John Tichy
email: tichyj@rpi.edu

Marine Menut

Camella Oumahi

Sandrine Muller

Benyebka Bou-Saïd

Grease Flow based on a Two-Component Mixture Model by

Keywords: grease, lubrication, non-Newtonian 1

Rolling element bearings are the most common type of bearings and about 90 % of them are lubricated with grease. In flow modeling grease has almost always been treated as a homogeneous single-component material described by a shear thinning rheological model. The present modeling paper attempts to incorporate key features not addressed by existing models: the two-component nature of the grease, and the fact that the thickener is a sort of delivery system of oil to the contact. In the present approach, the thickener is modeled as a medium containing the oil. Oil flows through the thickener according to the Darcy-Brinkman law. The thickener is regarded as a sort of porous medium, but the key feature is a linear interaction force between the two components. The model is conceptual and speculative but could provide inspiration for alternate ways of thinking about grease flow behavior. Three idealized cases are considered: flow in simple configurations of rheological devices, flow in a contact completely composed of grease, and flow in a contact with a layer of grease and a layer of oil. For illustrative purposes the rigid cylinder-plane contact is considered, and a beneficial effect of the grease is seen. Many extensions of the model itself are possible, and it can be applied to more realistic problems. Although validation is not provided,the conceptual framework could likely be tested by relatively straightforward experiments.

I. Introduction

Grease is a widely used complex material. Many authors point out that grease is a twocomponent mixture -a base oil and a thickener. Likewise, it is commonly noted that the thickener (usually a polymer fiber or a soap) may constitute a delivery system for the oil to the contact. The oil, in turn, provides the essential lubrication -be it hydrodynamic or boundary. Despite its industrial and economic importance, and despite the fact that understanding grease lubrication presents an interesting intellectual challenge, these aspects (the two-component nature and the oil delivery system) are rarely addressed in a mathematical model with predictive ability in flow problems. Indeed, the mechanisms may be so different from application to application, that it is not evident a unified conceptual approach can be applied. Clearly, better grease models would be desirable. Please note throughout that by 'modeling' we mean only flow modeling, not modeling of the bleed process, evaporation, aging, oxidation, etc. This paper owes a heavy debt to a 1999 paper by P Cann [START_REF] Cann | Understanding grease lubrication[END_REF]. She notes the importance of grease, and the lack of satisfactory models for grease lubrication. She goes on to present a heuristic physical description of grease lubrication in rolling contacts that we loosely try to implement in the present paper. In her description, the grease layer is deposited in the contact track, repetitive passes of the roller releases oil from the grease, which, in turn improves the hydrodynamic film.

There are numerous papers in the literature applying mathematical models of grease to lubrication contacts as a flow boundary value problem. However, as to predictive flow models, we could only find ones which consider the grease as a homogeneous single component material. The thickener is generally a very viscous non-Newtonian fluid, which may be purely viscous (so-called generalized non-Newtonian) or viscoelastic. Non-Newtonian fluid models and rheology principles are discussed in the text by Bird et al. [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF]. Typically, the grease at low shear is some 1000 times more viscous than the base oil, which is probably close to Newtonian in nature.

In the following background description, we do not try to be comprehensive but rather to give the reader a flavor as to what is in the current literature. Probably being discouraged by the problem complexity, the general nature of grease flow models has changed very little over 50 years. For example in 1965, Batra [START_REF] Batra | Rheodynamic lubrication of a journal bearing[END_REF] applied the Bingham model to a journal bearing. The Bingham model describes a fluid that behaves as a rigid solid when the shear stress magnitude is lower than a yield stress, and flows as a quasi-Newtonian material at higher stresses. Similarly, in 1973 Wada et al. [START_REF] Wada | Behavior of Bingham soild in hydrodynamic lubrication, part 1, general theory[END_REF] published a series of papers applying the Bingham model to several bearing configurations. Through the intervening years, there are a number of similar papers.

Jumping forward to more recent times, in 2008 Gertzos et al. [START_REF] Gertzoa | CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant[END_REF] applied computational fluid dynamics (CFD) methods for the Bingham material to journal bearings, while in 2017 Westerberg et al. [START_REF] Westerberg | Lubricating grease flow in a double restriction seal geometry: A computational fluid dynamics approach[END_REF] describe grease flow in seals by the Herschel-Bulkley fluid (another yield stress model) using CFD. Zhang et al. [START_REF] Zhang | Pressure distribution analysis of elastohydrodynamic journal bearing under grease lubrication[END_REF] apply several generalized Newtonian models to simulate grease flow to an elastohydrodynamic lubrication (EHL) journal bearing. A concentric cylinder geometry is both experimentally and theoretically studied by Li et al., [START_REF] Li | Lubricating grease shear flow and boundary layers in a concentric cylinder configuration[END_REF], using the Carreau model and the Herschel-Bulkley model. Augusto et al. in 2020 [START_REF] Augusto | A numerical investigation of grease friction losses in labyrinth seals[END_REF] apply sophisticated CFD methods and the Herschel-Bulkley model to the labyrinth seal problem.

There are a number of papers in the literature which can be retrieved searching under 'grease' and 'model' whose intention is to predict grease behavior in transient shear strain, strain relaxation and the like, rather than prediction of behavior in lubrication flow. See for example, the studies of Madiedo et al. [START_REF] Madiedo | Modeling of the non-linear rheological behavior of a lubricating grease at low-shear rates[END_REF]. Cyriac et al. [START_REF] Cyriac | Effect of thickener particle geometry and concentration on the grease EHL film thickness at medium speeds[END_REF], have written more recent papers on grease lubrication and flow.

A paper with some direct relevance to the present work is from 2010 due to Baart et al. [START_REF] Baart | Oil-bleeding model for lubricating grease based on viscous flow through a porous microstructure[END_REF]. The effect of porosity in grease is studied, but during 'bleeding' conditions, i.e., in storage as the oil separates from the thickener. A 2017 paper by Saatchi et al. [START_REF] Saatchi | A fundamental study of oil release mechanism in soap and non-soap thickened greases[END_REF] use a two-component flow analysis similar to that used here to describe the oil release mechanism in greases in somewhat the same fashion as the present work.

It can be safely stated that grease lubrication is still not thoroughly understood. There are a number of recent excellent papers on grease behavior which bear mentioning, to give the interested reader a feeling for the current state of affairs. In no special order, Cann and Lubrecht 2007 [START_REF] Cann | Bearing performance limits with grease lubrication: the interaction of bearing design, operating conditions and grease properties[END_REF] discuss grease performance and design, Cousseau et al. 2015 [START_REF] Cousseau | Grease aging effects on film formation under fully-flooded and starved lubrication[END_REF] study the grease aging process, Venner et al. 2012 [START_REF] Venner | Thin layer flow and film decay modeling for grease lubricated rolling bearings[END_REF] treat grease film behavior in an EHL contact, while Couronné et al. 2003, [17] correlate grease properties with EHL film thickness. Cousseau et al. 2012 [18] correlate grease flow rheology to frictional torque in thrust ball bearings.

