The Elastohydrodynamics of Nonconforming and Conforming Contacts J

PHYSICS OF FLUIDS 17, 092101 ͑2005͒

Soft lubrication: The elastohydrodynamics of nonconforming and conforming contacts J. M. Skotheim and L. Mahadevan Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge CB3 0WA, United Kingdom and Division of Engineering and Applied Sciences, Harvard University, Pierce Hall, 29 Oxford Street, Cambridge, Massachusetts 02138 ͑Received 13 December 2004; accepted 13 May 2005; published online 2 September 2005͒ We study the lubrication of fluid-immersed soft interfaces and show that elastic deformation couples tangential and normal forces and thus generates lift. We consider materials that deform easily, due to either geometry ͑e.g., a shell͒ or constitutive properties ͑e.g., a gel or a rubber͒, so that the effects of pressure and temperature on the fluid properties may be neglected. Four different system geometries are considered: a rigid cylinder moving parallel to a soft layer coating a rigid substrate; a soft cylinder moving parallel to a rigid substrate; a cylindrical shell moving parallel to a rigid substrate; and finally a cylindrical conforming journal bearing coated with a thin soft layer. In addition, for the particular case of a soft layer coating a rigid substrate, we consider both elastic and poroelastic material responses. For all these cases, we find the same generic behavior: there is an optimal combination of geometric and material parameters that maximizes the dimensionless normal force as a function of the softness parameter ␩=hydrodynamic pressure/elastic stiffness =surface deflection/gap thickness, which characterizes the fluid-induced deformation of the interface. The corresponding cases for a spherical slider are treated using scaling concepts. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1985467͔

I. INTRODUCTION tions, here we address a slightly different set of questions: How can one generate lift between soft sliding surfaces to The reduction of friction in practical applications has increase separation and reduce wear? What is the role of been studied since antiquity. Pictographs found in Uruk, lo- geometry in determining the behavior of such systems? How cated in modern day Iraq, have been dated to ca. 3000 B.C. do material properties influence the elastohydrodynamics? and illustrate the transition from sleds to wheels.1 While this Are there optimal combinations of the geometry and the ma- advance certainly reduced friction, further reductions were terial properties that maximize the lift force? Can the study possible upon the introduction of a viscous lubricating fluid of soft lubrication lead to improved engineering designs? in the axle joints. The theoretical underpinnings of fluid lu- And finally, how is soft lubrication relevant to the study of brication in such geometries can be traced back to the work real biological systems? of Reynolds,2 who studied the mechanics of fluid flow To address some of these questions, we start with the through a thin gap using an approximation to the Stokes’ simple case of two fluid-lubricated rigid nonconforming sur- equations, now known as lubrication theory. Recent efforts in faces sliding past one another at a velocity V as shown in this technologically important problem have focused on Fig. 1. The force on the slider is dominated by viscous modifications of Reynolds’ lubrication theory to account for stresses and pressure in the narrow contact zone. For a New- elastohydrodynamic effects ͑elastic surface deformation due tonian fluid, the Stokes equations ͑valid in the gap͒ are re- to fluid pressure͒, piezoviscous behavior ͑lubricant viscosity versible in time t, so that the transformation t→−t implies change due to high pressure͒, and thermoviscous behavior the transformations of the velocity V→−V and the normal ͑lubricant viscosity change due to frictional heating͒.1,3–6 force L→−L. In the vicinity of the contact region noncon- Inspired by a host of applications in physical chemistry, forming surfaces are symmetric which implies that these polymer physics, and biolubrication, in this paper we focus flows are identical and therefore L=0. Elastohydrodynamics on the elastohydrodynamics of soft interfaces, which deform alters this picture qualitatively. In front of the slider the pres- easily thereby precluding piezoviscous and thermoviscous sure is positive and pushes down the substrate, while behind effects. There have been a number of works in these areas in the slider the pressure is negative and pulls up the substrate. the context of specific problems such as cartilage As the solid deforms, the symmetry of the gap profile is biomechanics,7–9 the motion of red blood cells in broken, leading to a normal force that pushes the cylinder capillaries,10–18 the elastohydrodynamics of rubber-like away from the substrate. elastomers,19,20 polymer brushes,21,22 and vesicles.23,24 An- This picture applies naturally to soft interfaces, which other related phenomenon is that of a bubble rising slowly arise either due to the properties of the material involved, as near a wall; the fluid pressure deforms the bubble’s surface, in the case of gels, or the underlying geometry, as in the case which leads to a hydrodynamic force repelling the bubble of thin shells. In Sec. III, we study the normal-tangential from the wall.25,26 Instead of focusing on specific applica- coupling of a nonconforming contact coated with a thin com-

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FIG. 1. A solid cylinder moves through a liquid of viscosity ␮ above a thin gel layer of thickness Hl that covers a rigid solid substrate. The asymmetric pressure distribution pushes down on the gel when the fluid pressure in the gap is positive while pulling up the gel when the pressure is negative. The FIG. 2. Overview of the different geometries and elastic materials consid- asymmetric traction breaks the symmetry of the gap thickness profile h͑x͒ ered. G and ␭ are the Lamé coefficients of the linear elastic material, where ␭ thus giving rise to a repulsive force of hydrodynamic origin. The pressure G/ =0 corresponds to an incompressible material, Hl is the depth of the ␶ profile and the gap thickness shown here are calculated for a thin elastic elastic layer coating a rigid surface, lc is the contact length, p is the time layer ͑Sec. III͒ for a dimensionless deflection ␩=10. scale over which stress relaxes in a poroelastic medium ͑a material composed of an elastic solid skeleton and an interstitial viscous fluid͒,and␶ =lc /V is the time scale of the motion. Section III treats normal-tangential coupling of nonconforming contacts coated with a thin compressible elastic pressible elastic layer. If the gap profile prior to elastic de- layer. Section IV treats normal-tangential coupling of higher order degenerate contacts coated with a thin compressible elastic layer. Section V treats formation is parabolic in the vicinity of the contact, the con- normal-tangential coupling of nonconforming contacts coated with a thick tact is nonconforming. However, if a parabolic description compressible elastic layer. Section VI treats normal-tangential coupling of prior to the deformation is insufficient we refer to the contact nonconforming contacts coated with an incompressible elastic layer. Section as conforming, e.g., the degenerate case considered in Sec. VII treats normal-tangential coupling of nonconforming contacts coated with a thin compressible poroelastic layer. Section VIII treats normal- IV. Section V treats normal-tangential coupling of noncon- tangential coupling of nonconforming contacts between a rigid solid and a forming contacts coated with a thick compressible elastic cylindrical shell. Section IX treats elastohydrodynamic effects due to coat- layer. In Sec. VI, we consider the normal-tangential coupling ing a journal bearing with a thin compressible elastic layer. Section X treats of nonconforming contacts coated with an incompressible elastohydrodynamic effects for three-dimensional flows using scaling analysis. In the Appendix, we consider rotation-translation coupling of noncon- elastic layer. In Sec. VII, we treat the normal-tangential cou- forming contacts coated with a thin compressible elastic layer. A poroelastic pling of nonconforming contacts coated with a thin com- material is composed of an elastic solid skeleton and an interstitial fluid. In pressible poroelastic layer which describes a biphasic mate- the general case, where the fluid is free to flow relative to the solid, a rial composed of an elastic solid matrix and a viscous fluid.27 composite material cannot be both poroelastic and incompressible. In Sec. VIII, we study the normal-tangential coupling of a nonconforming contact where one solid is rigid and the other is a deformable cylindrical shell. In Sec. IX, we study a ric and material parameters that maximizes the dimension- conforming contact: a journal bearing coated with a thin less lift. Whether or not the dimensional lift has a maximum compressible elastic layer. Finally, in Sec. X, we treat the depends on the control parameter: the normal force increases elastohydrodynamics of three-dimensional flows using scal- monotonically with the velocity, but has a maximum as a ing analysis. Figure 2 provides an overview of the different function of the effective elastic modulus of the system. This geometries and elastic materials considered. In the Appendix, suggests that a judicious choice of material may aid in the we study the coupling between rotation and translation in a generation of repulsive elastohydrodynamic forces thereby soft contact and uncover an elastohydrodynamic analog of reducing friction and wear. the reciprocal theorem for viscous flow. Our detailed study of a variety of seemingly distinct physical systems allows us to clearly observe their similari- ties and outline a robust set of features, which we expect to II. FLUID LUBRICATION THEORY see in any soft contact. In all the cases studied, the normal force=contact areaϫcharacteristic hydrodynamic pressure We consider a cylinder of radius R moving at a velocity ϫL͑␩͒, where L͑␩͒ is the dimensionless lift and the V, rotating with angular frequency ␻, and immersed com- softness parameter ␩=͑hydrodynamic pressure/elastic pletely in a fluid of viscosity ␮ and density ␳ as shown in stiffness͒f͑geometry͒ϭelastic surface displacement/ Fig. 1. The surfaces are separated by a distance h͑x͒, the gap characteristic gap thickness. Tables I and II summarize our profile, where the x direction is parallel to the solid surface in results for ␩Ӷ1. Increasing ␩ increases the asymmetry of the direction of motion and the z direction is perpendicular to the gap profile which results in a repulsive elastohydrody- the solid surface. We assume that the velocity and pressure namic force, i.e., in the generation of lift forces. However, field are two dimensional and in the region of contact we use increasing ␩ also decreases the magnitude of the pressure a parabolic approximation, valid for all nonconforming con- distribution. The competition between symmetry breaking, tacts, for the shape of the cylindrical surface in the absence which dominates for small ␩, and decreasing pressure, domi- of any elastic deformation. Then the total gap between the nant at large ␩, produces an optimal combination of geomet- cylinder and the solid is given by

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TABLE I. Summary of results for two-dimensional ﬂows with small surface deﬂections.

Geometry Material Surface displacement Lift force/unit length

Thin layer Compressible ␮V H R1/2 3ͱ2␲ ␮2V2 H R3/2 elastic solid ͱ2 l l 2G + ␭ 3/2 ␭ 7/2 h0 4 2G + h0

Thin layer with Compressible 3/4 2 2 9/4 a ␮V H R 351␲ ␮ V H R degenerate contact elastic solid l l ␭ 7/4 ͱ ␭ 17/4 2G + h0 784 2 2G + h0

␮ 5/6 2 2 5/2 V HlR ␮ V H R 0.8859 l ␭ 11/6 ␭ 9/2 2G + h0 2G + h0

Soft slider Elastic solid 1 ␮V͑␭ +2G͒ R 3␲2 ␮2V2͑␭ +2G͒ R2 ␲ ͑␭ ͒ ͑␭ ͒ 3 2 G + G h0 8 G + G h0

Thickness ϳͱRh Incompressible 0 1 ␮V R C ͑␨͒ ␮2V2 R2 elastic solid i ␲ ␲ 3 2 G h0 2 G h0

Thin layer Poroelastic ␮V H R1/2 ␮2V2 H R3/2 ͱ2͑1−␣͒ l C ͑␥͒͑1−␣͒ l ␭ 3/2 p ␭ 7/2 2G + h0 2G + h0

Cylindrical shell Elastic solid ␮V͑␭ +2G͒ R7/2 h ␮2V2͑␭ +2G͒ R9/2 3ͱ2␲2 6ͱ2␲2C ͩ 0 ͪ ͑␭ ͒ 3 1/2 s ͑␭ ͒ 3 5/2 G + G hs h0 R G + G hs h0

Journal bearing Elastic solid H ␮R2␻ ␮␻2R2 H R3 thin layer l C ͑⑀ ͒ l 2͑ ␭͒ j x ␭ 5 h0 2G + 2G + h0

aUpper row corresponds to n=2, while the lower row corresponds to n=3 and the undeformed dimensionless gap thickness proﬁle is h=1+x2n.

