to the inadaptability of classical methods to changes of orientation without a particular learning. The remainder of this paper is organized as follows.

Next section provides a survey on work made using virtual environments as a data source. Section 3 introduces the proposed framework to generate omnidirectional images. Section 4 presents the OmniScape dataset. Finally, Section 6 concludes the paper.

II. RELATED WORK

In the literature, there are several works conducted on virtual environments for the development or validation of autonomous driving systems. Virtual environments have several advantages, mainly the inexpensiveness to generate realistic data, as well as the variety of the nature of the data that can be generated, such as depth maps, semantic segmentation or details of the dynamic properties of the vehicle. These virtual environments allow to simulate different sensors. We also do not have to deal with the problem of data protection and privacy of individuals. Currently several datasets were generated from virtual environments, such as SYNTHIA [START_REF] Ros | The synthia dataset: A large collection of synthetic images for semantic segmentation of urban scenes[END_REF], VEIS [START_REF] Saleh | Effective use of synthetic data for urban scene semantic segmentation[END_REF], and Playing for benchmark [START_REF] Richter | Playing for benchmarks[END_REF].

Thanks to advantages offered by the reverse engineering and modding tools, several recent works have been carried out on the generation of data from GTA V. We can mention the method proposed by Doan et al. in [START_REF] Doan | G2D: from GTA to data[END_REF] for generating perspective images using a virtual camera with six degrees of freedom. In [START_REF] Richter | Playing for data: Ground truth from computer games[END_REF], Richter et al. used GTA V to capture pixel-by-pixel semantic segmentation using an open source middleware called renderdoc between the game and the GPU. In [START_REF] Angus | Unlimited road-scene synthetic annotation (URSA) dataset[END_REF], Angus et al. also extracted semantic segmentation images by changing the textures and shaders of the game in the game files. Richter et al. generated in [START_REF] Richter | Playing for benchmarks[END_REF] a benchmark of several data types from GTA V, all annotated with ground truth data for low-level and high-level vision tasks, including optical flow, instance segmentation, detection and objects tracking, as well as visual odometry.

Johnson-Roberson et al. used in [START_REF] Johnson-Roberson | Driving in the matrix: Can virtual worlds replace human-generated annotations for real world tasks[END_REF] synthetic data generated by GTA V, to show that state-of-the-art algorithms trained only by this data, work better than if they are driven on manually annotated real-world data when tested on the KITTI dataset for vehicle detection. We can note that all these works considered perspective images and, until now, there is neither a dataset for omnidirectional images, nor a dataset for motorcycles or any powered two-wheeler. The present paper seeks to fill this gap, by proposing a framework for omnidirectional data generation from a virtual environment, and generating specifically motorcycles datasets.

III. PROPOSED FRAMEWORK

The proposed framework generates omnidirectional images from a virtual environment using 360°cubemap images. To create 360°images, six images are extracted in the six different directions. Using the appropriate omnidirectional camera model, each pixel of the omnidirectional image can be associated with a 3D direction on the unit sphere. We then compute the cube that presents the six images under the cubemap projection. Using ray tracing, we construct a lookup table that stores correspondences between the omnidirectionnal image and the cubemap images. It corresponds to the intersection of the 3D direction associated to each pixel in the omnidirectional image with the cubemap images. Then we just need to affect to each pixel in the omnidirectional image the corresponding relevant information from the cubemap image (RGB, depth or semantic segmentation), as sketched in Fig. 2. The parameters of the model are calculated from a calibrated camera. To generate omnidirectional images, the proposed framework can use well-known camera models, such as the models proposed by Geyer and Daniilidis [START_REF] Geyer | A unifying theory for central panoramic systems and practical implications[END_REF], Barreto and Araujo [START_REF] Barreto | Issues on the geometry of central catadioptric image formation[END_REF], Mei and Rives [START_REF] Mei | Single view point omnidirectional camera calibration from planar grids[END_REF] and Scaramuzza et al. [START_REF] Scaramuzza | A toolbox for easily calibrating omnidirectional cameras[END_REF]. Without loss of generality, we detail in the following the model proposed by Scaramuzza et al. in [START_REF] Scaramuzza | A toolbox for easily calibrating omnidirectional cameras[END_REF]. It is a calibration model for omnidirectional cameras, considering the omnidirectional imaging system as a compact and unique system composed by a pinhole camera and a mirror. It allows to compute the intrinsic parameters of the omnidirectional camera. This means that it provides the relation between a given 2D pixel and the corresponding 3D vector, from the point of view of the unit sphere, as illustrated in Fig. 3. Let (u, v) be the metric coordinates of a pixel p with respect to the center of the omnidirectional image, and (x, y, z) those of the corresponding 3D vector P , according to