Rheological behavior of many polymer greases was reported in 2015 by Gonçalves et al. 2015 [START_REF] Gonçalves | Formulation, rheology and thermal ageing of polymer greases -part i: Influence of the thickner content[END_REF] II. Analysis

A. Grease flow modeled as a mixture

The proposed model assumes that the flow of grease consists of an oil component, denoted by the superscript 'O', and a thickener component which we denote by superscript 'T '. We use the precepts of mixture theory, outlined in the 2017 review article of Siddique et al. [START_REF] Siddique | A review of mixture theory for deformable porous media and applications[END_REF] for porous media. The components have different properties that interact, but the mixture may be treated as single fluid with homogeneous properties. The thickener can consist of materials such as a polymer fiber matrix, or lithium soap. Thus we describe a two-component material with a linearly weighted average:

v = φ v O + (1 -φ) v T , τ = φ τ O + (1 -φ) τ T v O = v O x i + v O y j + v O z k, τ O = τ O xx ii + τ O xy ij + τ O xz iz + τ O yx ji + . . . , etc. (1)
The components v O and v T represent the velocity of a volume element if it were entirely oil O or thickener T . Instead, each volume element contains the two components treated as an equivalent single component homgeneous material. The symbol φ denotes the oil component volume fraction, and thus 1 -φ is the thickener volume fraction. The volume-averaged velocity of the mixture is v. The stress tensor is τ shown in dyadic form. The primary flow direction along the film is x and the direction across the thin film is z. The surfaces separated by the film reside primarily in the xy plane. In the present paper, we restrict ourselves to two-dimensional flow (a one-dimensional contact) to illustrate the basic concepts, although the model can be extended to the three-dimensional case.

At the boundaries of the mixture, the individual mixture components match the velocity of the opposing material. For example, at a boundary z = z b , where

v x = V b and τ zx = τ b v O x (z b ) = V b , v T x (z b ) = V b , φ τ O zx (z b ) + (1 -φ) τ T zx (z b ) = τ b (2)
1. Thickener modeled as a generalized Newtonian fluid, oil as a Newtonian fluid

The purely viscous or generalized Newtonian case, is much simpler than the viscoelastic case, often used to describe grease. The shear stresses are obtained from a model such as Carreau-Yasuda as described in the text of Bird et al. [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF]:

τ T zx = η T ∂v T x ∂z , γT = ∂v T x ∂z 2 η T = η T ∞ + (η T 0 -η T ∞ ) 1 + λ 2 γT 2 (n-1)/2 (3)
where γT is the thickener shear rate, η T is the thickener non-Newtonian viscosity; and η T 0 and η T ∞ are the thickener low and high shear viscosities, respectively. The symbols λ and n are model parameters, the former having dimensions of time, the latter being dimensionless.

The thickener has very high viscosity at low shear (η T ∞ large). In the limit the behavior of an extremely viscous fluid approaches that of a a rigid solid. The particular model for the thickener is not crucial to the paper, only that its viscosity is much greater than that of the oil.

Grease modeled as a porous medium mixture

We treat the oil phase as Newtonian, obeying the Darcy-Brinkman law [START_REF] Brinkman | A caclulation of the viscous force exerted by a flowing fluid on a dense swarm of particles[END_REF]. Likewise in what follows we treat the thickener as Newtonian. We assume that the thickener shearing is slow and its viscosity can be considered to be the low-shear viscosity η T = η T 0 . Both bulk soap and an array of fibers, although solid-like materials, can evidence flow properties, see for example Castro et al. [START_REF] Castro | Comparison of methods to measure yield stress of soft solids[END_REF] (on soft soaps) and du Roure et al. [START_REF] Du Roure | Dynamics of flexible fibers in vviscous flows and fluids[END_REF] (on fiber arrays).

∂p O ∂x = ∂p T ∂x = ∂p ∂x (4)
∂p ∂x = η O k v O x -v T x + η O ∂ 2 v O x ∂z 2 (5)
∂p ∂x = - η O k v O x -v T x + η T ∂ 2 v T x ∂z 2 (6)
The pressure is identical for the oil and thickener components, Eq. ( 4). The thickener component is considered to be a porous network causing a drag on the oil flow passing through it, Eq. ( 5). The drag is proportional to the velocity difference v O x -v T x , the oil viscosity η O and inversely proportional to the permeability k. Permeability has units of m 2 and √ k roughly corresponds to the pore size. There is an equal and opposite force of the oil on the thickener, Eq. ( 6). The second derivative terms are due to Brinkman and account for shear stress at the boundaries. These terms allow boundary conditions to be imposed in shear flow.

In the end, the important thing is that the local interaction force between the oil and thickener is linearly proportional to the oil viscosity and the component relative velocity and inversely proportional to a characteristic pore size. It is not required that the pore be of a certain size or shape.

B. Rheological Behavior

In a viscometric shear flow, each fluid element undergoes an identical simple shearing motion, regardless of the constitutive behavior. Through various arguments made in rheology texts (see Bird et al., page 100, for example), Couette (simple shearing) flow is found to be viscometric. The present case is that of two parallel plates separated by a constant gap H, one of which is sliding at velocity V . The material is undergoing a viscometric flow of shear rate γ = V /H. Cone and plate flow is also viscometric in that the shear rate likewise constant throughout. The gap is thin H ≪ L, where L is the characteristic length in the direction of flow. Strictly speaking, viscometric flow is required for rheological measurements because any homogeneous fluid (Newtonian or otherwise) will have a known velocity field. Other configurations, such as Poiseuille flow between parallel plates driven by a pressure gradient, are sometimes used for viscosity measurement (similar to the the capillary viscometer) but to a lesser degree of certitude because the velocity field may vary depending on non-Newtonian fluid type.

Couette or Simple Shearing Flow

As described above, there is a lower stationary surface at z = 0 and an upper surface at z = H sliding at velocity V , with uniform pressure. Wall boundary conditions for the mixture are treated just as for a single-component fluid. For these conditions, the solutions to Eqs. ( 4)-( 6) are

v T x = V z H , v O x = V z H (7)
According to the model, the shear stress is,

τ O zx = η O V H , τ T zx = η T V H τ zx = φ τ O zx + (1 -φ) τ T zx = η ef f V H , η ef f = φ η O + (1 -φ) η T . (8)
The effective viscosity η ef f is a weighted sum of the viscosities of the two components. Since η T ≫ η O if the model is to be believed, such a viscometer measures an effective viscosity proportional close to that of the thickener.