4͒͑ . ץ + ץ=x2 0 ͑ ͒ ͩ ͪ ͑ ͒ ͑ ͒ xvx zvz h x = h0 1+ + H x , 1 2h0R with H being the additional elastic deformation and h0 the The associated boundary conditions are characteristic gap thickness in the absence ofͱ the solid deformation. The size of the contact zone, lc = 2h0R, characterizes the horizontal size over which the lubrication forces are ͉v ͉ =−V, ͉v ͉ ͑ 2 ͒ =−␻R, important, consistent with the parabolic approximation in x z=−H x z=h0+ x /2R ͑ ͒ Ӷ 1 .Ifh0 R, the gap Reynolds number Reg ͑␳ 2 ͒ ͑␮ 2͒ϳ␳ 3/2 ␮ 1/2Ӷ␳ ␮ = V /lc / V/h0 Vh0 / R VR/ =Re, the nomi- Ӷ nal Reynolds number. Then, if Reg 1, we can neglect the 2 H, ͉v ͉ ͑ 2 ͒ =−x␻, ͑5͒ ץv ͉ = V͉ inertial terms and use the lubrication approximation to de- z z=−H x z z=h0+ x /2R scribe the hydrodynamics. For a two-dimensional velocity ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ field v= vx x,z ,vz x,z and a pressure field p x,z , the fluid stress tensor is ͉ ͉ ͉ ͉ ,T͒ ͑ ͒ p x→ϱ =0, p x→−ϱ =0 ١ ␴ ␮͑١ f = v + v − pI. 2 ␴ ٌ Stress balance in the fluid, · f =0, yields -where we have chosen to work in a reference frame translat ץ 0= zp, ing with the cylinder. We note that the boundary conditions ͑3͒ ͑5͒ assume that the elastic layer is impermeable to fluid. We . v ץp + ␮ ץ−=0 x zz x make the variables dimensionless with the following defini- Mass conservation implies tions:

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TABLE II. Summary of results for three-dimensional ﬂows with small surface deﬂections.

Geometry Material Lift force

␮2 2 2 Thin layer Compressible V HlR elastic solid ␭ 3 2G+ h0 ␮2 2 4−͑2/n͒ Thin layer with Compressible V HlR degenerate contact elastic solid 2G+␭ 5−͑2/n͒ h0 Soft slider Elastic solid ␮2V2 R5/2 5/2 G h0 տͱ ␮2 2 5/2 Thickness Rh0 Incompressible V R elastic solid 5/2 G h0 Ӷͱ ␮2 2 3/2 Thickness Rh0 Incompressible V HlR elastic solid 5/2 G h0 ␮2 2 2 Thin layer Poroelastic V HlR ␭ 3 2G+ h0 Cylindrical shell Elastic solid ␮2V2 R4 h ӷh 5/2 3/2 s 0 G hs h0 Cylindrical shell Elastic solid ␮2V2 R5/2 Շ 3/2 hs h0 G h h s 0 FIG. 3. Schematic diagrams of mathematically identical configurations. ␮␻2 2 4 Configuration ͑a͒ is treated in the text. Journal bearing Elastic solid R HlR thin layer ␭ 5 2G+ h0 related to the compliance of the elastic material. The dimensionless version of the fluid stress tensor ͑2͒ is ץ␧3 ץ␧ ץ␮͑ ␻ ͒ ␧2 lc V − R − p +2 xvx zvx + xvz x = l xЈ, z = h zЈ, p = p pЈ = pЈ, ␴ = ͩ ͪ, ͑8͒ ץ␧2 ץ␧3 ץc 0 0 2 f ␧ h0 zvx + xvz − p +2 zvz where ␧=ͱh /2R. Solving ͑7͒ gives the Reynolds equation28 l ␮͑V − ␻R͒ 0 ␴ = p ␴Ј = c ␴Ј, h = h hЈ, H = H HЈ, ͑6͒ p͒, ͑9͒ ץ6h + h3͑ ץ=f 0 f 2 f 0 0 0 h0 x x subject to ͑V − ␻R͒h ͑ ␻ ͒ Ј 0 Ј vx = V − R vx, vz = vz . p͑ϱ͒ = p͑− ϱ͒ =0. ͑10͒ lc Note that ␻Ј is scaled away. Here, h is the gap profile given Here, H ͑p /EЈ,l /l ,l /l ,...͒ is the characteristic scale of 0 0 2 1 3 1 by ͑1͒ in dimensionless terms the deflection, where EЈ is the effective elastic modulus of ͑ ͒ 2 ␩ ͑ ͒ ͑ ͒ the medium and li are the length scales of the system. The h x =1+x + H x . 11 pressure scaling follows from ͑3͒ and the fact that xϳl c To close the system we need to determine ␩H͑x͒, the elastic =ͱ2h R, the size of the contact zone. Then, after dropping 0 response to the hydrodynamic forces. This depends on the the primes the dimensionless versions of Eqs. ͑3͒–͑5͒ are detailed geometry and constitutive behavior of the cylindri- ץ 0= zp, cal contact. In the following sections, we explore various configurations that allow us to explicitly calculate ␩H, thus ץ ץ 0=− xp + zzvx, allowing us to calculate the normal force on the cylinder,

ץ ץ 0= xvx + zvz, L = ͵ pdA, ͑12͒ contact area −1 − ␻Ј ͉ ͉ ␩ = , ͉ ͉ 2 = , ͑7͒ and determine the elastohydrodynamic tangential-normal vx z=− H ␻Ј vx z=1+x ␻Ј 1− 1− coupling. We note that when ␩=0, the contact is symmetric ͑11͒ so that the form of ͑9͒ implies that p͑−x͒=−p͑x͒ and ␻Ј ץ␩ xH −2x L=0. ͉v ͉ ␩ = , ͉v ͉ 2 = . z z=− H 1−␻Ј z z=1+x 1−␻Ј III. ELASTIC “LUBRICATION” THEORY: ͉ ͉ ͉ ͉ p x→ϱ =0, p x→−ϱ =0. DEFORMATION OF A THIN ELASTIC LAYER Here ␻Ј=␻R/V characterizes the ratio of rolling to sliding, In our ﬁrst case, we consider a thin elastic layer of thick- ␩ the softness parameter, =H0 /h0, characterizes the scale of ness Hl coating the cylinder or the rigid wall or both, all of the elastic deformation relative to the gap thickness, which is which are mathematically equivalent ͑Fig. 3͒. Our descrip-

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͒ ͑ ␴ ١ tion of the layer as thin means that the layer is thin compared · s =0. 14 to other length scales in the problem such as the contact We make the equations dimensionless using length lc. The layer thickness Hl is smaller than the length ͱ Ј Ј Ј Ј ␴ ␴Ј scale of the hydrodynamic contact zone, lc = 2h0R, i.e., Hl z = Hlz , x = lcx , ux = h0ux, uz = h0uz s = p0 s . Ӷ lc. We first turn our attention to determine the surface de- ͑15͒ flection of the layer for an arbitrary applied traction. Throughout the analysis, we assume that the surface deflec- We note that the length scale in the z direction is the depth of Ӷ the layer H ; the length scale in the x direction is the length tion H0 Hl so that a linear elastic theory suffices to describe l ␴ ͱ the material response. The stress tensor s for a linearly scale of the hydrodynamic contact zone, lc = 2h0R; the dis- elastic isotropic material with Lamé coefficients G and ␭ is placements ux and uz have been scaled with the characteristic gap thickness h0; and the stress has been scaled using the hydrodynamic pressure scale following ͑6͒. We take the uI, ͑13͒ thickness of the solid layer to be small compared to the · ١ uT͒ + ␭ ١ + ␴ = G͑١u s ␨ Ӷ length scale of the contact zone with =Hl /lc 1 and restrict our attention to compressible elastic materials, where Gϳ␭. ͑ ͒ where u= ux ,uz is the displacement field and I is the iden- Then, after dropping primes, the dimensionless two- tity tensor. Stress balance in the solid implies dimensional form of the stress tensor ͑13͒ is

␭ G G u ץ u + ␨ ץ u ץu + ␨ ץ 1 2G + ␭ z z x x 2G + ␭ z x 2G + ␭ x z ␴ = . ͑16͒ s ␩΂ G G ␭ ΃ u ץ u + ␨ ץ u ץ u + ␨ ץ 2G + ␭ z x 2G + ␭ x z z z 2G + ␭ x x

͑ ͒ ͑ ͒ ͑ ͒ Here, uz x,− 1 =0, ux x,− 1 =0. 23 ␮͑ ␻ ͒ 1/2 p Hl V − R HlR ͑ ͒ ͑ ͒ ͑ ͒ ␩ = 0 = ͱ2 ͑17͒ Solving 19 , 22 , and 23 gives us the displacement of the ␭ ␭ 5/2 2G + h0 2G + h0 solid-fluid interface ͑ ͒ ͑ ͒ ␩ ␩ ͑ ͒ ͑ ͒ is the softness parameter, a dimensionless number governing ux x,0 =0, uz x,0 =− p =− H x . 24 the relative size of the surface deflection to the undeformed gap thickness. Stress balance ͑14͒ yields This linear relationship between the normal displacement and fluid pressure is known as the Winkler or “mattress” ␭ ␭ 29 elastic foundation model. In light of ͑24͒, we may write the ץͪ ␨2ͩ ץͪ ␨ͩ ץ zzux + 1+ xzuz + 2+ xxux =0, G G gap profile ͑11͒ as ͑ ͒ 18 2 ␩ ͑ ͒ G + ␭ G h =1+x + p. 25 ץ ␨2 ץ ␨ ץ zzuz + xzux + xxuz =0, 2G + ␭ 2G + ␭ Equations ͑9͒, ͑10͒, and ͑25͒ form a closed system for the elastohydrodynamic response of a thin elastic layer coating a so that to O͑␨͒ the leading order balance is rigid cylinder. We note that Lighthill13 found a similar set of ͒ ͑ ץ ץ zzux =0, zzuz =0. 19 equations while studying the flow of a red blood cell through a capillary. However, his model’s axisymmetry proscribed The normal unit vector to the soft interface is n the existence of a force normal to the flow. u ͉ ,1͒, which in dimensionless form is ץ−͉͑= x z z=0 When ␩Ӷ1, we can employ a perturbation analysis to ͒ ͑ ͒ ͉ ץ␧ ͉͑ n = − xuz z=0,1 , 20 find the lift force experienced by the cylinder. We use an expansion of the form ␧ ͱ where = h0 /2R. The balance of normal traction on the solid-fluid interface yields p = p͑0͒ + ␩p͑1͒ + O͑␩2͒, h = h͑0͒ + ␩h͑1͒ + O͑␩2͒, ͑26͒ ͉␴ ͉ ͉␴ ͉ ͑ ͒ f · n z=0 = s · n z=0, 21 to find ͒ ͑ 0͖͒͑ ץ0͔͒3͑ ͓ 0͒͑ ͕ ץ so that ␩0 : x 6h + h xp =0, 27 ͒ ͑ ␩ ͉ ץ͉ ͉ ץ͉ zux z=0 =0, zuz z=0 =− p. 22 ͒ ͑ 1͖͒͑ ץ0͔͒3͑ ͓ 0͒͑ ץ1͒͑ 0͔͒2͑ ͓ 1͒͑ ͕ ץ ␩1 : x 6h +3 h h xp + h xp =0, 28 At the interface between the soft film and the rigid substrate, the no-slip condition yields where