  x y z   =   u v f (w)   , (1)
with w = √ u 2 + v 2 . Within this model, the function f (w) is considered to be a polynomial function, namely of the following form

f (w) = a 0 + a 1 w + a 2 w 2 + a 3 w 3 + a 4 w 4 + ... (2)
The calibration parameters a i are estimated by the leastsquares method on data acquired with a real camera, as described in [START_REF] Dupuis | Robust radial face detection for omnidirectional vision[END_REF].

Since the above model is general for omnidirectional images, not just fisheye images, we use the same method to generate also catadioptric images. We mean by catadioptric images, in this paper, the images taken by a camera composed by a pinhole camera (perspective camera) and a hyperboloidal mirror [START_REF] Geyer | A unifying theory for central panoramic systems and practical implications[END_REF], [START_REF] Mei | Single view point omnidirectional camera calibration from planar grids[END_REF]. The catadioptric images we generate in this work are made in a way that includes all Fig. 3: The omnidirectional camera model proposed in [START_REF] Scaramuzza | A toolbox for easily calibrating omnidirectional cameras[END_REF]. directions of the road in the images, which means that the pinhole camera is on the top of the mirror or the contrary.

Fig. 4 shows the mapping of the six images in the fisheye images and the catadioptric images. The colors red, orange, yellow, blue, green and purple represent respectively the six sides, front, back, left, right, up and down.

The computational cost to render one omnidirectional frame does not exceed 4.6ms on Ubuntu 18.04.3 64-bit running on an Intel Core i7-8750H CPU @ 2.20GHz.

IV. OMNISCAPE DATASET

The OmniScape2 dataset contains, for each capture, fisheye and catadioptric stereo RGB images from the two front sides of a motorcycle, with semantic segmentation and depth map ground truth, as well as the dynamics of the vehicle with its velocity, angular velocity, acceleration and orientation. See Fig. 1 for an overview. The OmniScape dataset will be progressively augmented with more omnidirectional data using the described framework with different vehicles, modalities and environments. The dataset contains data generated from GTA V and CARLA, and can be extended to other simulators. However, due to space limitation, we present in the following data extracted only from CARLA.

For more insights on the extraction of data from GTA V, we refer the interested reader to our previous work [START_REF] Sekkat | Génération d'images omnidirectionnelles à partir d'un environnement virtuel[END_REF].

To generate the images, we used 5 towns available in CARLA Simulator. An example of images generated from a single capture is given in Fig. 5. The RGB images are available for 14 different weather conditions and time of the day and this for each capture. Fig. 6 shows an example of a capture with 4 different weather conditions in fisheye and catadioptric. CARLA Simulator gives a semantic segmentation into 13 classes, namely Building, Fence, Other, Pedestrian, Pole, Road line, Road, Sidewalk, Vegetation, Vehicle, Wall, Traffic sign, Unlabeled. Fig. 7 shows the distribution of pixel of all images in the dataset for both fisheye images and catadioptric images. In complement to these omnidirectional images, the Om-niScape dataset contains also the dynamics of the vehicle at each capture, such as velocity, angular velocity, acceleration and orientation. As we explained before, the case of twowheelers is more challenging because of the dynamics of these vehicles. We computed statistics concerning these dynamics in CARLA Simulator. As presented in TABLE I, we can see that the roll and the pitch change considerably. These alterations will surely affect classical tasks such as visual odometry and semantic segmentation. This is due to the fact that most computer vision and machine learning tasks are often trained on perspective data acquired with cars as autonomous vehicles, while these vehicles do not suffer from modifications in these dynamics.