Poiseuille or Simple Channel Flow

Consider the case of an infinitely wide channel of constant gap height H. The channel is of length L ≫ H, so thin film flow conditions are in force. The flow is thus steady and fully developed. A pressure p 0 is imposed on the left hand entry side, while the exiting pressure on the right hand side, p L = 0, thus ∂p/∂x = -p 0 /L. As usual for thin film flows, the pressure does not vary across the gap, ∂p/∂y = 0. Classical no-slip conditions are assumed to be in force on both surfaces for both v T x and v O x . This flow through a thin slit is the rectilinear analogue to flow through a capillary which is commonly used as a viscometer, especially in adverse conditions such as at high temperatures or pressure.

We further assume that the flow is slow (low Reynolds number) such that inertia can be neglected. The no-slip boundary conditions on the two phases are

v T x (0) = v O x (0) = v T x (H) = v O
x (H) = 0. Equations ( 4)-( 6) can be solved in closed form for the two velocity components:

v T x = - p 0 2η T 0 L (z 2 -z H) (9)
z * = z √ k , H * = H √ k (10) v O x = - p 0 exp(-z * ) 2L η O (1 + exp H * ) (exp z * + exp(z * + H * )) η O η T 0 z * (z * -H * ) + A A = 2 [-exp H * + exp z * -exp 2z * + exp(H * + z * )] k η O η T 0 -1 (11)
An expression for the permeability in terms of the porosity is

k = φD 2 p 32T , (12)
where D p is an effective pore size and T is the tortuosity (arc length to chord length) [START_REF] Happel | Low Reynolds Number Hydrodynamics[END_REF]. This equation is derived by modeling the thickener medium as an array of thin tubes oriented in the direction of flow.

For the limiting case of very small oil viscosity, the equations simplify considerably:

v O x = - p 0 k L η O exp z * 1 + exp H * (1 -exp z * ) (exp z * -exp H * ) (13)
For the following conditions, the velocity field is shown in Fig. 1: H= 1 mm, L= 100 mm, p 0 = 10 kPa, η O = 1 Pa.s, η T 0 = 1000 Pa.s, φ = 0.7, and k = 10 -8 m 2 . The thickener flows slowly in a parabolic Newtonian fashion, while the oil is flowing much faster. Most of the oil flow resistance comes from the porosity effect. The volume-averaged flow is much closer to that of the oil than the thickener. The densities of typical oils and polymer thickeners are not that different, thus mass-averaged and volume average quantities would be nearly the same.

The flow behavior is strongly dependent on the ratio √ k/H which roughly corresponds to the pore size relative to the lubricant gap size. If this dimensionless parameter is small (relative to one), the oil encounters high resistance to flow through the thickener and the two would have nearly identical velocities. By contrast, if this ratio is large, meaning of order one, the oil flows easily through the thickener and the oil flow profile is nearly Newtonian. These latter conditions are portrayed in Fig. 1. We are not aware of permeability measurements for grease thickener, so for the purposes of this paper we have selected values of k to best illustrate the orders of magnitude and the trends predicted by the proposed model for the various cases.

The oil flow rate q O and thickener flow rate q T (per unit depth in the y-direction) are shown below. These quantities represent the flow rate of thickener or oil if the volume element contained the pure components. The mixture volume flow rate q is also given

q O = H 0 v O x dz, q T = H 0 v T x dz, q = φ q O + (1 -φ) q T (14) q T = H 3 12η T p 0 L , q O = - k η O p 0 L 1 1 + exp H * H (1 + exp H * ) -2 √ k (-1 + exp H * ) (15)
For channel flow, effective viscosity based on the Newtonian profile is as follows,

η ef f = - H 3 12q p 0 L . (16)
For the above parameter values, the effective viscosity as measured in this 'thin slit Poiseuille viscometer' would be η ef f = 14.8 Pa.s. The slit viscometer takes into account the flow of oil through the thickener as well as flow of the thickener itself. The viscosity of the same grease measured in a Couette viscometer would be η ef f = (1 -φ) η T = 300 Pa.s. In the limit of large √ k/H the effective viscosity becomes simply the viscosity of the base oil.

C. The Flow of Grease in a Lubricated Contact

The case of a full grease contact

This situation is schematically shown in Fig. 2. The film thickness for a rigid cylinder-plane contact is described by:

h = h 0 + x 2 2R (17)
Note, we are not attempting to model an EHL contact with elastically deforming surfaces. Due to pure rolling conditions the contact points of the two surfaces are equal, and by suitable placement of the coordinates, the velocity of both surfaces can be represented as V . This is highly simplified illustrative example just to show to how the model behaves.

Further refinements such as a layer attached to each surface could be added. We admit to many unrealistic assumptions (such as flooded conditions, half-Sommerfeld exit boundary conditions, a sort of rebound of the grease layer after the minimum point, etc.). We are not trying to solve a complex EHL problem, although there is no reason, in principle, this could not be done.

The viscosity of the thickener is often some 1000 times greater than that of the oil. It can be shown formally through a regular perturbation of a small parameter η O /η T , that Eqs. ( 6) and ( 5), respectively, reduce to:

0 = η T ∂ 2 v T x ∂z 2 (18) ∂p ∂x = η O k v O x -v T x + η O ∂ 2 v O x ∂z 2 (19)
It is reasonable that the Darcy drag of the oil on the thickener (proportional to η O ) is much less than the boundary shear stress (proportional to η T ). The general solution to Eq. ( 18) is

v T x = V 1 + (V 2 -V 1 ) z h(x) (20)
however, for this rolling contact 19) and solving the differential equation by standard methods we obtain,

(V 2 = V 1 ) = V and thus v T x = V . Substituting v T x = V into Eq.(
v O x = dp dx k η O exp z * 1 + exp h * (1 -exp h * ) (exp z * -exp h * ) + v T x (21)
and integration across the film for the oil volume flow rate gives

q O = k η O dp dx 1 1 + exp h * -h (1 + exp h * ) + 2 √ k (-1 + exp h * ) + h 2 (V 1 + V 2 ) (22)
The incompressible oil flow rate is constant. Respecting global continuity as in all developments of Reynolds equation, we differentiate q O with respect to x, and a modified Reynolds equation is obtained:

q O = -f P [h(x), k, η O ] dp dx + f V [h(x)]V, V = V 1 + V 2 2 dq dx = 0, d dx f P dp dx = V f V , (23)
Typical pressure profiles are shown in Fig. 3. The parameter values used are representative of conditions at a position along a high speed train wheel/rail contact are as follows: h 0 = 50 µm, R= 0.5 m, V = 80 m/s, η O = 0.1 Pa.s, η T = 100 Pa.s. The lower and upper curves are for permeability k = 10 -6 m 2 and k = 10 -8 m 2 , respectively. The effect of the grease is to increase lubricant flow resistance and thus increase the hydrodynamic pressure generated.