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FIG. 5. ͑a͒ Dimensionless lift per unit length L plotted against ␩, the softness parameter. L has a maximum at ␩=2.06 which is the result of a competition between symmetry breaking ͑dominant for ␩Ӷ1͒ and decreasing pressure ͑dominant for ␩ӷ1͒ due to increasing the gap thickness. For small ␩ asymptotic analysis yields L=͑3␲/8͒␩, which matches the numerical so- FIG. 4. ͑a͒ Pressure p͑x͒ as a function of ␩. ͑b͒ Gap thickness proﬁle lution. ͑b͒ The dimensional lift force, Lift, is quadratic in the velocity for h͑x͒=1+x2 +␩p as a function of ␩. The symmetry of the initially parabolic small velocities while being roughly linear for large velocities. V +␻ R gap thickness proﬁle is broken and the maximum value of the pressure 0 0 =͓h5/2͑2G+␭͒/ͱ2RH ␮͔ is the velocity and rate of rotation at which ␩=1, decreases as ␩ increases. 0 l and L0 is corresponding lift.

h͑0͒ =1+x2, h͑1͒ = p͑0͒, ͑29͒ subject to the boundary conditions p͑−9␩/10͒=0.As ␩ increases, h increases and the asymmet- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ric gap proﬁle begins to resemble that of a tilted slider bear- p 0 ͑ϱ͒ = p 0 ͑− ϱ͒ = p 1 ͑ϱ͒ = p 1 ͑− ϱ͒ =0. ͑30͒ ing, which is well known to generate lift forces. However, an Solving ͑27͒–͑30͒ yields increase in the gap thickness also decreases the peak pressure ϳ␮VR1/2 h3/2 ͓ ͑ ͔͒ 2x 3͑3−5x2͒ / 0 see Fig. 4 a . The competition between sym- p = + ␩ + O͑␩2͒, ͑31͒ metry breaking, dominant for ␩Շ1, and decreasing pressure, ͑ 2͒2 ͑ 2͒5 1+x 5 1+x dominant for ␩տ1, produces a maximum scaled lift force at so that ␩=2.06. In dimensional terms, this implies that the lift as a function of the effective modulus 2G+␭ will have a maxi- 3␲ L = ͵ pdx = ␩. ͑32͒ mum, however, the lift as a function of the relative motion 8 between the two surfaces V−␻R increases monotonically ͓ ͑ ͔͒ ͑ ͒ In dimensional form, the lift force per unit length is see Fig. 5 b . In fact, 33 shows that the dimensional lift increases as ͑V−␻R͒2 for ␩Ӷ1. 3ͱ2␲ p2 H ͱR 3ͱ2␲ ␮2͑V − ␻R͒2 H R3/2 L = 0 l = l , ␭ ͱ ␭ 7/2 8 2G + h0 4 2G + h0 ͑33͒ IV. DEGENERATE CONTACT as reported in Ref. 30. The same scaling was found in Ref. 22 but with a different prefactor owing to a typographical In this section, we consider the case where the parabolic error. When ␩=O͑1͒, the system ͑9͒, ͑10͒, and ͑25͒ is solved approximation in the vicinity of the contact breaks down. numerically using a continuation method31 with ␩ as the con- Since rotation changes the nature of the contact region for tinuation parameter. In Fig. 4, we show the pressure distri- such interfaces, we consider only a purely sliding motion bution p͑x͒ and the gap h͑x͒ as a function of ␩. For ␩Ӷ1, with ␻=0. We assume that the gap thickness is described by

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x2n ͩ ͪ ͑ ͒ ͑ ͒ h = h0 1+ 2n−1 + H x , 34 h0R

where n=2,3,... characterizes the geometric nature of the ϳ͑ 2n−1͒1/2n contact and the contact length is lc h0R . We note that we always focus on symmetric contacts. We make the variables dimensionless using the following scalings:

Ј * Ј 1/2n 1−1/2n Ј h = h0h , x = l x = h0 R x , ͑35͒ R1−1/2n * Ј ␮ Ј p = p p = V 2−1/2n p . h0

In Sec. III, we have seen that for a thin compressible soft layer the pressure and surface deflection can be linearly re- ͑ ͒ ␩ lated by 24 so that the scale of the deflection is h0 ͑ ␭͒ =p0Hl / 2G+ . To find the scale of the deflection for the ͑ ͒ FIG. 6. Pressure distribution and gap thickness profile for degenerate con- degenerate contact described by 34 , we replace p0 with the 2n ␩ ͑ ͒ * tacts corresponding to a gap thickness profile of h=1+x + p with n=2 a appropriate pressure scale p , so that and ͑c͒, and n=3 ͑b͒ and ͑d͒.

p* ␮V H R1−1/2n H = H HЈ = l , ͑36͒ ␭ l ␭ 2−1/2n 2G + 2G + h0 experienced by the slider arises from the ﬂuid pressure rather and the size of the deformation relative to the gap size is than the shear force because the normal to the surface no governed by the dimensionless group longer passes through the center of the object as it would for a cylinder or sphere. The ratio of the torque due to the shear p* H ␮V H R1−1/2n ␩ = l = l . ͑37͒ to the torque due to the pressure is ␭ ␭ 3−1/2n 2G + h0 2G + h0 shear torque ␮VR/h h 1−1/n Then the dimensionless version of the gap thickness proﬁle ϳ 0 ϳ ͩ 0 ͪ Ӷ 1, ͑42͒ ͑34͒ is pressure torque plc R

h =1+x2n + ␩p. ͑38͒ provided nϾ1 as for degenerate contacts. Hence, the dominant contribution to the torque is due to the pressure and As in Sec. III, we employ a perturbation expansion for the pressure field in the parameter ␩ ͑Ӷ1͒ to solve ͑9͒, ͑10͒, and ͑38͒ and find the pressure field and the lift for small ␩. This ⌫ = ͵ pxdA. ͑43͒ ␩ yields p=p0 + p1 and gives the following dimensionless result: ␩Ӷ ͑␩͒ For 1, p=p0 +O so that 351␲ ␩ ␩ ͑ ͒ Ln=2 = , Ln=3 = 0.8859 , 39 784ͱ2

where we have not shown the pressure distribution due to its unwieldy size. In dimensional terms, the normal force reads as

351␲ p*2 H l* 351␲ ␮2V2H R9/4 L = l = l , ͑40͒ n=2 ͱ ␭ ͱ ͑ ␭͒ 17/4 784 2 2G + h0 784 2 2G + h0

p*2 H l* ␮2V2H R5/2 L = 0.8859 l = 0.8859 l . ͑41͒ n=3 ␭ ͑ ␭͒ 9/2 2G + h0 2G + h0

The results for n=2 and n=3 are shown in Figs. 6 and 7. One FIG. 7. Dimensionless lift for h=1+x2n +␩p where n=1,2,3. The curves noteworthy feature of a degenerate contact is that the torque are similar, however, they cannot be rescaled to a universal curve.

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␭ +2G H͑x,y͒ = 4␲G͑␭ + G͒ ϱ l dyЈ ϫ͵ ͫp͑xЈ͒͵ ͬdxЈ ͱ 2 2 −ϱ −l ͑xЈ − x͒ + ͑yЈ − y͒ ϱ ␭ +2G 4͑l2 − y2͒ = ͵ logͫ ͬp͑xЈ͒dxЈ. ␲ ͑␭ ͒ ͑ Ј͒2 4 G + G −ϱ x − x ͑46͒

FIG. 8. Schematic diagram of a gel cylinder moving through a liquid over a To make the equations dimensionless, we employ the follow- rigid solid substrate. The asymmetric pressure distribution pushes on the gel ing scalings: when the fluid pressure in the gap is positive while pulling on the gel when the pressure is negative. This breaks the symmetry of the gap thickness x = ͱ2h RxЈ, y = lyЈ, h = h hЈ, profile h͑x͒ and gives rise to a repulsive force of elastohydrodynamic origin. 0 0 The dashed line in the lower right hand denotes the undeformed location of ͑47͒ the gel cylinder. ͱ ␮͑ ␻ ͒ Ј Ј 2R V − R Ј H = h0H , p = p0p = 3/2 p , h0 so that the dimensionless gap thickness ͑11͒ now reads 6x 3␲ n =2:p = , ⌫ = , ϱ 0 7͑1+x4͒2 ͱ Y 14 2 h =1+x2 + ␩͵ dxЈp͑xЈ͒logͫ ͬ, ͑48͒ ͑ Ј͒2 −ϱ x − x 6x ␲ n =3:p = , ⌫ = , 0 11͑1+x6͒2 11 where ͑44͒ 1 ␮͑V − ␻R͒͑␭ +2G͒ R 2l2 ␩ = , Y = ͑1−y2͒. ͑49͒ 6x ␲ ͑␭ ͒ 2 n = m:p = , 2 G + G h0 Rh0 0 ͑4m −1͒͑1+x2m͒2 Comparing with our softness parameter for a thin section, we 3␲ see that 3␲͑2m −3͒csc 2m ␩ ⌫ = . thin layer G͑G + ␭͒ H H 2͑ ͒ =2␲ͱ2 l ϳ l Ӷ 1, ͑50͒ m 4m −1 ␩ ͑ ␭͒2 ͱ ͱ soft slider 2G + h0R h0R i.e., a thin layer is stiffer than a half space made from the ͱ ␩ same material by the geometric factor h0R/Hl. For small V. SOFT SLIDER ␩ ͑ 2͒2 ͑ ͒ we write p=p0 + p1, where p0 =2x/ 1+x as in 31 . Sub- ͑ ͒ To contrast our result for a soft thin layer with that for a stituting into 48 yields soft slider, we consider the case where the entire cylindrical 2␲x 2 ␩ ͑␩2͒ ͑ ͒ slider of length 2l and radius R is made of a soft material h =1+x + 2 + O . 51 with Lamé coefficients G and ␭. Equivalently, we could have 1+x a rigid slider moving above a soft semi-infinite half space. To order ␩, ͑9͒ and ͑10͒ yield the equations for the pertur- Since the deformation is no longer locally determined by the bation pressure p , pressure, we use a Green’s function approach to determine 1 the response to the hydrodynamic pressure. We note that we 24␲x͑x2 −1͒ ,ͬ + p ץ1+x2͒3͑ͫ ץ = ␩ :0 have considered a cylinder of finite length in the direction x x 1 ͑1+x2͒2 perpendicular to the flow, rather than the apparently simpler ͑52͒ two-dimensional case, in order to avoid a logarithmic diver- p ͑− ϱ͒ = p ͑ϱ͒ =0, gence in the deformation. Following Ref. 32, we use the 1 1 Green’s function for a point force on a half space since the which has the solution ϳͱ Ӷ scale of the contact zone, lc h0R R, the cylinder radius. ͑ ͒ 2␲͑2x2 −1͒ Deformation at the surface due to a pressure field p x,y will p = . ͑53͒ be33 1 ͑1+x2͒4 ␭ +2G p͑xЈ,yЈ͒dxЈ dyЈ H͑x,y͒ = ͵ , ͑45͒ Hence the dimensionless lift force is 4␲G͑␭ + G͒ ͱ͑xЈ − x͒2 + ͑yЈ − y͒2 2␲͑2x2 −1͒ 3␲2 L = ␩ ͵ p dx = ␩ ͵ dx = ␩. ͑54͒ with x, y as defined in Fig. 8. Neglecting the end effects, so 1 ͑1+x2͒4 8 that the pressure is a function of x only, we integrate ͑45͒ over yЈ to get In dimensional terms, the lift force is