In complement to the images given in this paper, more examples from CARLA Simulator and GTA V can be found in OmniScape GitHub. V. CONCLUSION This paper presented a general framework to generate datasets of omnidirectional images from virtual environments, and provided the OmniScape dataset. We demonstrated the relevance of this framework by generating fisheye and catadioptric images with depth map, semantic segmentation and dynamic parameters. Two simulators were investigated with success, GTA V and open-source CARLA Simulator.

There are many possible extensions to this application, including the generation of other types of datasets, using different types of omnidirectional camera models and different vehicles like drones. These datasets can be used as evaluation credentials for different vision and deep learning applications, whose algorithms applied to perspective images have limited performance on omnidirectional images. A wide variety of applications include Simultaneous Localization And Mapping (SLAM), visual odometry, depth estimation, object recognition and classification, detection and tracking. Moreover, they can also be used to evaluate or even train semantic segmentation algorithms developed for omnidirectional images.

Fig. 1 :

 1 Fig. 1: Recording platform and a representation of the different modalities.

Fig. 2 :

 2 Fig. 2: Lookup table construction to set the omnidirectional image pixel values.

Fig. 4 :

 4 Fig. 4: Mapping of the six images in the fisheye (left) and catadioptric (right) images.

  RGB left image. (b) RGB right image. (c) RGB left image. (d) RGB right image. (e) Semantic segm. left image. (f) Semantic segm. right image. (g) Semantic segm. left image. (h) Semantic segm. right image. (i) Depth map left image. (j) Depth map right image. (k) Depth map left image. (l) Depth map right image.

Fig. 5 :

 5 Fig. 5: Examples of fisheye (left panel) and catadioptric (right panel) images generated from a single capture.

Fig. 6 :

 6 Fig. 6: Examples of fisheye and catadioptric images generated from a single capture with four different weather conditions and time. The motorcycle in this capture undergoes rotations.

Fig. 7 :

 7 Fig. 7: Percentage of pixels representing each class in the dataset for both fisheye and catadioptric images.

TABLE I :

 I Statistics

		mean	std	min	median	max
	Yaw	1.15	110.54	-179.99	0.85	179.99
	Pitch	-0.15	1.75	-10.26	-0.07	18.94
	Roll	0.14	3.62	-24.83	0.00	25.91

concerning the dynamics (yaw, pitch and roll, in degrees) of the motorcycle in a tested route

https://github.com/ARSekkat/OmniScape

https://github.com/ARSekkat/OmniScape

VI. ACKNOWLEDGEMENTS

The authors would like to thank Vincent Vauchey for his valuable suggestions on improving the optimisation of the computational cost. This work was mainly supported by a RIN grant, Région Normandie, France. It was partially supported by ANR CLARA (ANR-18-CE33-0004-02) and DAISI project funded with the support from the European Union with the European Regional Development Fund (ERDF) and from the Regional Council of Normandy.


Anton Musickhin

Andrey Yu Zubarev

Maxim Raboisson-Michel

Gregory Verger-Dubois

Pavel Kuzhir

Field-induced circulation flow in magnetic fluids

Keywords: Magnetic fluid, Oscillating magnetic field, Field-induced flow

We present results of theoretical study of circulation flow in ferrofluids under the action of an alternating inhomogeneous magnetic field. The results show that the field with the amplitude about 17 kA/m and frequency 10s -1 can induce the mesoscopic flow with the velocity amplitude about 0.5mm/s. This mechanism can be used for intensification of drag delivery in blood vessels.