The case of a contact with a grease layer

V V V R x z H h 0 h(x)
oil flow q U grease layer thickener flow oil flow q L q i Figure 4: Schematic of rolling contact with a grease layer.

The following analysis builds on a ideas put forth in a paper in 1996 by P. Cann [START_REF] Cann | Understanding grease lubrication[END_REF], as mentioned. She idealizes a grease lubricated EHL rolling contact as a grease layer (thickener plus base oil) deposited in the contact track and a layer of bled oil released by the grease. Repeated passes of the cylinder serve to extract the base oil from grease mixture, and the oil serves as the lubricant.

We consider a two-dimensional incompressible flow (1-D bearing) with rigid surfaces. The caveats regarding over-simplification discussed in the previous section are still in force. The grease layer (oil-thickener mixture) of thickness H(x) is attached to the lower surface. These conditions are noted schematically in Fig. 4 for the case of a cylinder-plane rolling contact.

The oil in the upper region obeys,

dp dx = ∂τ U ∂z , τ U = η O ∂v U ∂z . (24)
Likewise the oil in the lower porous medium region (i.e., in the grease) obeys,

dp dx = - η O (v L -V ) k + ∂τ L ∂z , τ L = η O ∂v L ∂z . (25)
By the usual lubrication assumptions, it can be shown that the pressure gradient is constant across the film. The solution of these equations has the form

v U = z 2 2η O dp dx + z C 4 + C 3 (26)
v L = k η O dp dx + C 1 exp(z/ √ k) + C 2 exp(-z/ √ k) + V (27)
The boundary conditions on the oil flow are,

z = 0 : v L = V ; z = h : v U = V U ; z = H : v L = v U ; τ L = τ U (28)
For pure rolling contact the upper surface boundary condition v U = V U = V , while for pure sliding V U = 0. Solving for the four constants is tedious but straightforward, and the resulting expressions are complicated, take up a lot of space, and are not shown.

Some computed results are shown in Fig. 5. The parameter values used are representative of conditions at a position along a high speed train wheel/rail contact, and noted are as follows:

L = √ 2Rh 0 , x = -L/2, h 0 = 50 µm, R = 0.5m, V = 80 m/s, h = 62 µm, H = 31 µm, η O = 0.1Pa.s, k = 2.0 10 -8 m 2 ,
and dp/dx=-1.0 10 9 N/m 3 . The negative pressure gradient propels the oil forward relative to the sliding speed. Grease is present in the lower half of the figure, where the oil velocity is retarded due to the Darcy porous medium effect.

A steady 1-D incompressible modified Reynolds equation is now developed for this grease layer configuration. The volume flow rate is,

q = H 0 v L dz + h H v U dz = -f P [H(x), h(x), k, η O ] dp dx + f V [H(x), h(x), k]V (29)
Reynolds equation is determined from the constancy of the flow rate:

dq dx = 0, d dx f P dp dx = V f V = V dh dx , (30)
As above, determination of f P = f P (H, h, k, η0) is straightforward but complicated, as are the resulting expressions.

An approximate version of Reynolds equation for the grease layer can be written as follows:

d dx H 3 12η T dp dx + H 3 12η T p W 2 = V dH dx (31)
The second term on the left hand side approximately represents loss of grease from the ends (into and out of the plane of the paper). This flow of grease is assumed to be governed by the thickener viscosity at low rates of shear.

In performing the computations that follow, Eq. ( 31) is rewritten as

dH dx = dp 2 dx 2 + p W 2 / 12 η T V H 3 - 1 3H dp dx (32)
and the right hand side terms are treated as known from the conditions upstream. The solution thus involves numerical integration of Eqs. (31) and (32), subject to initial conditions:

x = x in : H = H in , p = 0, dp/dx = p g,in (33)
This is a so-called 'shooting problem' in which p g,in is guessed until p(x out ) = 0.

III. Results

A. Cylinder-plane rolling contact, constant grease layer thickness

To portray the overall model behavior, we use the same simplified representative problem for rigid surfaces and an isoviscous oil. For the first example, we assume the grease layer height H is fixed. In the calculations, we decouple Eq. (32) from Eq. (31). The boundary conditions used are,

L = 2Rh 0 , x = -3L : p = 0; x = 0, p = 0 ( 34
)
The inlet is taken sufficiently far to the left such that the local film thickness h in is much greater than the minimum thickness at the midplane h 0 . Because our goal is to portray the basic model behavior, we assume the pressure in the diverging region is zero (ambient), i.e., the half-Somerfeld condition. Again, we realize these conditions are oversimplified, but they The lower pressure curve is the solution of Reynolds equation for a film entirely of oil, while the upper is for a grease layer with layer thickness H= 45 µm, and permeability coefficient k = 2.0 10 -8 m 2 . The pressure, and thus the load carrying capacity, is considerably enhanced by the grease layer. If the film is same order as the pore size, the model effective viscosity becomes that of the base oil. This result is consistent with the experimental observation of Cousseau et al. 2012 that for EHL (very thin films) the base oil viscosity is a reasonable predictor friction and load.

Another interesting result is shown in Fig. 7. For the same conditions, the figure shows the flow supply from the grease layer to the upper pure bled oil region (q i ), as a fraction of total oil flow (q O ). In the region of positive pressure gradient toward the left side, the grease is supplying oil to the upper layer, while toward the minimum film, oil is being squeezed back into the grease.

B. Cylinder-Plane Rolling Contact, varying grease layer thickness

In this case, we use Eqs.

q i /q
[-] and k = 10 -11 m 2 (upper curve). As in the cases above, porous flow resistance in the grease causes increased pressure. Figure 9 shows the variation of the grease layer under the bearing conditions for the same two cases. In the case of lower permeability, the grease layer thickness is reduced into the contact.

IV. Conclusions

A new model for grease lubrication is proposed, which takes into account the fact that grease is a two-component mixture consisting of a thickener and a base oil. The thickener, rather than providing lubrication itself, is a sort of delivery system for the base oil. The thickener and the base oil interact according to the Darcy-Brinkman porous media model. The goal of the effort is to develop a model with predictive ability in engineering design, rather than to describe grease lubrication in scientific detail for numerical simulations. It is possible that grease lubrication is too complex and too varied from application to application and from lubricant to lubricant that anything other than highly empirical approaches will prove futile. The one-dimensional bearing case (two-dimensional thin film shearing flow) is addressed to focus on the basic physical aspects of the model, but it can be readily extended to the three-dimensional case.