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tures discussed previously, i.e., for ␩Ӷ1, Lϳ␩; L shows a maximum at ␩=0.25 and decreases when ␩Ͼ0.25. The rea- sons for this are the same as before, i.e., the competing effects of an increase in the gap thickness and the increased asymmetry of the contact zone.

VI. INCOMPRESSIBLE LAYER

In contrast to compressible layers, an incompressible layer ͑e.g., one made of an elastomer͒ can deform only via shear. For thin layers, incompressibility leads to a geometric stiffening.29 To address this problem in the most general case, we use a Green’s function approach. The constitutive behavior for an incompressible linearly elastic solid is

͒ ͑ T͒ ١ ١͑ ␴ = G u + u − psI, 56 ͑ ͒ where u= ux ,uy ,uz is the displacement and ps is the pressure in the solid. Mechanical equilibrium in the solid implies ,.␴=0, i.e·١

2ٌ ץ 0=− xps + G ux,

͒ ͑ 2ٌ ץ 0=− yps + G uy, 57

FIG. 9. Gap thickness h and pressure p as a function of ␩ for a soft cylin- 2ٌ ץ drical gel slider. We note that while the pressure distribution is localized to 0=− zps + G uz. the region near the point of closest contact, the change in gap thickness is spread out due to the logarithmic nature of the Green’s function of a line Incompressibility of the solid implies contact: h=1+x2 +␩͐dxЈp͑xЈ͒log͓Y /͑x−xЈ͒2͔. ͒ ͑ ץ ץ ץ ١ 0= · u = xux + yuy + zuz. 58

3␲ ␮2͑V − ␻R͒2͑␭ +2G͒ R2 For the Green’s function associated with a point force, L = . ͑55͒ ͑␭ ͒ 3 ͉␴ ͉ =−f␦͑x͒␦͑y͒, where ␦ is a delta function, the bound- 8 G + G h0 zz z=0 ary conditions are Comparing this expression with that for the case of a thin ͑ ͒ elastic layer, Eq. 33 , we see that confinement acts to reduce ux = uy = uz =0 atz =−Hl, the deformation and hence reduce the lift in the small deflection, ␩Ӷ1, regime. When ␩=O͑1͒, we solve ͑9͒, ͑10͒, and ␴ = ␴ =0 atz =0, ͑59͒ ͑48͒ for p͑x͒ using an iterative procedure. First, we guess an xz xy initial gap profile hold and use a shooting algorithm to calcu- ␴ ␦͑ ͒␦͑ ͒ late the pressure distribution. The new pressure distribution zz =−f x y at z =0. is then used in Eq. ͑48͒ with Y =1000 ͑corresponding to a ͒ ͑ ͒ ͑ ͒ very long cylinder to calculate a new gap profile, hnew.If We solve the boundary value problem 57 – 59 by using a ͐10 ͑ ͒2 Ͻ −6 two-dimensional Fourier transforms defined as −10 hold−hnew dx 10 the calculation is stopped, else we set hold=hnew and iterate. The results are shown in Figs. 9 and ϱ ϱ ͑ ͒ 10, and not surprisingly they have the same qualitative fea- ͵ ͵ ͑ ͒ −i kxx+kyy ͑ ͒ ux = uˆ x kx,ky,z e dkxdky, ... . 60 −ϱ −ϱ

Then Eqs. ͑57͒–͑59͒ in Fourier space are

͒ ץ 2 2 ͑ 0=ikxpˆ s + G − kxuˆ x − kyuˆ x + zzuˆ x ,

͒ ץ 2 2 ͑ 0=ikypˆ s + G − kxuˆ y − kyuˆ y + zzuˆ y , ͑61͒ ͒ ץ 2 2 ͑ ץ 0=− zpˆ s + G − kxuˆ z − kyuˆ z + zzuˆ z ,

FIG. 10. Dimensionless lift as a function of ␩ a measure of the increase in ץ ␩Ӷ ͑ ␲2 ͒␩ gap thickness for a soft cylindrical gel slider. For 1, L= 3 /8 . 0=−ikxuˆ x − ikyuˆ y + zuˆ z,

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͑ ͱ 2 2͒ sinh 2Hl kx + ky 2Hl − ͱk2 + k2 uˆ ͑k ,k ,0͒ = f x y . z x y ͓ 2͑ 2 2͒ ͑ ͱ 2 2͔͒ 2G 1+2Hl kx + ky + cosh 2Hl kx + ky ͑63͒ Since this corresponds to a radially symmetric integral ker- ͱ 2 2 nel, we can take q= kx +ky and use the Hankel transform, ␲͐ϱ ͑ ͒ 34 FIG. 11. Green’s function for a point force ͑a͒ and a line load ͑b͒ acting on uz =1/2 0 J0 rq quˆ zdq, to find the surface displacement, ␨ an incompressible layer of dimensionless thickness =Hl /lc =1. ϱ f 2H q − sinh͑2H q͒ ͉u ͑r͉͒ = ͵ J ͑rq͒ l l dq z z=0 ␲ 0 2 2 4 G 0 1+2H q + cosh͑2H q͒ subject to the boundary conditions l l f uˆ − ik uˆ at z =0, = W͑r͒, ͑64͒ ץ=␴ˆ =0 xz z x x z ␲ 4 GHl ץ ␴ ˆ yz =0= zuˆ y − ikyuˆ z at z =0, where r=ͱx2 +y2 and J ͑rq͒ is a Bessel function of the first ͑62͒ 0 kind. To see the form of the surface displacement, we inte- ,uˆ at z =0 ץ␴ˆ =−f =−pˆ +2G zz s z z grate ͑64͒ numerically and show the dimensionless function W͑r͒ in Fig. 11͑a͒. The surface displacement H for a pressure uˆ = uˆ = uˆ =0 atz =−H . x y z l distribution p͑x͒ is found using the Green’s function for a ͑ ͒ ͑ ͒ ͑ ͒ Solving 61 and 62 for uˆ z z=0 ,wefind point force ͑64͒, so that

ϱ ϱ ϱ p͑x*͒ 2H q − sinh 2H q H =−͵ ͭ͵ ͫ͵ J ͑rq͒ l l dqͬdy*ͮdx*, ͑65͒ ␲ 0 2 2 −ϱ 4 G −ϱ 0 1+2Hl q + cosh 2Hlq

where r=ͱ͑x−x*͒2 +͑y−y*͒2. In terms of dimensionless vari- h =1+x2 + ␩H, ables with the following definitions: where Ј q ͱ * ͱ * p = p0pЈ, q = , x = 2h0RxЈ, x = 2h0RxЈ , 1 p R 1 ␮͑V − ␻R͒ R ͱ2h R ␩ = 0 ͱ = . ͑68͒ 0 ͱ ␲ 2 2 2␲ G h0 2 G h0 ͱ * ͱ * ͱ ͑ ͒ y = 2h0RyЈ, y = 2h0RyЈ , r = 2h0RrЈ, 66 For ␩Ӷ1, we can expand the pressure p in powers of ␩ and carry out a perturbation analysis as in Sec. III. This yields a p ͱ2h R ␮͑V − ␻R͒ R linear relation between the dimensionless lift L and the scale h = h hЈ, H = 0 0 HЈ = HЈ, 0 ␲ ␲ of the deformation: L=C ͑␨͒␩, where C ͑␨͒ is shown in Fig. 4 G 2 G h0 i i 12͑d͒. The dimensional lift per unit length is Eq. ͑65͒ may be rewritten ͑after dropping primes͒ as C ͑␨͒ ␮2͑V − ␻R͒2 R2 ϱ L = i . ͑69͒ ␲ 3 H =−͵ p͑x*͒ 2 G h0 −ϱ As ␨→ϱ, we approach the limit of an infinitely thick layer, ϱ ϱ 2␨q − sinh 2␨q in which case there is no stiffening due to incompressibility, ϫ ͵ ͵ ͑ ͒ * * ͭ ͫ J rq dqͬdy ͮdx , 2 0 ␨2 2 ␨ so that C͑␨͒→3␲ /8, which is the result for an infinitely −ϱ 0 1+2 q + cosh 2 q thick layer. To study the effects of confinement on a thick ͑ ͒ 67 layer, i.e., ␨ӷ1, we approximate ͑67͒ as where r=ͱ͑x−x*͒2 +͑y−y*͒2 and the dimensionless group ␨ ϱ lЈ ϱ ͱ H Ϸ − ͵ p͑x*͒͵ ͭ͵ J ͑rq͒ =H / 2h R is the ratio of the layer thickness to the size of 0 l 0 ϱ Ј the contact zone. To understand the form of ͑67͒, we con- − −l 0 sider the response of a line force p͑x*͒=␦͑x*͒ and show the ϫ͓1−͑2+4␨q + ␨2q2͒e−2␨q͔dqͮdy*dx*, ͑70͒ results graphically in Fig. 11͑b͒. Moving on to the case at hand, that of a parabolic contact of a cylinder of length 2l, we write the dimensionless gap thickness as where we are now integrating y* over the dimensionless