Introduction

The main problem of treatment of brain strokes is related to very slow diffusion of the thrombolytic drugs toward blood clots through blocked vessels. An American company Pulse Therapeutics has proposed a smart solution to this problem using magnetic micro-or nanoparticles entrained in motion by alternating magnetic fields and able to create recirculating flows in the blocked vessels [START_REF] Creighton | Magnetic-based systems for treating occluded vessels[END_REF]. These recirculating flows considerably enhance convective transport of the drug towards the clots. Only a few works have been published on this topic [START_REF] Clements | A mathematical model for magnetically-assisted delivery of thrombolytics in occluded blood vessels for ischemic stroke treatment[END_REF][START_REF] Gabayno | Controlled manipulation of Fe 3 O 4 nanoparticles in an oscillating magnetic field for fast ablation of microchannel occlusion[END_REF][START_REF] Li | Thrombolysis Enhancing by Magnetic Manipulation of Fe3O4 Nanoparticles[END_REF], so, the physical understanding of the origin of the recirculating or oscillatory flows created by moving and rotating magnetic particles is still lacking.

To shed more light onto this problem, in this paper, we propose a theoretical model considering ferroparticle motion *Author for correspondence (A.J.Zubarev@urfu.ru). †Present address: Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin Ave, 51, Ekaterinburg, 620083, Russia, and induced flows inside a channel under applied alternative non-homogeneous magnetic fields. The obtained amplitudes of the velocity and fluid displacement are compared to those required for efficient drug delivery to the blood clots in a real situation.

The aim of this work is theoretical study of circulation flows induced by an alternating magnetic field in a magnetic fluid filling a flat gap, which size in its plane is much more than the distance between the gap walls.

Mathematical model and the main approximations.

We consider an infinite flat gap of the thickness l filled with a ferrofluid, containing identical spherical ferroparticles. The alternating magnetic field is created by four solenoids, illustrated in Fig. 1. Fig. 1. Illustration of the model system .

Let m and M be the particle magnetic moment and saturated magnetization of its material; 𝑉 𝑝 is the particle volume.

It is supposed that the gap thickness l is much less than the solenoid diameter 𝐷 (𝑙 << 𝐷). We suppose that, initially, some drop of a ferrofluid is injected in the center of the considered model. We will denote the local volume concentration of the particles as Ф(x,z,t) and, for maximal simplification of mathematics, consider the 2D approximation, when all physical events take place in the plane (x,z) shown in Fig. 1. Additionally, we will neglect effects of the particles Brownian rotation. This means, we suppose that the Zeemen energy of the particle interaction with the field H is significantly more than the thermal energy kT. Note that, from the practical point of view, this case is the most interesting. Note, for the magnetite particles with diameter about 10-20nm this condition is fulfilled if the local H exceeds 10kA/m. That is easy achievable range of the field. Equations of flow of the magnetizable fluid at low Reynolds number can be presented as (see [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Landau | Electrodynamics of Continuum Media[END_REF] 𝜕𝑧 2 is the Laplace operator, 𝛤 = 𝜇 0 𝑀Φ(𝐻 𝑧 𝑠𝑖𝑛𝜃 -𝐻 𝑥 cos 𝜃) is the magnetic torque, acting by unit volume of the ferrofluid; H is the local magnetic field in the fluid; 𝜃 is the angle between the particle magnetic moment m and the axis Oz, normal to the gap plane (see Fig. 1). The third terms in the first two equations of (1) present the stress, which appear because of the magnetic torque 𝛤; the fourth one is the ponderomotive force, acting on the fluid from the side of the non-uniform field H.