The basic equations of the model are developed consisting of two coupled second order differential equations, one each for the thickener and the oil. If a Newtonian fluid is assumed, the equations are linear and can be solved in closed form. The linear equations are applied to Couette and Poiseuille flow, each of which is used as a device to measure flow properties. A Poiseuille flow device would be the more useful for grease as it would shed some light on the porous flow interactions between the components.

The model is then applied to highly idealized and simplified lubricated contacts. Two cases are noted, one in which the entire contact is flooded with grease, and a second in which the contact contains both a layer of grease and a layer of oil. The layered system loosely follows Cann's physical description of lubricated rolling contact. An increase in pressure (and consequently load carrying capacity) due to the grease is predicted. This increase is less notable as the film thickness approaches the pore size, and in the limit, the base oil behavior is obtained.

Only very basic and highly simplified demonstrative cases are addressed in the present work.

Extensive parametric studies would best await a more highly evolved state of affairs. In particular, experiments to determine the permeability k would seem to be feasible. The basic model physics should be testable by viscosity determination of pure thickener and pure oil, followed by relatively straightforward measurements of grease behavior in Poiseuille flow.

Any number of extensions to the model can be foreseen (3D flow, non-Newtonian flow, property change with pressure and temperature, aging of the thickener properties, etc.), and many additional problems can be attacked (other types of bearings, elastic compliance of surfaces, etc.)

Whether such further studies prove worth the effort is yet to be determined. However, the present model does provide a new framework about which to consider grease modeling.

Figure 1 :Figure 2 :

 12 Figure 1: Velocity profiles for channel flow. Parameters: p 0 = 10 kPa, L= 100 mm, η O = 1 Pa.s, η T 0 = 1000 Pa.s, φ = 0.7, and k = 10 -8 m 2 .

Figure 3 :

 3 Figure 3: Pressure profiles, full grease contact. Parameter values: h 0 = 50 µm, R= 0.5 m, V = 80 m/s, η O = 0.1 Pa.s, η T = 100 Pa.s. The lower and upper curves are for permeability k = 10 -6 m 2 and k = 10 -8 m 2 , respectively.

Figure 5 :

 5 Figure 5: Oil velocity profile, film with grease layer. Parameter values: h 0 = 50 µm, R = 0.5 m, V = 80 m/s, h = 62 µm, H = 31 µm, η O = 0.1 Pa.s, k = 2.0 10 -8 m 2 , and dp/dx = -1.0 10 9 N/m 3 .

Figure 6 : 6 .

 66 Figure 6: Rolling contact pressure distribution, with and without grease layer. Parameter values: h 0 = 50 µm, R = 0.5 m, V = 80 m/s, η O = 0.1 Pa.s. The lower curve is for no grease layer, the upper curve is for H = 45 µm, and k = 2.0 10 -10 m 2 .

  (31)-(32) and (34), along with the left-hand-side boundary condition x = -3L : H = H in . Parameter values are as follows: L = √ 2Rh 0 , x = -L/2, h 0 = 50 µm, R=0.5m, V=80 m/s, η O = 0.1Pa.s. An inlet condition on the grease layer is needed: H in = 0.9h 0 = 45 µm.

Figure 8

 8 Figure8shows the pressure behavior for two cases: permeability k = 10 -9 m 2 (lower curve)

Figure 7 :

 7 Figure 7: Rolling contact, oil flow from grease layer to upper oil layer. Parameter values: h 0 = 50 µm, R = 0.5m, V = 80 m/s, η O = 0.1 Pa.s, H = 45 µm, and k = 2.0 10 -10 m 2 .

Figure 8 :

 8 Figure 8: Rolling contact pressure distribution in which the grease layer has variable thickness, effect of permeability. Parameter values: h 0 = 50 µm, R = 0.5 m, V = 80 m/s, η O = 0.1 Pa.s, H in = 45 µm, permeability k = 10 -9 m 2 (lower curve) and k = 10 -11 m 2 (upper curve).

Figure 9 :

 9 Figure 9: Rolling contact, grease layer thickness variation, effect of permeability. Parameter values: h 0 = 50 µm, R = 0.5 m, V = 80 m/s, η O = 0.1 Pa.s, H in = 45 µm. Permeability k = 10 -9 m 2 (lower curve) and k = 10 -11 m 2 (upper curve).

Accepted Manuscript


Joseph Paturel

Clément Quinson

Motionlab Ml

Martin Quinson

Simon Rokicki

SmolPhone: a smartphone with energy limits

This paper explores the smartphone design space to achieve energy sobriety. At its core, the proposed platform has a compute-limited but energy efficient microcontroller (MCU). Another processor can be attached to execute legacy Linux applications. Some tasks are offloaded to the Wi-Fi network controller. Our design prefers an energy-efficient variant of the 4G cellular network, which is turned off as often as possible. We are in the process of designing a hardware evaluation board to implement the vision presented in this paper. Raspberry Pi Zero RP2040 Screen Keyboard i2c + irq SPI USB Quectel 4G Cat 1 UART Raspberry Pi Zero RP2040 Screen Keyboard i2c + irq SPI USB Quectel 4G Cat 1 UART Raspberry Pi Zero RP2040 Screen Keyboard i2c + irq SPI USB Quectel 4G Cat 1 UART Sleep mode Main operating mode Compatibility / performance mode

I. INTRODUCTION

Commercialized in 1994, the IBM Simon was the first device that blended together a mobile phone and a Personal Digital Assistant (PDA). Later recognized as the first smartphone, it included a calendar, an address book, a notepad and an email client along with voice call support. One could argue that the smartphone history since then has consisted in growing the list of features added to this initial design. For example, the connectivity has seen tremendous improvements with Wi-Fi, GPS, Bluetooth, NFC and 5G to name a few. While historical devices could not handle any video, streaming and augmented reality applications, they are now commonplace. Modern devices also support AI acceleration allowing them to implement features such as voice recognition or image enhancement on-board.