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ϱ length of the cylinder lЈ=l/ͱ2h Rӷ1, since the interactions 2x* 4͑lЈ2 − y2͒ 0 H Ϸ ͵ ͭlogͫ ͬ for deep layers are not limited by conﬁnement effects. Evalu- ͑ *2͒2 ͑ *͒2 −ϱ 1+x x − x ating ͑70͒ yields ͑x − x*͒2 +4␨2 8␨2͓͑x − x*͒2 +12␨2͔ ϱ + 2 logͫ ͬ − ͮdx*, H Ϸ ͵ p͑x*͒ 4͑lЈ2 + y2͒ ͓͑x − x*͒2 +4␨2͔2 ϱ − ͑72͒ lЈ 1 2r4 +20r2␨2 +96␨4 ϫͭ͵ ͫ− − ͬdy*ͮdx*. ͑71͒ ͑ 2 ␨2͒5/2 −lЈ r r +4 where we have used the leading order pressure ͑31͒ to evalu- Integrating ͑71͒ with respect to y* and keeping the leading ate p͑x*͒=2x* /͑1+x*2͒2 +␩p͑1͒. We integrate ͑72͒ with re- order terms in lЈ yields spect to x* to ﬁnd

24␲x͑x2 −1͒ 4␲x͓x4 +2x2͑1+6␨ +6␨2͒ + ͑1+2␨͒2͑1+8␨ +24␨2͔͒ H Ϸ − . ͑73͒ ͑1+x2͒2 ͓x2 + ͑1+2␨͒2͔3

As in Sec. III, Eqs. ͑28͒ and ͑30͒ yield the system of equa- ␲͑ 2 ͒ ͑1͒ 2 2x −1 ͑1͒ p = tions to be solved for the pressure perturbation p : ͑1+x2͒4 ␲͓12␨2 −20␨ +30−x2͑45 − 60␨ +36␨2͒ +45x4͔ − , 2␨4͑1+x2͒3 ,p͑1͔͒ =0 ץ6H +3͑1+x2͒2H + ͑1+x2͒3͓ ץ x x ͑ ͒ ͑74͒ 75 p͑1͒͑− ϱ͒ = p͑1͒͑ϱ͒ =0.

which we integrate with respect to x to ﬁnd the dimension- Solving ͑73͒ and ͑74͒ yields less lift

3␲2 30 L = ␩p͑1͒dx = ␩C = ␩ ͩ1− ͪ + O͑␨−5͒. ͑76͒ i 8 ␨4 On the other hand, as ␨→0 we approach the limit where the contact length is much larger than the layer thickness leading to the geometric stiffening. Due to solid incompressibility the layer may only deform via shear and the dominant ١ ϳ ١ term of the strain is u ux /Hl. Since ·u=0, we see that ϳ → ϳ 2 ux /lc H/Hl ux /Hl lcH/Hl . Balancing the strain energy ϳ ͑ 2͒2 2͒ ٌ͑ ͐ per unit length G u dA G lcH/Hl Hllc with the work ϳ 3 2 done by the pressure p0Hlc yields H p0Hl /Glc so that ␩ ϳ 3 2 ␨→ϱ p0Hl /Glch0. For a thin incompressible layer, the characteristic deflection is reduced by an amount ␩␨→ϱ /␩ ϳ␨−3 ͑␨͒ϳ␨3 ͑ ͒ so that lim␨→ϱCi . Ci is displayed in Fig. 12 a . For intermediate values of ␨, we computed the results numerically. The nonlinear problem arising for ␩=O͑1͒ is solved using ͑9͒, ͑10͒, ͑67͒, and ͑68͒. We first guess an initial gap profile hold, then a shooting algorithm is employed to calculate the pressure distribution. The new pressure distribution ␩Ӷ ͑␨͒␩ ͑␨͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ FIG. 12. For 1, L=Ci where Ci is shown in a . b shows the is then used in 67 and 68 to calculate a new gap profile dimensionless lift per unit length L as a function of ␩ for ␨=H /l =1. ͑c͒ ͐10 ͑ ͒2 Ͻ −5 l c hnew.If −10 hold−hnew dx 10 then the calculation is and ͑d͒ show pressure p and gap thickness h as a function of ␩. As the ␨ stopped, else we iterate with hold=hnew. For =1, the results thickness of the layer decreases the presence of the undeformed substrate ͑ ͒ ͑ ͒ below is increasingly felt and the layer stiffens. In the linear regime a stiffer are shown in Figs. 12 b –12 d . Not surprisingly, we find the layer results in a smaller deformation and concomitant decrease in lift. same qualitative features discussed previously: L has a

Downloaded 09 Sep 2005 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp 092101-12 J. M. Skotheim and L. Mahadevan Phys. Fluids 17, 092101 ͑2005͒ maximum due to the competing effects of an increase in the volume fraction ␣, the stress tensor ␴ is given by the con- gap thickness and the increased asymmetry of the contact stitutive equation zone. ͒ ͑ ␣ ١ T͒ ␭ ١ ١͑ ␴ = G u + u + · uI− pgI. 79 VII. POROELASTIC LAYER The equations of equilibrium are

Motivated in part by the applications to the mechanics of ͒ ͑ ␴ ١ cartilaginous joints, we now turn to the case of a cylinder · =0, 80 moving above a fluid filled gel layer. This entails a different model for the constitutive behavior of the gel accounting for where we have neglected inertial effects. Mass conservation both the deformation of an elastic network and the concomi- requires that the rate of dilatation of a solid skeleton having −1 tant fluid transport. To describe the mechanical properties of a bulk modulus ␤ is balanced by the entering fluid: a fluid filled gel we use poroelasticity, the continuum description of a material composed of an elastic solid skeleton k u. ͑81͒ · ١ ץ + p ץ␤ = 2pٌ 27,35,36 and an interstitial fluid. Our choice of poroelasticity to ␮ g t g t model the gel is motivated by the following scaling denote gradients on the system ١ and ١ argument.37 Let lp Here ␤␭+2G/3, since the Lamé coefficients ␭ and G are scale and the pore scale respectively; p is the pressure vary- g for the composite material and take into account the micro- ing on the system scale due to the boundary conditions driv- structure, while ␤−1 is independent of the microstructure; for ing the flow, while p is the pressure varying on the micro- p cartilage ␤−1ϳ1 GPa, while Gϳ␭ϳ1 MPa. Equations scopic scale due to pore geometry. Fluid stress balance on ͑79͒–͑81͒ subject to appropriate boundary conditions de- the pore scale implies that the sum of the macroscopic pres- scribe the evolution of displacements u and fluid pressure p ١p , and the microscopic g ,sure gradient driving the flow g in a poroelastic medium. p is balanced by the viscous resistance ١ pressure gradient lp p We now calculate the response of a poroelastic gel to an of the fluid having viscosity ␮ and velocity v, ␮ٌ2 v, so that lp arbitrary time-dependent pressure distribution before consid- the momentum balance in the fluid yields ering the specific case at hand. To make the equations dimensionless, we use the following scalings: p =0. ͑77͒ ١ − p ١ − ␮ٌ2 v lp g lp p ͱ2h R x = ͱ2h RxЈ, z = H zЈ, t = ␶tЈ = 0 tЈ, When the pore scale lp and system size Hl are well separated, 0 l ␻ Ӷ ͑ ͒ V − R i.e., lp /Hl 1, Eq. 77 yields the following scaling relations: Ј Ј uz = h0uz , ux = h0ux , H ␮V ␮V ϳ l ӷ ϳ ͑ ͒ ͑82͒ pg 2 pp 78 lp lp 2R p = p pЈ = ␮͑V − ␻R͒ͱ pЈ, p = p pЈ, g 0 g h3 g 0 from which we conclude that the dominant contribution to 0 the fluid stress tensor comes from the pressure. The simplest 2R stress-strain law for the composite medium, proposed by ␴ = p ␴Ј = ␮͑V − ␻R͒ͱ ␴Ј. 27 0 3 Biot, is found by considering the linear superposition of the h0 dominant components of the fluid and the solid-stress tensors. If strains are small, the elastic behavior of the solid We take the thickness of the layer to be much smaller than ␨ ͱ Ӷ skeleton is well characterized by isotropic Hookean elastic- the length scale of the contact zone, =Hl / 2h0R 1, and ity. For a poroelastic material composed of a solid skeleton consider a compressible material, Gϳ␭. Then after dropping with Lamé coefficients G and ␭ when drained and a fluid primes, the stress tensor ͑79͒ becomes

1 ␭ ␨ 1 G ␨ G u ץ + u ץ u − ␣p ץ + u ץ ␩ 2G + ␭ z z ␩ x x g ␩ 2G + ␭ z x ␩ 2G + ␭ x z ␴ = . ͑83͒ ΂ 1 G ␨ G 1 ␨ ␭ ΃ u − ␣p ץ + u ץ u ץ + u ץ ␩ 2G + ␭ z x ␩ 2G + ␭ x z ␩ z z ␩ 2G + ␭ x x g