The boundary conditions for (1) are: 𝑣 𝑥 = 𝑣 𝑧 = 0 𝑎𝑡 𝑧 = 0, 𝑙

(2) 𝑣 𝑥 , 𝑣 𝑧 → 0 𝑎𝑡 𝑥 → ∞ In the "no Brownian" approximation, equation for the angle 𝜃 reads [START_REF] Creighton | Magnetic-based systems for treating occluded vessels[END_REF]

𝜕𝜃 𝜕𝑡 = 1 2 ( 𝜕𝑣 𝑥 𝜕𝑧 - 𝜕𝑣 𝑧 𝜕𝑥 ) - 1 6𝜂Φ Γ (3)
Note, the ratio Γ/Φ is magnetic torque with acts on the unite volume of the particle. Equation of continuity for the particles concentration can be written as:

𝜕 𝜕𝑡 Φ + 𝑑𝑖𝑣 [Φ (𝐯 - 1 3𝜋𝜂𝑑 ∇𝑈)] = 𝐷∆Φ (4) 𝑈 = -𝜇 0 𝑀𝑉 𝑝 (𝐻 𝑧 cos 𝜃 + 𝐻 𝑥 sin
Here d is the particle diameter; U is the particle potential energy in the field H. Below we will use the following estimates for the system parameters. We suppose that the particles are magnetite, therefore 𝑀 ≈ 500𝑘𝐴/𝑚; diameter of the particle 𝑑 ≈ 15 -20𝑛𝑚; volume concentration of the particles Φ~0.01; the angular frequency of the alternating field, 𝑐𝑟𝑒𝑎𝑡𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑐𝑜𝑙𝑒𝑛𝑜𝑖𝑑 𝜔~10𝑠 -1 ; the solenoid provides the field H inside the gap with the strength absolute value 𝐻 > 10𝑘𝐴/𝑚. The viscosity and density of the host medium are close to those of water ones, i.e. we suppose 𝜂~10 -3 𝑃𝑎 • 𝑠; 𝜌~10 3 𝑘𝑔/𝑚 3 . The gap thickness is estimated as 𝑙~1𝑚𝑚.

Analytical solution of the whole system (1-4) is impossible. However, serious simplifications can be achieved by using the mentioned estimates for the system characteristics. First of all, from the mathematical point of view, the fluid flow is provoked by the last two terms in the Navier-Stocks equations of (1). These terms are proportional to the small concentrations Φ of the particles. Therefore, roughly, the scaling relation 𝑣~Φ must be held and the terms ) , Φ𝑣 can be considered as the smallest ones in eqs. ( 3), ( 4).We will neglect these terms.

Let us present the local concentration of the particles as Φ(𝑥, 𝑧, 𝑡) = Φ 0 (𝑥, 𝑧) + 𝜑(𝑥, 𝑧, 𝑡) where Φ 0 ~0.01 is the initial volume concentration of the particles in the drop. Estimates show that in the field with the strength 𝐻~10𝑘𝐴/𝑚 and the spatial scale 𝐷~10 -2 𝑚 the characteristic time 𝜏 = 3𝜋𝜂𝑑𝐿 2 𝜇 0 𝑀𝑉 𝑝 𝐻 of magnetophoretic migration of the magnetite particle with diameter 20nm at the distance 𝐷 is more than 5 • 10 5 𝑠 ≈ 140 ℎ𝑜𝑢𝑟𝑠. That significantly exceeds the time, presenting interest from the viewpoint of the drug delivery technology. Therefore, at least in the first approximation, one can neglect the term 𝜑 as compared with Φ 0 and suppose that for the time, presenting interest, the equality Φ = Φ 0 (𝑥, 𝑧) is held. In principle, the drop shape can be varied because of the convective motion of the ferrofluid particles in the generated circulation flow. Discussion of this factor is given in the Conclusion.