In pace with the feature offering, the smartphone hardware capabilities have dramatically increased over the last few decades. The Nokia 3310, released in 2000, had [START_REF] Systems | Esp32 wi-fi and bluetooth mcu[END_REF] Mb of storage with a 100 MHz ARM7 processor. The iPhone 3GS, released in 2009, had 256 Mb of RAM and 32 Gb of storage with a 600 MHz ARM8 processor and a GPU on chip. The iPhone 14, released in 2022, has 6 Gb of RAM, a SoC encompassing 6 compute core, 5 GPU cores, a neural processing unit and an image signal processor. These increases in hardware performance naturally come with corresponding increases in resource consumption. For example, an early Nokia phone had a carbon footprint of 13.6 kg against 95 kg for the iPhone6 [START_REF] Suckling | Redefining scope: the true environmental impact of smartphones?[END_REF] and up to 116 kg for the iPhone 14 [2].

Battery life is perhaps the only feature that has been steadily decreasing over time. Old devices could easily be used two days on a single charge while many modern phone users struggle with one charge per day, despite the increase of battery capacity over time. In terms of design goals for the existing smartphones, the battery time falls clearly behind the increased feature set. Battery life is merely optimized as a second thought after the primary development efforts, despite the risks of rebound effects inherent to this approach. This paper explores an alternate avenue in the smartphone design space: instead of increasing the number of features possibly at the cost of reduced battery life, we aim for an increased battery life possibly at the cost of a reduced set of available features. Increasing the time between battery charges may also improve the overall device durability, as the performance decrease of batteries due to their aging is a function of charge cycles [START_REF] Cordella | Durability of smartphones: A technical analysis of reliability and repairability aspects[END_REF]. While some efforts achieve a very low energy consumption allowing battery-free calls to be made, they do not offer any features a user might expect from a smartphone-like device [START_REF] Talla | Battery-free cellphone[END_REF]. This solution also relies on custom bridges between their infrastructure and the overall cellular network. On the other hand of the spectrum, other approaches are based around "real" smartphone hardware and lower power consumption by limiting the interaction between the user and the applications [START_REF] Xu | Ultra-lowpower mode for screenless mobile interaction[END_REF].

The objective of this work is not to describe a new product intended for a specific market, but rather to evaluate how such a platform could be designed and built using recent technical and scientific advances.

In the following paper we'll first discuss the energy tradeoffs that one could make in order to improve re-balance the features/sustainability of a smartphone. In Section II, we will present our vision for what we call "SmolPhone", an energy limited smartphone research platform. We will then go over the current state of the projects and our ideas for its future.

II. SMARTPHONE DESIGN ENERGY TRADE-OFFS

The most power-hungry components of a smartphone are the display, the cellular modem and the CPU [START_REF] Carroll | Understanding and reducing smartphone energy consumption[END_REF]. A typical OLED screen consumes 800 mW, a 4G modem (LTE Cat 4) consumes between 600 mW (idle) and 1200 mW (transmitting) and a Wi-Fi modem consumes from 80 mW (idle) to 110 mW (transmitting) [START_REF] Huang | A close examination of performance and power characteristics of 4g lte networks[END_REF]. 5G modems consume over 3000 mW while transmitting [START_REF] Xu | Understanding operational 5g: A first measurement study on its coverage, performance and energy consumption[END_REF]. The 4G variants intended for IoT sensors such as LTE Cat M1 can be used to reduce the power usage (transmitting at 200 mW [START_REF] Masek | Implementation of 3gpp lte cat-m1 technology in ns-3: System sim-ulation and performance[END_REF]), noting that their networking performance is vastly reduced compared to LTE Cat 4 (from Mbps to kbps).

Several technologies are available for low-power screens. Typical OLED screens consume from 3 mW/cm 2 for a black screen to 20 mW/cm 2 at maximal brightness [START_REF] Carroll | Understanding and reducing smartphone energy consumption[END_REF]. Electrophoretic displays (EPD) typical of e-reader devices are compelling in this context because they are bi-stable: Displaying a still image does not consume any energy while changing the content consumes 10 mW/cm 2 at 2 Hz on the screen itself [START_REF] Grosse-Puppendahl | Exploring the design space for energy-harvesting situated displays[END_REF]. The screen controller induces an idle consumption of 2 mW/cm 2 (300 mW in total for 6 in) [START_REF] Waveshare | 6inch e-ink display specifications[END_REF]. EPDs are unfortunately difficult to drive above 3 Hz [START_REF] Yang | E-Paper Displays[END_REF], hindering their usability in a smartphone. Dot-matrix memory LCDs embed memory cells in an LCD to remove the need to constantly refresh the screen, reaching a consumption of 2 µW/cm 2 for a monochrome screen or 0.1 mW/cm 2 in full color, along with fast refresh rates [START_REF] Grosse-Puppendahl | Exploring the design space for energy-harvesting situated displays[END_REF].

It is very common to offload computations to remote servers in an attempt to reduce the power consumption of the user's device [START_REF] Nguyen | A smartphone perspective on computation offloading-a survey[END_REF]. Computations can either be offloaded to a cloud datacenter or closer to the user in an edge compute node. Either way, offloading proves challenging since the latency induced by the network may greatly hinder the performance. Moreover, the energy footprint of the communication often offsets any energy savings on the local CPU, even when the energy consumption of the remote computation is not taken into account.

Cloud-based executions yields better energy savings when the data is handled remotely before being downloaded on the smartphone, instead of being uploaded from the smartphone to the cloud [START_REF] Barbera | Mobile offloading in the wild: Findings and lessons learned through a real-life experiment with a new cloud-aware system[END_REF]. This approach is particularly adapted when the data to download needs to be filtered or transformed remotely.

III. THE SMOLPHONE VISION

In this project, we are tackling two key challenges:

• Minimizing System Power Consumption: We are exploring the extent to which we can reduce the system's power consumption by strategically shutting down the application CPUs, peripherals and phone connectivity. Our approach involves utilizing various sensors to precisely measure the costs associated with different operations and duty cycles in real-world usage scenarios.

• Trimming Phone Functionality While Preserving Core

Features: Another aspect of our ongoing prototype development involves investigating how much we can trim down the phone's functionality while still maintaining its essential features. The aim here is to create a tool that allows us to experiment with different levels of service quality degradation to determine what is deemed acceptable and what is not. By addressing these challenges, we aim to develop innovative solutions that optimize power efficiency and user experience on mobile devices, ultimately contributing to a more sustainable and user-centric mobile technology ecosystem.

Our goal is to design a device that offers many smartphonelike features within tight energy limits, we call it "Smol-Phone".