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͒ ͑ ץ␦ ץ␥ ץ Here, tpg − zzpg = tp, 94 p H ␮͑V − ␻R͒ H R1/2 ␩ = 0 l = ͱ2 l ͑84͒ ␭ ␭ 5/2 where 2G + h0 2G + h0 is the dimensionless number governing the relative size of ͱ ͑ ␭͒ ␶ k 2Rh0 2G + the surface deflection to the undeformed gap thickness, i.e., ␥ = = , ␶ H2͑V − ␻R͓͒␤͑2G + ␭͒ + ␣͔ the material compliance. Stress balance ͑80͒ yields p l ͑95͒ G + ␭ 2G + ␭ 1 .p ͪ ␦ = ϳ O͑1͒ ץ ␣u − ␩ ץ u + ␨ͩ ץ=0 zz x G xz z G x g ␤͑2G + ␭͒ + ␣ 2G + ␭ ͱ ץ ␨2 + xxux, Here ␶=l /Vϳ h R/V is the time scale associated with mo- G c 0 tion over the contact length, ␶ ϳ␮H2 /kG is the time scale ͑ ͒ p l 85 associated with stress relaxation via fluid flow across the G + ␭ G thickness of the gel, and ␦ϳ1/␣ the inverse of the fluid ץ ␨2 ץ ␨ ץ␣␩ ץ 0= zzuz − zpg + xzux + xxuz, 2G + ␭ 2G + ␭ volume fraction. Equation ͑94͒ corresponds to the short time limit discussed in Ref. 38. The boundary conditions for ͑94͒ and continuity ͑81͒ yields are determined by the fact that at the solid-gel interface there k␶ h is no fluid flux and at the fluid-gel boundary there is no .͒ u ץu + ␨ ץ͑ ץ p + 0 ץ = ͒ p ץp + ␨2 ץ͑ ␮ 2␤ zz g xx g t g ␤ t z z x x pressure jump so that Hl Hlp0 ͑86͒ ͒ ͑ ͒ ͑ ͒ ͑ ץ zpg x,− 1,t =0, pg x,0,t = p, 96 To leading order ͑85͒ and ͑86͒ reduce to where both conditions are a consequence of Darcy’s law for ץ zzux =0, flow through a porous medium. ͑87͒ Although there is flux through the gel-fluid interface, the ,p =0 ץu − ␣␩ ץ zz z z g Reynolds equation ͑9͒ for the fluid pressure will remain valid and if the fluid flux through the gel is much less than the flux through the thin gap. A fluid of viscosity ␮ flows with a ␶ k h0 ͑ ͒ u . ͑88͒ velocity v= vx ,vz through a porous medium of isotropic ץ + p ץ = p ץ ␮ 2␤ zz g t g ␤ tz z Hl Hlp0 permeability k according to Darcy’s law, so that v ϳk/␮١p. Hence, the total flux through a porous medium of The normal unit vector to the soft interface is n ͐0 ␮ͱ thickness Hl is vxdz, which will scale as Hlkp / h R Hl 0 0− ͒ ͉ ץ ͉͑ = − xuz z=0,1 , which in dimensionless form is ϳ ␮ 2 HlkV/ h0. Comparing this with the flux through the thin 3 ͒ ͑ ͒ ͉ ץ␧ ͉͑ n = − xuz z=0,1 , 89 gap h0V leads to the dimensionless group QR =Hlk/h0.If Ӷ ͱ QR 1, we can neglect the flow through the porous medium. where ␧= h /2R. The balance of normal traction on the ϳ ϳ −14 2 0 For cartilage, Hl 1 mm and k 10 mm , and flow solid-fluid interface yields ӷ through the porous medium can be neglected if h0 10 nm. ͉␴ ͉ ͉␴ ͉ ͑ ͒ This implies that the Reynolds lubrication approximation f · n z=0 = s · n z=0, 90 embodied in ͑9͒ remains valid in the gap for situations of so that biological interest. In response to forcing, a poroelastic material can behave ,u ͉ =0 ץ͉ z x z=0 in three different ways depending on the relative magnitude ͑91͒ ␶ of the time scale of the motion =lc /V and the poroelastic u ͉ − ͉␣␩p ͉ =−␩p. ␶ ␶ӷ␶ ץ͉ z z z=0 g z=0 time scale p.If p, the fluid in the gel is always in At the interface between the soft film and the underlying equilibrium with the surrounding fluid and a purely elastic ␶ϳ␶ rigid substrate the no-slip condition yields theory for the deformation of the gel suffices; if p, the ␶Ӷ␶ gel will behave as a material with a “memory;” if p, the ͑ ͒ ͑ ͒ ͑ ͒ uz x,− 1 =0, ux x,− 1 =0. 92 fluid has no time to move relative to the matrix and the poroelastic material will again behave as a solid albeit with a ͑ ͒ ͑ ͒ ͑ ͒ Solving 87 , 91 , and 92 yields higher elastic modulus. In the physiological case of a carti- u =0, lage layer in a rotational joint, the poroelastic time scale for x ␶ ͑93͒ bovine articular cartilage is reported to be between p Ϸ ͑ ͒ ␶ Ϸ 9 s Ref. 7 and p 500 s. For time scales on the order 20 ץ ␩ ␣␩ − p =− pg + zuz. of 1 s, the cartilage should behave as a solid but with an ͑ ͒ To calculate the displacement at the surface uz x,0,t ,we elastic modulus greater than that measured by equilibrium ͑ ͒ need to determine the fluid pressure in the gel pg. Using 93 studies. We consider three different cases which are as fol- in ͑88͒ yields lows.

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͒ ͑ ץ͒ ␦͑ ͒ ͑ ץ␥ ͒ ͑ ץ A. Low speed: ␶ š␶ m p t pg − p − zz pg − p = −1 tp, 103 When the cylinder moves slowly, ␶ ӷ␶ , the time scale m p with the boundary conditions of the motion is much larger than the time scale over which ͒ ͑ ͒ ͑͒ ͑ ͒ ͑͒ ͑ ץ ͒ ͑ ͒␦ӷ ϳ␥ ͑ the pressure diffuses i.e., 1 so that 94 becomes z pg − p x,− 1,t =0, pg − p x,0,t =0. 104 ͒ ͑ ץ zzpg =0. 97 We expand pg −p in terms of the solution of the homoge- Solving ͑97͒ subject to ͑96͒ yields neous part of ͑103͒ and ͑104͒: p = p, ͑98͒ ϱ g ͑ ͒ ␲͑ 1 ͒ ͑ ͒ pg − p = ͚ An t sin n + 2 z. 105 i.e., at low speeds the fluid pressure in the gel is the same as n=0 the fluid pressure outside the gel. Equations ͑92͒ and ͑93͒ can be solved to yield Inserting the expansion into ͑103͒,wefind ץ͒ ␦͑ ͒ ␲2͑ 1 ͒2 ͔ ␲͑ 1␥ ץ͓ u ͑x,0,t͒ =−͑1−␣͒␩p͑x,t͒. ͑99͒ z ͚ tAn + n + 2 An sin n + 2 z = −1 tp. n=0 We see that this limit gives a local relationship between the ͑ ͒ displacement of the gel surface and the fluid pressure in the 106 gap, exactly as in the case of a purely elastic layer treated in 1 Multiplying ͑106͒ with sin ␲͑m+ ͒z and integrating over the Sec. III. 2 thickness yields ␶ ™␶ B. High speed: m p 1 2 2͑1−␦͒ p. ͑107͒ ץ = A + ␥␲2ͩn + ͪ A ץ t n n ␲͑ 1 ͒ t When the time scale of the motion is much smaller than 2 n + 2 the time scale over which the pressure diffuses, i.e., ␥Ӷ1 ͑ ͒ ͑ ͒ ϳ␦, ͑94͒ becomes Solving 107 for An t yields t ͒ ͑ ץ␦ ץ tpg = tp. 100 2͑1−␦͒ 2 2 pdtЈ. ͑108͒ ץA ͑t͒ = ͵ e−␥␲ ͑n +1/2͒ ͑t−tЈ͒ n ␲͑ 1 ͒ tЈ Since the gel is at equilibrium with the external fluid before n + 2 −ϱ the cylinder passes over it, p ͑x,z,−ϱ͒=p͑x,−ϱ͒=0, and Eq. g ͑ ͒ ͑ ͒ ͑100͒ yields Substituting 108 into 105 yields the fluid pressure in the gel, ͑ ͒ ␦ ͑ ͒ ͑ ͒ pg x,z,t = p x,t . 101 2͑1−␦͒sin ␲͑n + 1 ͒z Inserting ͑101͒ into ͑93͒ and integrating yields p = p + ͚ 2 g ␲͑ 1 ͒ n=0 n + 2 − ␤͑2G + ␭͒␩ u ͑x,0,t͒ = p͑x,1,t͒. ͑102͒ t ͒ ͑ Ј ץz ␤͑2G + ␭͒ + ␣ ϫ͵ −␥␲2͑n +1/2͒2͑t−tЈ͒ e tЈpdt . 109 −ϱ In this limit, the fluid has no time to flow through the pores and the only compression is due to bulk compressibility of Finally, ͑92͒, ͑93͒, and ͑109͒ yield the composite gel, which now behaves much more rigidly. The effective elastic modulus of the solid layer is now G eff u ͑1͒ = ␩ͫ͑−1+␣͒p ϳ␤−1ϳ ϳ z 1 GPa rather than Geff 1 MPa. However, the relationship between the pressure and the displacement remains t 2͑␦ −1͒ 2 2 .pdtЈͬ ץlocal as in Sec. III. + ␣͚ ͵ e−␥␲ ͑n +1/2͒ ͑t−tЈ͒ ␲2͑ 1 ͒2 tЈ ͒ ͑ ١ We note that if ·u=0, 81 has no forcing term and n=0 n + 2 −ϱ p =0. Poroelastic theory does not take into account shear g ͑110͒ deformations since these involve no local change in fluid volume fraction in the gel. In this case, all the load will be Since the higher order diffusive modes ͑nϾ0͒ decay more borne by the elastic skeleton. However, shear deformation in rapidly than the leading order diffusive mode ͑n=0͒, a good a thin layer will involve geometric stiffening due to incom- approximation to ͑110͒ is pressibility so that the effective modulus will be Geff ϳ͑h R/H2͒G.30 Hence, if ͑h R/H2͒G␤Ӷ1, the deformation 0 l 0 l − ␩H͑x͒ = u ͑0͒ = ␩ͫ͑−1+␣͒p should be treated as an incompressible layer as in Sec. VI. If z ͑ 2͒ ␤ӷ h0R/Hl G 1, the layer should be treated as in Sec. III −1 t with an effective modulus ␤ . 8͑␦ −1͒ 2 .pdtЈͬ ץe͑−␥␲ /4͒͑t−tЈ͒ ͵ ␣ + 2 tЈ ␲ ϱ ␶ È␶ − C. Intermediate speeds: m p ͑111͒ When ␦ϳ␥ϳ1, rewriting ͑94͒ for the difference between the fluid pressure inside and outside the gel, This approximation is similar to that used in Ref. 37. To ͑ ͒ ͑ ͒ ͑ ͒ pg x,z,t −p x,t , yields simplify 111 for the case of interest, we define

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t ͒ ͑ Ј ץ␰ ͵ ͑−␥␲2/4͒͑t−tЈ͒ = e tЈpdt , 112 −ϱ so that ␥␲2 p. ͑113͒ ץ + ␰ =− ␰ ץ t 4 t

In the reference frame of the steadily moving cylinder ␰͑x,t͒=␰͑x−t͒, so that ␥␲2 p. ͑114͒ ץ + ␰ = ␰ ץ x 4 x