Neglecting the term ) and by using the explicit form for the torque Г, one can rewrite eq. ( 3) as:

𝜕𝜃 𝜕𝑡 = - 𝜇 0 𝑀 6𝜂 (𝐻 𝑧 𝑠𝑖𝑛𝜃 -𝐻 𝑥 cos 𝜃) (6)
For convenience one can present the first two equations of (1) as: It follows from eq.( 6) that, in the order of magnitude, the characteristic time 𝜏 𝐻 of the angle 𝜃 relaxation to the given field H is 𝜏 𝐻 ~𝜂/𝜇 0 𝑀H . Suppose that absolute value of the field in the gap has the typical magnitude 𝐻~10𝑘𝐴/𝑚, the particles are magnitude (𝑀~500𝑘𝐴/𝑚) and the carrier liquid is water (𝜂~ 10 -3 𝑃𝑎 • 𝑠 ). In this case 𝜏 𝐻 ~10 -6 s . The typical value of the angular frequency for the experiments like that is 𝜔~10𝑠 -1 , therefore, the field alternation period is about 0.1s -1 . Therefore, the rate of the field alternation is about five decimal orders less than the rate of 𝜃 relaxation. This means that in any moment of time the angle 𝜃 has practically the value 𝜃 0 equilibrium with respect to the field H at the given time t; this angle is determined by the relations

𝜌 𝜕𝑣 𝑥 𝜕𝑡 = - 𝜕𝑝 𝜕𝑥 + 𝜂Δ𝑣 𝑥 + 𝜇 0 𝑀ΦG 𝑥 (7)
cos 𝜃 0 = 𝐻 𝑧 𝐻 , sin 𝜃 0 = 𝐻 𝑥 𝐻 (9) 𝐻 = √𝐻 𝑥 2 + 𝐻 𝑧 2
Taking into account, we use 𝜃 = 𝜃 0 in eq. ( 9). In this approximation, with account of the standard magnetostatic relation 𝑟𝑜𝑡𝑯 = 0, one gets from (9):

𝐺 𝑥 = 𝜕 𝜕𝑥 𝐻; 𝐺 𝑧 = 𝜕 𝜕𝑧 𝐻 (10)
This is convenient to introduce the current function Ψ so that

𝑣 𝑥 = 𝜕 𝜕𝑧 Ψ, 𝑣 𝑧 = - 𝜕 𝜕𝑥 Ψ (11)
Combining ( 7) and ( 11), taking into eq. ( 10), after transformations we come to the following equation with respect to Ψ :

𝜌 𝜕𝑄 𝜕𝑡 = 𝜂Δ𝑄 + 𝜇 0 𝑀 [ 𝜕 𝜕𝑥 𝐻 • 𝜕 𝜕𝑧 Φ - 𝜕 𝜕𝑧 H • 𝜕 𝜕𝑥 Φ] (12

Solution of equation (12); results and discussions.

We suppose that the spatial scale of the function f in the x direction is much more than that in z direction. This is justified by configuration of the solenoids, which create the field H and the ferrofluid drop distribution in the gap. Indeed, the spatial scale of the function f is determined by the solenoid diameter D and their distances a,b from the ferrofluid drop. All these scales usually are much more than diameter of the blood vessel, which is modeled here by the gap with the thickness l. Thus, at the distance l ., the function f changes very weakly and, in the first approximation, one can neglect dependence of f on the coordinate z.

Taking into account that the function f is periodical with the period 2𝜋/𝜔 , we can expand this function in the Fourier series as (see, for example, ref. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]): 𝑓 = By using ( 16) in ( 14) and ( 15), we come to the relations:

𝛹𝐶 𝑛 = -𝐴 𝑛 𝑒 𝑘𝑧 𝑐𝑜𝑠𝑘𝑧 2𝑘 2 -𝐵 𝑛 𝑒 𝑘𝑧 𝑠𝑖𝑛𝑘𝑧 2𝑘 2 -𝐶 𝑛 𝑒 -𝑘𝑧 𝑐𝑜𝑠𝑘𝑧 2𝑘 2 -𝐷 𝑛 𝑒 -𝑘𝑧 𝑠𝑖𝑛𝑘𝑧 2𝑘 2 + 𝐿1 𝑛 𝑧 + 𝐿2 𝑛 - 𝑓𝑆 𝑛 2(𝜔𝑛) 𝑧 2 (17)
𝛹𝑆 𝑛 = 𝐴 𝑛 𝑒 𝑘𝑧 𝑠𝑖𝑛𝑘𝑧 2𝑘 2 -𝐵 𝑛 𝑒 𝑘𝑧 𝑐𝑜𝑠𝑘𝑧 2𝑘 2 -𝐶 𝑛 𝑒 -𝑘𝑧 𝑠𝑖𝑛𝑘𝑧 2𝑘 2 + 𝐷 𝑛 𝑒 -𝑘𝑧 𝑐𝑜𝑠𝑘𝑧 2𝑘 2 + 𝑁1 𝑛 𝑧 + 𝑁2 𝑛 + 𝑓𝐶 𝑛 2(𝜔𝑛) 𝑧 2 𝑘 = √ 𝜔 2𝜈
Here 𝐴 𝑛 ,…, 𝑁2 𝑛 are constants of integration, to be determined.

The boundary conditions (13) can be rewritten in the form z=0,l

𝜕Ψ𝐶 𝑛 𝜕𝑧 = 0, 𝜕Ψ𝑆 𝑛 𝜕𝑧 = 0, ΨC 𝑛 = 0, ΨS 𝑛 = 0 . (18)
Combining eqs. ( 17) and (18), we come to a system of eight linear equations with respect to eight variations An,…,N2n. Analytical solution of this systems is too cumbersome; however, the numerical calculations can not present a problem. Having the coefficients An,…,N2n found, by using eq. ( 11), we determine the velocity components 𝑣 𝑥,𝑧 .

Results and discussion.

For definiteness we supposed the the spatial distribution of the particles obey to the Gaussian distribution

Φ = Φ 0 𝑒𝑥𝑝 [- (𝑥 2 +(𝑧- 𝑙 2 ) 2 ) 𝜎 2 ] (19)
This formula means that in the vicinity of the middle of the gap there is a dense cloud of ferrofluid with the maximal 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛Φ 0 at point (0, 𝑙/2) ; 𝜎 is dispersion of this distribution. It should be noted that the spatial distribution ( 19) is used here just as a model one. Any other, more relevant to a concrete experimental situation, can be used for the calculations in the frame of the suggested approach.

To calculate the components of amplitude 𝑯 01 of the field, created by the solenoid 1 in Fig. where 𝐼 is the current in the solenoid, ℎ is its height, 𝑁 is the number the solenoid coils and 𝑎 is the distance from solenoid to the gap. This formula represents the field components for the lower solenoid, for the other solenoids these components have been determined similarly

X, mm

Note that for the system parameters, corresponding ti Figs.2,3, the amplitude of absolute value of the field in the gap center is about 17kA/m

Conclusions

We present theoretical model of magnetically induced circulation flow in a flat channel filled by a Newtonian liquid and a drop of ferrofluid. The results show that oscillating magnetic field with amplitude about 17 kA/m and frequency about 10s -1 in the channel with the wideness 1mm can induce circulating flow with the the velocity amplitude about 0.5mm/s. Let us discuss now the approximation of the permanent shape of the ferrofluid drop. Since the magnetophoretic displacement of the ferrofluid particles is neglected, the drop shape can be changed only because of the convective effects. In the order of magnitude, the displacement 𝛿𝑟 of the drop particles for the field period satisfies to the relation 𝛿𝑟 ≤ 𝑣 𝑚𝑎𝑥 2𝜋/𝜔 , where 𝑣 𝑚𝑎𝑥 is maximal value of the velocity. Our results give 𝑣 𝑚𝑎𝑥 ~0.5𝑚𝑚/𝑠 . By using 𝜔~10𝑠 -1 , one gets 𝛿𝑟 ≤ 0.3 𝑚𝑚 . This is much less than the used characteristic size 𝜎~1𝑐𝑚 of the drop. Therefore, the simplification of the permanent shape of the drop is justified, at least, as the first approximation.