Instead of offloading computations to remote servers for uncertain energy savings, the core of our proposal is to offload computations to other chips on the platform. Our design uses a small microcontroller (MCU) as the central compute core. Such MCUs usually only have hundreds of kb in RAM, limited computational power and no memory management unit (MMU). The lack of MMU forbids the use of any OSes like Linux, so when the user requests a computation that requires MMU support (for instance to execute a legacy application), it is offloaded to a more power-hungry application processor physically connected to the MCU. Our goal is to restrict the usage of the application processor to further save energy while providing some access to applications that cannot be executed on the MCU. In some sense, this design decision opposes the classical Big-Little architecture [START_REF] Li | Energy-efficient execution for repetitive app usages on big.little architectures[END_REF] that sometimes offloads computation from a fast but energy-consuming compute core to a nearby slower but sparing cores. In comparison, we call our design Tiny-Small, where applications are offloaded to high-performance cores when needed and ran on energyefficient ones by default.

Because of the very limited performance of the central MCU, we want to leverage all the computing power that can be found around the board. For example, Wi-Fi communications are typically implemented using a dedicated chip that embeds both a Wi-Fi modem as well as a support MCU such as the Espressif ESP32 [START_REF] Lab | [END_REF] or the Bouffalo Lab BL602 [START_REF] Systems | Esp32 wi-fi and bluetooth mcu[END_REF]. Similarly, cellular chips often include a processor core and subsystem that can decode mp3 files (e.g., to play ringtones), access an SD card and encode/decode audio streams efficiently. Since some dedicated chips can run arbitrary code on their MCU, our vision is to replace the large System-on-Chips that are typical in smartphones with a collection of specialized chips. For example, the TCP stack will only be implemented in the networking chips, relieving the central MCU of all network protocol handling, and thus making it possible to implement the whole system on low-power and very limited hardware. This design borrows ideas both from the Computing Continuum paradigm (envisioning huge federations of computing nodes, from tiny microcontrollers up to the clouds [START_REF] Beckman | Harnessing the Computing Continuum for Programming Our World[END_REF]), and of the microservices paradigm (where applications are split in a set of interacting small software components [START_REF] Butzin | Microservices approach for the internet of things[END_REF]).

In addition, we plan to use an input proxy, a dedicated program located in the cloud whose goal is to reduce the load on the device as in [START_REF] Barbera | Mobile offloading in the wild: Findings and lessons learned through a real-life experiment with a new cloud-aware system[END_REF]. For example, HTML pages accessed by the device would be rendered in the cloud before being downloaded, to alleviate the fact that the device's central MCU cannot host a complete HTML5 rendering engine. This mechanism could also be used to increase the accessibility of the web through other appropriate filters. The input proxy will also make it possible to shut down the network connectivity when the user is not pulling information from the Internet, as proposed in [START_REF] Zhong | Smert: Energy-efficient design of a multimedia messaging system for mobile devices[END_REF]. Push notifications such as incoming emails or chat messages could trigger SMS notifications from the proxy, that are conveyed on the cellular network even when the mobile data connectivity is turned off to save energy on the device. Upon reception, these notifications would wake up the networking support system to fetch the incoming data.

One may wonder whether an MCU is a sensible choice for the central element of the SmolPhone. First, these chips are highly optimized for energy consumption: a RP2040 consumes 100 mW per busy core at 133 MHz and can very quickly enter a sleep mode in which the power consumption is negligible [START_REF] Zhao | Understanding energy efficiency in IoT app executions[END_REF]. In addition, modern MCUs actually present performance figures that are comparable to ones of full-blown Fig. 1. First version of the SmolPhone prototype. The system has different operating modes which depend on its utilization (Sleep, Main operating, and Compatibility). For each mode, components and busses are either turned 'off' (red -dashed), or 'on' (green -plain), or used partially (yellow -plain). application processors of years past. A RP2040 achieves 246 CoreMark per core @133 MHz while the Pentium 1 CPU (released in 1993) achieved 281 CoreMark. The 2040 can easily be overclocked to 400 Mhz to achieve 859 CoreMark per core, while the Pentium II (1997) with its 1080 CoreMark used to run Windows 98. By comparison, the Raspberry Pi Zero, which we plan to use as our application CPU for its MMU abilities, achieves 2084 CoreMark. This is why we think that this MCU will deliver the power needed to offer many more features than the IBM Simon.

The size and complexity of the needed software present challenges for our SmolPhone design. Larger and more complex systems were designed for older devices, but they usually required an inordinate development effort. In addition, embedded programming poses specific challenges due to the lack of an operating system. Recent works show the benefits of the Rust language in this context: compile-time verification partially compensates the lack of a MMU and runtime safety guards [START_REF] Nilsson | Abstraction layers and energy efficiency in tockos, a rust-based runtime for the internet of things[END_REF]. Furthermore, resolving memory management issues during compilation results in more optimized applications, in particular when these applications perform complex data management. Our effort will also build upon many current efforts in the Rust community to provide industrial-class solutions for embedded development.

The device will be fully open-sourced (hardware and software) to ease the exploration of the tradeoffs toward digital sobriety by ourselves and others.

IV. CURRENT STATE AND FUTURE WORK

We are currently designing a hardware prototype around a RP2040 dual-core microcontroller, an LCD memory display, a Wi-Fi chip, and a mechanical keyboard from a Blackberry Q20 phone. Our design is modular: neither the application CPU nor the cellular modem are included on the board. They come instead as extension boards connected through USB or other communication busses, so that several alternatives can be evaluated. For the application CPU we plan to use a Raspberry Pi Zero processor, which is a small, low-power processor that can run Linux and Android applications in a sandbox [START_REF] Waydroid | android in a linux container[END_REF]. The modem used is a Quectel EC200, a low-power 4G modem.

The system has three operating modes. In sleep mode, both the RP2040 and the Raspberry Pi Zero are turned off, waiting for a wake-up signal coming from the UI or the network. The modem only listens for GSM and can wake the system up if the cloud-based input-proxy sends a signal. In the main operating mode, only the RP2040 is turned on and offers the main functionality needed by the end user. The compatibility/performance mode is used when the user wants to execute a legacy application, or any task too heavy for the RP2040. The RP2040 only acts as a driver for the keyboard and the main CPU takes control of the system.

The goal is to maximize the smartphone functionalities handled by the main operating mode, reducing the active time of the Raspberry Pi Zero.

Additionally, to help us model and record the energy consumption of each feature/component of the SmolPhone, the current drawn by every major component on the board is recorded. This will allow us to correlate the use of features to their energy impact and understand what needs to be prioritized in the design of applications (does it matter if one receives their text-message notification at the very second it was received for example).

Another important aspect of any modern electronic device is security. While we can rely on Rust to ensure memory safety on the RP2040, and sandboxing for applications running on the CPU, we still need to look into the communication channels between the different board components. The interfaces used to connect the application processor to the low-power MCU should be secure in case personal tokens or any kind of credentials are exchanged between the two.