Integrating the above yields

ϱ ͒ ͑ Ј ץ␰ ͵ ͑−␥␲2/4͒͑xЈ−x͒ =− e xЈpdx . 115 x

Consequently, the distance between the gel and the cylinder is found from ͑11͒, ͑111͒, ͑112͒, and ͑115͒ to be

h͑x͒ =1+x2 + ␩͑1−␣͒ ϱ 8 2 pdxЈͬ, ͑116͒ ץϫͫp + ͵ e͑−␥␲ /4͒͑xЈ−x͒ ␲2 xЈ x where

p H ͱ2RH ␮͑V − ␻R͒ ␩ = 0 l = l , ͑117͒ ͑ ␭͒ 5/2 ␩Ӷ ͑␥͒␩ ␥ 2G + h0 h ͑2G + ␭͒ FIG. 13. For 1, L=Cp where is the ratio of translational to po- 0 ͑␥͒ ͑ ͒ ͑ ͒ ␩ ␩ roelastic time scales, and Cp is shown in a . b max, the value of at ␥ ͑ ͒ where we are considering the case in which the bulk modu- which the lift is maximum, plotted against . c The maximum lift Lmax as ␥ ␩ ͓␩ ␩ ͑␥͔͒ ͑␥͒ a function of . After scaling L and , L / max /Lmax , the curves can lus of the skeletal material is much larger than the modulus be almost perfectly collapsed onto a single curve. of the elastic matrix ␤GӶ1, so that ͑95͒ implies ␦Ϸ1/␣. This leaves us a system of equations ͑9͒, ͑10͒, and ͑116͒ for the pressure with two parameters: ␩ characterizes the deformation ͑softness͒ and ␥ is the ratio of translational to diffu- ͑␲2 −8͒2x 2␥ ␥Ӷ ␥ӷ ͑ ͒ h =1+x2 + ␩͑1−␣͒ͫ + ͬ. ͑121͒ sive timescales. The two limits 1 and 1of 116 can ␲2͑1+x2͒2 ͑1+x2͒ both be treated using asymptotic methods. For ␥ӷ1, ͑116͒ yields We see that increasing ␥ increases the gap thickness and lowers the pressure without increasing the asymmetry, thus 2 ␩͑ ␣͒ ͑ ͒ h =1+x + 1− p, 118 decreasing the lift. In the small deflection limit, ␩Ӷ1, the ͑␥͒␩ and we recover the limit of a thin compressible elastic layer dimensionless lift force is L=Cp and the lift force in treated in Sec. III with ␩→͑1−␣͒␩. For ␥Ӷ1, ͑116͒ yields dimensional terms is 2 2 3/2 ␮ V HlR 2 8 L = C ͑␥͒͑1−␣͒ , ͑122͒ h =1+x + ͩ1− ͪ͑1−␣͒␩p, ͑119͒ p ␭ 7/2 ␲2 2G + h0 ␥ which is the result for a thin compressible layer with ␩ where Cp is a function of and shown graphically in Fig. →͑1−8/␲2͒͑1−␣͒␩. When ␩Ӷ␥Ӷ1, we expand the pres- 13͑a͒. ͑ ͒ ␩ ͑ When ␩=O͑1͒, we use a numerical method to solve ͑9͒, sure field as in 26 writing p=p0 + p1, where p0 =2x/ 1 +x2͒2 as in ͑31͒. Inserting this expression into ͑116͒ yields ͑10͒, and ͑116͒ on a finite domain using the continuation software AUTO2000 ͑Ref. 31͒ with ␩ and ␥ as the continuation 8 parameters. The initial solution from which the continuation h =1+x2 + ␩͑1−␣͒ͭp + 0 ␲2 begins is with ␩=␥=0, corresponding to h=1+x2, and p =2x/͑1+x2͒2. The form of the lift force as a function of ␩, ϱ ␲2␥ L͑␩,␥͒ for various ␥ can be almost perfectly collapsed onto p dxЈͮ + O͑␥␩͒, ͑120͒ ץϫ͵ ͫ1+ ͑x − xЈ͒ͬ xЈ 0 ␩ x 4 a single curve after appropriately scaling the , L axes using ͓␩ ␩ ͑␥͔͒ ͑␥͒ the position of the maximum, i.e., L / max /Lmax ␩ ͑ ͒ ͑ ͒ which can be integrated to give where Lmax and max are shown in Figs. 13 b and 13 c .

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taken with respect to x are interchangeable with those taken ץϷ ץ ץϷ ץ with respect to s, i.e., xh sh and xp sp, which allows for a consistent framework for the ﬂuid and the solid equilibrium equations. This yields G͑␭ + G͒h3 ␪ = n sin ␪ − n cos ␪, ͑123͒ ץ s 3͑␭ +2␮͒ ss x z

where the stress resultants nx and nz are determined by the equations n + e · ␴ · n, ͑124͒ ץ=n + e · ␴ · n,0 ץ=0 FIG. 14. Schematic diagram of a half cylinder of radius R, thickness hs, and s x x f s z z f Lamé coefficients ␮ and ␭ moving at a velocity V while completely immersed in a fluid of viscosity ␮. The edges of the half cylinder are clamped i.e., ␪ p ץat a distance R+h0 from the surface of an undeformed solid. denotes the ␮V h ͒ ͑ ץ ץ s ץ angle between the tangent to the surface and the x axis. ͓X͑s͒,Z͑s͔͒ are the snx = + + p sh, snz =−p, 125 laboratory frame coordinates of the half cylinder as a function of the arc- h 2 length coordinate s. ͒ ͑ ␴ ͒ ץ͑ where n= sh,−1 , and f is given by 2 . Then, the pressure in the fluid is governed by the Reynolds equation ͑9͒, h ץ − VIII. ELASTIC SHELL ͒ ͑ ͒ ץs ͑ ␮ 2 ץ ssp = 6 V +3h sp . 126 For elastic layers attached to a rigid substrate the effec- h3 tive stiffness increases with decreasing thickness. However, Finally, since cylindrical deformations are inextensional, we for free elastic shells the effective stiffness increases with must complement ͑123͒–͑126͒ with the kinematic equations increasing thickness. To see the effects of this type of geom- ͒ ͑ ␪ ץ ␪ ץ etry in biolubrication problems, we turn our attention to a sX = cos , sh = sin , 127 configuration in which a surface is rendered soft through its where the position of the surface of the elastic cylinder is geometry rather than its elastic moduli. We consider a half- ͑X,h͒. Equations ͑123͒–͑127͒ are made dimensionless with cylindrical elastic shell of thickness hs and radius R moving the following scalings: with constant velocity V parallel to the rigid substrate, while ͱ 1/2 completely immersed in fluid of viscosity ␮, as shown in 2␮VR p = p pЈ = pЈ, s = ␲RsЈ, h = RZ, Fig. 14. The shell is clamped at its edges, which are at a 0 3/2 h0 R h height + 0 above the rigid solid. The shape of the elastic ͑128͒ half cylinder is governed by the elastica equation ͑for the X = RXЈ, n = p h nЈ, n = p ͱ2h RnЈ. history of the elastica equation as well as its derivation see x 0 0 x z 0 0 z Ref. 39͒ for ␪͑s͒, the angle between the horizontal and the After dropping primes, the dimensionless forms of Eqs. tangent vectors where s is the arc-length coordinate. ͑123͒–͑127͒ are written as Balancing torques about the point O in Fig. 15 Z h 3/2 ץ − ͒ ͑ ͬ ץs ͫ ͱ ␲ͩ 0 ͪ 2 ץ gives: M͑s+ds͒−M͑s͒−dsn sin ␪+dsn cos ␪+ds2 /2͑p cos ␪ x z ssp = 3 2 +3Z sp , 129 Z3 R ץ → ␪͒ ץ ␪ ץ␮ − zu sin −p sh sin =0. In the limit ds 0, xM ␪ ␪ ͑ ͒ −nxsin +nzcos =0. The x-force balance is nx s+ds p h ␲ ץR Z ץ → ͒ ץ␮ ץ ͑ ͒ ͑ Z + s ͪ + ͱ 0 , ͑130͒ ץn = ͩp ץ nx s −ds p sh+ xu =0, which as ds 0 yields snx− s x s ͒ ͑ ͒ ͑ ץ ץ␮ = xu+p sh. The z-force balance is nz s+ds −nz s +pds h0 2 2R Z ץ → =0, which as ds 0 yields snz =−p. We note that external forces are applied in the contact region where derivatives R ͒ ͑ ␲ͱ ץ snz =− p, 131 2h0

͒ ͑ ␲ ␪ ץ sX = cos , 132

͒ ͑ ␲ ␪ ץ sZ = sin , 133

␮V͑␭ +2G͒ ␪ =3ͱ2␲2 ץ ss G͑␭ + G͒ R5/2 2R ϫ ͩ ␪ ͱ ␪ͪ ͑ ͒ 3 1/2 nx sin − nz cos . 134 hs ho h0 FIG. 15. Schematic of the torque and the force balance for a bent cylindrical To find the scale of the elastic deformation ␩=H /h , where ␭ 0 0 shell of thickness hs and Lamé coefficients G and subject to a traction the maximum displacement of the cylinder is of order H0,we ͒ ץ ץ␮ ͑ − zu−p xh,p applied by a viscous fluid. x and z are coordinates in the 2 reference frame of the rigid solid, while s is the arc-length coordinate in the note that the change in curvature is of order H0 /R so that ␪ ⑀ 2 ץ͔͒ ␭ ͒ 3 ͑␭͑ ͓ ͒ ͑ shell. M s = G +G hs /3 +2G ss is the bending moment. the bending strain is =hsH0 /R and the elastic energy per

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͐ ⑀2 ϳ͐ ͑ 2͒2 unit length, therefore, scales as G dA G hsH0 /R dA ϳ 3 2 3 Ghs H0 /R . The work done by the ﬂuid is due to a local- ͐ ϳ ized torque and scales as pxdx p0h0R, acting through an ⌬␪ϳ angle H0 /R. Balancing the work done by the ﬂuid 3 2 3 torque p0h0H0 with the elastic energy Ghs H0 /R yields H ␮V R7/2 ␩ ϳ 0 ϳ ͑ ͒ 3 3/2 , 135 h0 G hs h0 so that ͑134͒ can be written as

h ␩ 2R ͒ ͑ ␪ 0 ͩ ␪ ͱ ␪ͪ ץ ss = nx sin − nz cos . 136 R h0 ͑ ͒ ͑ ͒ ͑ ͒ ␪ The system 129 – 133 and 136 for , p, nx, nz, x, and z has two dimensionless parameters, ␩=3ͱ2␲2͓␮V͑␭ ͒ ͑␭ ͔͒ 7/2 3 3/2 +2G /G +G R /hs h0 and h0 /R, and is subject to eight boundary conditions FIG. 16. ͑a͒ and ͑b͒ pressure distribution p͑X͒ as a function of the softness, ␩.As␩ increases the asymmetry of the pressure distribution increases and p͑0͒ = p͑1͒ =0, the maximum pressure decreases. ͑c͒ and ͑d͒ shape of the sheet, where X͑s͒ and Z͑s͒ are the coordinates of the center line in the laboratory frame. We see that the point of nearest contact is pulled back and the symmetry of the X͑0͒ =−1, X͑1͒ =1, profile is broken by the forces exerted by the fluid on the cylindrical shell. h /R=10−3. ͑137͒ 0 h Z͑0͒ = Z͑1͒ =1+ 0 , R IX. JOURNAL BEARING So far, with the exception of Sec. IV, we have dealt only − ␲ ␲ ␪͑0͒ = , ␪͑1͒ = . with nonconforming contacts. In this section, we consider an 2 2 elastohydrodynamic journal bearing: a geometry consisting of a cylinder rotating within a larger cylinder that is coated The first two are a consequence of lubrication theory, while with a soft solid. The journal bearing is a conforming contact the rest are kinematic boundary conditions on the cylinder’s and is a better representation of biolubrication in mammalian lateral edges, which are assumed to be clamped. We note that joints in which synovial fluid lubricates bone coated with if the cylinder has a natural curvature this will not affect the thin soft cartilage layers. Previous analyses of the elastohy- ͑ ͒ ͑ ͒ system of equations 123 – 127 , but may change the bound- drodynamic journal bearing have focused on situations ary conditions. where fluid cavitation needs to be taken into account.3,6 As We solve ͑129͒–͑133͒, ͑136͒, and ͑137͒ numerically using the continuation software AUTO2000 ͑Ref. 31͒ with ␩ as the continuation parameter. For ␩Ͻ1, the dimensionless lift ␩ ͑ ͒ ͑ ͒ L=Cs , where the constant Cs h0 /R is shown in Fig. 17 b . However, we see that even for ␩Ͻ100 the linear relationship remains valid. The dimensional lift force for ␩Ͻ100 is