Funding Statement

The work was supported by the Russian Fund of Basic Researches, project 19-31-90003 and by French "Agence Nationale de la Recherche", Project Future Investments UCA JEDI, No. ANR-15-IDEX-01 (projects ImmunoMag and MagFilter)

. Red line: at time 𝑡 = 0.5 𝑠; blue one: 𝑡 = 1 𝑠. The other parameters are the same as in fig. 2 Note that for the system parameters, corresponding ti Figs.2,3, the amplitude of absolute value of the field in the gap center is about 17kA/m

Figure captions

1 .

 1 We will suppose that the solenoids create the field with the same angular frequency 𝜔 and the components 𝐻 𝑥 = (𝐻 01𝑥 (𝑥, 𝑧) -𝐻 04𝑥 (𝑥, 𝑧))𝑐𝑜𝑠𝜔𝑡 + (𝐻 02𝑥 (𝑥, 𝑧) -𝐻 03𝑥 (𝑥, 𝑧))𝑠𝑖𝑛𝜔𝑡 (8) 𝐻 𝑧 = (𝐻 01𝑧 (𝑥, 𝑧) -𝐻 04𝑧 (𝑥, 𝑧))𝑐𝑜𝑠𝜔𝑡 + (𝐻 02𝑧 (𝑥, 𝑧) -𝐻 03𝑧 (𝑥, 𝑧))𝑠𝑖𝑛𝜔𝑡 Here 𝑯 01 , 𝑯 02, 𝑯 03 and 𝑯 04 are amplitudes of the fields created by the lower, upper, left and right solenoids in Fig.The relations (8) mean that the north pole of the left solenoid in Fig.1 is situated in front of the south pole of the right one. The same is for the magnets which are above and below the gap with the fluid.

  Taking into account the relation 𝐷 >> 𝑙 between the velocity 𝑣 scales in the x (D) and z (l) directions, we get the strong inequalities

Fig. 2 . 1 .

 21 Fig. 2. Longitudinal velocity component 𝑣 𝑥 vs. 𝑧 coordinate at a fixed 𝑥 = 10 𝑚𝑚 𝑖𝑛 𝐹𝑖𝑔. 1. Red line: time 𝑡 = 0.5 𝑠; blue one: 𝑡 = 1 𝑠. The field angular frequency 𝜔 = 10𝑠 -1 . Volume concentration of the particles in the drop center Φ 0 = 0.01; the dispersion 𝜎 = 1 𝑐𝑚; the solenoid characteristics are identical for all solenoids: diameter D=1cm; current 𝐼 = 10 𝐴; height h=1 𝑐𝑚; number of coils 𝑁 = 10 4 . The distances, shown in Fig. 1: 𝑎 = 𝑏 = 5 𝑐𝑚; the gap thickness l=1mm.

Fig. 3 . 1 . 2 -

 312 Fig. 3. Longitudinal velocity component 𝑣 𝑥 vs. 𝑥 coordinate at a fixed 𝑧 = 𝑙/8 in Fig. 1.Red line: time 𝑡 = 0.5 𝑠; blue one: 𝑡 = 1 𝑠. The other parameters are the same as in Fig.2

Figure 1

 1 Figure captionsFigure 1 Illustration of the model system Figure 2. Longitudinal velocity component 𝑣 𝑥 vs. 𝑧 coordinate at a fixed 𝑥 = 10 𝑚𝑚 𝑖𝑛 𝐹𝑖𝑔. 1.

Figure 3 .

 3 Longitudinal velocity component 𝑣 𝑥 vs. 𝑥 coordinate at a fixed 𝑧 = 𝑙/8 𝑖𝑛 𝐹𝑖𝑔. 1.

Data Accessibility

Competing Interests

We have no competing interests We have no competing interests.