V. CONCLUSION

In this paper, we have introduced our vision for an energylimited smartphone-like device: the SmolPhone. At its core, the SmolPhone philosophy is based on a power consumption conscious connection of execution layers that can be turned on and off function of the level of features an application needs. We have presented a solution that relies on firmware and OS-level mechanics, enabling an efficient energy usage and studies the benefits of trimming down the user experience. Our future work include a security survey of our solution and an integration of bio-sourced components into the hardware of the device.


GULLY FORMATION ON MARS A Noblet
email: anoblet@uwo.ca

G R Osinski

S J Conway

GULLY INCISION DEPTHS IN MANTLE AND BEDROCK IN UTOPIA PLANITIA: IMPLICATIONS FOR

Introduction: Gullies on Mars are geologically young landforms that resemble terrestrial gullies [1]. Martian gullies are often carved within a mantling unit in the mid-to high-latitudes of Mars [START_REF] Dickson | Lunar Planet Sci Conf[END_REF]. This mantle is a relief-muting unit [START_REF] Kreslavsky | [END_REF] that is proposed to be composed of a mixture of dust and water ice that recurrently deposited during Mars' periodic high-obliquity excursions [4]. The formation mechanism of gullies is key to understand Mars' recent climate: previous works, focused on gullies carved into the mantling unit, have suggested meltwater sourced from the volatile-rich surface as one plausible gully-forming scenario (e.g. [5,6]). Other works, focused on the mantle, have studied gullies to infer the thickness of the mantle deposits [7] and provide constraints on Mars' near surface water ice volume. To test the effect of the substrate nature and slope-orientation on gully formation, we inspect individual gully alcoves in Utopia Planitia in the northern plains of Mars, record their orientation, the nature of the substrate, and calculate their incision depths.

Approach: We consider gullied hillslopes in Utopia Planitia, a region with abundant occurrences of ice-rich mantling material [8]. We used HiRISE images at 25-50cm/pixel [9] in a sinusoidal projection as a basemap. We inspected every gullied hillslope that we previously mapped in a previous abstract [START_REF] Noblet | Lunar Planet Sci Conf[END_REF] using [START_REF] Harrison | [END_REF]'s gullied landforms global dataset (Figure 2). We focused our study on individual gully alcoves displaying a V-shaped incision profile with straight walls (Figure 1). We recorded the substrate nature, either mantle deposits or bedrock, and we measured the width of the alcove wall at its widest part, We calculated the associated incision depth, assuming a 20° cross-profile slope as observed by [12]. Finally, we record the upslope to downslope orientation value of each studied alcove.

Results:

We have mapped and calculated incision depths for 542 gully alcoves, of which 306 carved in mantle substrate and 236 in bedrock. The orientation distribution of alcoves is bi-modal (Figure 3), depending on the substrate: mantle carved alcoves are pole-facing while bedrock carved alcoves are almost exclusively equator-facing. This trend reflects a globalscale trend of the mantling unit being preferentially observed on pole facing slopes [13,14].

We observe that alcove wall widths in adjacent gullies in the same type of substrate can vary by a factor of 10, which explains the important scatter of our individual data points (Figure 4). Incision depths derived for a 20° alcove wall slope are comparable in bedrock and mantle alcoves (respectively 2-46 m and 2-59 m), though they are deeper overall in mantle alcoves (median=15.6 m) than in bedrock (median=13 m). This result was expected, as the mantle deposits provides a more erodible substrate than bedrock, which results in more dramatic topographic expressions of gullies [15]. However, we note that on the contrary between ~43°N and 52°N bedrock alcoves are on average deeper than mantle alcoves. This latitudinal band of shallow mantle alcoves also overlaps with a region between 37° and 47°N where only 8 occurrences of mantle alcoves are observed.

For gully alcoves emplaced in mantle material, incision depths values do not show clear positive correlation with increasing latitude as opposed to what [12] noted. Instead, we observe maximum depths (up to 59m) centered around 37°N and 52°N, and shallower depths (up to 20m) between 37°N and 47°N, as well as poleward of 55°N.

Discussion: We have shown that alcove incision depth in mantle material in Utopia Planitia does not increase with increasing latitude, contrary to the global trend of increasing mantle thickness with increasing latitude noted by [12], and have observed a region with few and shallow mantle alcoves between 37° and 52°N. We interpret this as indicative of local variations in the mantling material's physical properties on hillslopes in this latitudinal band, with shallower and less widespread mantle between 37°N and 52°N. This interpretation is in line with the identification of a stratified ice-rich sedimentary unit between ~38°N to 47°N in Utopia Planitia by [16]. This mid-latitude unit is older (~11Ma age), more degraded, and is interpreted to have different geological characteristics from the Latitude Dependent Mantle (~1.5Ma age).

We have shown that gullied slopes in Utopia Planitia follow a global trend in terms of substrate and orientation, with mantle carved alcoves occurring preferentially on pole-facing slopes. In addition, we observed that gully alcoves are found in almost equal parts in the mantling unit and in the bedrock. Our results neither prove or disprove the top-down melting scenario proposed by [6], where gully formation initiates via the melting of the mantling unit and ultimately results in the exposure of the underlying bedrock. Indeed, during mapping we observed degraded mantle remnants on the lower portion of equator-facing slopes, which could suggest that the mantling unit was removed in the upper portion of these slopes. However, we also observed 'pristine' equator-facing gullied slopes with no evidence of past mantling, as well as fresh-looking channels within bedrock carved alcoves. We suggest that another process is required to explain these morphologies in the absence of a volatile-rich mantling unit, for example the melting of recent snowpack or CO2-ice driven mechanisms [17].

Conclusion and further work:

We suggest that gully alcoves' average depths reflect the ice-rich mantle physical properties variations at the regional scale of Utopia Planitia, but large variability at the local scale suggest local controls and potentially local processes for gully development.

Figure 2 :

 2 Figure2: Map of gullied hillslopes in Utopia Planitia mapped by[13].

Figure 1 :

 1 Figure 1: Mapping method for recording the orientation of gully alcoves in white, and the alcove wall length in red. a) Mantle alcove, ESP_017181_2405. b) Bedrock alcove, ESP_017720_2200.

Figure 3 :

 3 Figure 3: a) Gully alcoves count as a function of alcove direction. b) Average incision depth for a 20° wall slope as a function of alcove direction.

Figure 4 :

 4 Figure 4: Incision depth for bedrock alcoves non-bedrock as a function of latitude. Individual data points correspond to depths calculated for an alcove wall slope of 20°. The thick lines correspond to the average incision depth in 5° latitude bins for a 20° wall slope value.

Acknowledgments: We acknowledge the support of the Canadian Space Agency Mission Co-Investigator Program and the French space agency, CNES.