␮2V2͑␭ +2G͒ R9/2 h L =6ͱ2␲2 C ͩ 0 ͪ. ͑138͒ ͑␭ ͒ 3 5/2 s G + G hs h0 R Figures 16 and 17͑a͒ show the gap thickness proﬁle and ␩ pressure distribution for h0 /R=0.001. As increases the elastic deformation breaks the symmetry of the gap thickness proﬁle and results in an asymmetric pressure distribution and corresponding lift. However, the concomitant increase in gap thickness decreases the magnitude of the pressure. As for the previous systems considered, the competition between symmetry breaking ͑dominant for small ␩͒ and decreasing pressure ͑dominant for large ␩͒ produces a maximum in the lift. ␩ ͑␩ ͒ The form of the lift force as a function of , L ,h0 /R for ͑ ͒ −3 various h /R can be almost perfectly collapsed onto a single FIG. 17. a Dimensionless lift on a cylindrical shell with h0 /R=10 . For 0 ␩Ͻ ͑ ͒␩ ͑ ͒ ͑␩ ͒ curve after appropriately scaling the ␩, L axes; i.e., 100. L=Cs h0 /R , where Cs is shown in b . The form of L ,h0 /R can be almost perfectly collapsed onto a single curve after appropriately L͓␩/␩ ͑h /R͔͒/L ͑h /R͒ where L and ␩ are ␩ ͓␩ ␩ ͑ ͔͒ ͑ ͒ ␩ max 0 max 0 max max scaling the , L axes, i.e., L / max h0 /R /Lmax h0 /R , where max and ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ shown in Figs. 17 c and 17 d . Lmax are shown in c and d respectively.

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H p͑␪͒ H͑␪͒ = l , ͑143͒ 2G + ␭ so that ͑142͒ yields H p h = h + ⑀ cos ␪ + ⑀ sin ␪ + l . ͑144͒ 0 x z 2G + ␭ Using the following primed dimensionless variables: ␻ Ј ␻ Ј Ј Ј vr = h0vr , v␪ = Rv␪, r = h0r , h = h0h , ͑145͒ ␮R2␻ * Ј Ј ⑀ ⑀Ј ⑀ ⑀Ј p = p p = 2 p , x = h0 x, z = h0 z , h0 we write ͑139͒–͑141͒ and ͑144͒, after dropping the primes, as ץ FIG. 18. Schematic diagram of the modiﬁed journal bearing geometry in rp =0, which the larger cylinder of radius R+Hl +h0 has been coated by a soft solid ␭ ץ ץ of thickness Hl having Lamé coefﬁcients G and . The larger cylinder’s axis ⑀ ⑀ ␪p = rrv␪, is located a distance x in the x direction and a distance z in the z direction from the axis of the inner cylinder of radius R. The average gap thickness is ͑146͒ ץ ץ h0. rvr + ␪v␪ =0,

⑀ ␪ ⑀ ␪ ␩ h =1+ x cos + z sin + p, before, we restrict our attention in the case where the surface subject to deforms appreciably before the cavitation threshold is R reached so that the gap remains fully flooded. A schematic vr =0, v␪ =−1 atr = , diagram is shown in Fig. 18. h0 We take the center of the inner cylinder, Oi,tobethe origin; the center of the outer cylinder, O , is located at x R o v =0, v␪ =0 atr = + h, ͑147͒ ⑀ ⑀ r = x, z= z. The inner cylinder of radius R rotates with angular h0 ␻ velocity ; the stationary outer cylinder of radius R+h0 Ӷ +Hl is coated with a soft solid of thickness Hl R and Lamé p͑0͒ = p͑2␲͒, ␭ coefficients and G. Here, h0 is the average distance be- 28 where tween the inner cylinder and the soft solid. Following Leal, we use cartesian coordinates to describe the eccentric geom- H p* ␮␻R2H ␩ = l = l ͑148͒ etry and applied forces, but use polar coordinates to describe h ͑2G + ␭͒ ͑2G + ␭͒h3 Ӷ 0 0 the fluid motion. When h0 /R 1, the lubrication approximation reduces the Stokes equations in a cylindrical geometry is the softness parameter. As in Sec. II, we use ͑146͒ to for a fluid of viscosity ␮, pressure p, and velocity field v derive the system of equations for the fluid pressure, ͑ ͒ 28 = vr ,v␪ to 3 ,␪p͒ =0ץ ␪͑6h + hץ 1 ͒ ͑ ץ␮ ץ ץ rp =0, ␪p = rrv␪, 139 ⑀ ␪ ⑀ ␪ ␩ ͑ ͒ R h =1+ x cos + z sin + p, 149

subject to the boundary conditions p͑0͒ = p͑2␲͒. v =0, v␪ =−␻R at r = R, r In addition, fluid incompressibility implies that the average ͑ ͒ deflection must vanish: vr =0, v␪ =0 atr = R + h, 140 2␲ 2␲ ͵ ␪ ␲ → ͵ ␪ ͑ ͒ p͑0͒ = p͑2␲͒. hd =2 pd =0. 150 0 0 Since h /RӶ1, the continuity equation simplifies to28 0 The forces on the inner cylinder are ͒ ͑ ץ ץ R rvr + ␪v␪ =0, 141 2␲ 2␲ ͵ ␪ ␪ ͵ ␪ ␪ ͑ ͒ and the gap thickness profile simplifies to p sin d = Lz, p cos d = Lx. 151 0 0 ͑␪͒ ⑀ ␪ ⑀ ␪ ͑␪͒ ͑ ͒ h = h0 + x cos + z sin + H , 142 Here, Lz is the vertical force and Lx is the horizontal force. where H͑␪͒ is the elastic interface displacement due to the We begin with the classical solution for a rigid journal 28 ␩ ⑀ ⑀ fluid forces. As in Sec. III, bearing: Lz =1, Lx = = z =0, x =0.053.

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␩ ͑⑀ ͒␩ FIG. 19. For small the dimensionless horizontal force, L=Cj x , where the coefﬁcient Cj is plotted above.

As in previous sections, we investigate how elastohydrodynamics alters this picture by specifying the eccentricity ͑⑀ ⑀ ͒ ͑ ͒ x , z and calculating the forces Lx ,Lz as a function of the softness parameter ␩. Solutions to ͑149͒–͑151͒ are computed numerically using the continuation software AUTO2000 ͑Ref. 31͒ with ␩ as the continuation parameter and the solution for ␩=0 as the initial guess. Just as for different geometries analyzed in previous sections, the deflection of the surface of the soft solid breaks the symmetry and leads to the genera- Ͼ tion of a horizontal force in the x direction: Lx 0. For small deformations ͑␩Ӷ1͒, the dimensionless horizontal force L ͑⑀ ͒␩ ͑⑀ ͒ =Cj x , where the coefficient Cj x is shown in Fig. 19. In dimensional terms, the horizontal force per unit length for small deformations is p*H ␮2␻2R5H L = C ͑⑀ ͒ l p*R = C ͑⑀ ͒ l . ͑152͒ x j x ͑ ␭͒ j x ͑ ␭͒ 5 2G + h0 2G + h0 ⑀ Ӷ ͑⑀ ͒ ⑀ For nearly concentric cylinders, x 1, Cj x =115 x ͑⑀2͒ ⑀ → +O x . For large eccentricities, x 1, the lubrication pres- ͑⑀ ͒Ϸ ͑ ⑀ ͒−3 ␩Ӷ sure diverges and Cj x 12.3 1− x . For 1, we show FIG. 20. ͑a͒ Dimensionless horizontal force L acting on the inner cylinder as L, h, and p in Fig. 20. a function of ␩, a measure of the surface deflection; ͑b͒ the corresponding ͑ ͒ ⑀ ⑀ gap thickness profiles; c the corresponding pressure profiles. z =0, x =0.053. X. THREE-DIMENSIONAL LUBRICATION FLOW The analysis of the three-dimensional problem of a sphere moving close to a soft substrate is considerably more ␮Vl ␮Vl3 involved. Here, we restrict ourselves to scaling arguments to ␩ ͵ ͑1͒ ϳ ␩ ͵ c ϳ ␩ c ͑ ͒ Ls = p dA 2 dA 2 . 155 generalize the quantitative results of previous sections to h0 h0 spherical sliders. The results are tabulated in Table II. In the ␩ fluid layer separating the solids, balancing the pressure gra- To compute L, we need a prescription for the softness and dient with the viscous stresses yields the contact radius lc for each configuration. ␮ ␮ ͑ ͒ ͑ ͒ p V Vlc a For a thin compressible elastic layer Sec. III ,we ϳ → p ϳ , ͑153͒ ϳͱ ␩ϳ͓␮͑ ␻ ͒ ␭͔ l h2 h2 substitute lc h0R and V− R /2G+ c ϫ͑ 1/2 5/2͒ ͑ ͒ HlR /h0 into 155 to find where lc is the size of the contact zone. Substituting h=h0 Ӷ 2 2 2 +H0 with H0 h0, we find that the lubrication pressure is ␮ V H R L ϳ l . ͑156͒ s ␭ 3 ␮Vl ␮Vl H ␮Vl 2G + h0 p ϳ c ϳ c ͩ1+ 0 ͪ = c ͑1+␩͒. ͑154͒ ͑ ͒2 2 2 h0 + H0 h0 h0 h0 ͑b͒ For a thin elastic layer with a degenerate axisymmet- ͑ ͒ ϳ͑ 2n−1͒1/2n ␩ The reversibility of Stokes equations and the symmetry of ric conforming contact Sec. IV , lc h0R and ϳ͑␮ ␭͒͑ 1−1/2n 3−1/2n͒ ͑ ͒ paraboloidal contacts implies that the lift force L=0 when V/2G+ HlR /h0 so that 155 yields ␩ ␩Ӷ ͑0͒ =H0 /h0 =0. For 1, we expand the pressure as p=p 2 2 4−͑2/n͒ ͑ ͒ ͑ ͒ ␮ V H R +␩p 1 . Since p=p 0 will not generate vertical forces, the lift L ϳ l . ͑157͒ s 2G + ␭ h5−͑2/n͒ on a spherical slider Ls will scale as 0

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͑ ͒ ͑ ӷ 2 2 3 2 c For a soft spherical slider or thick layer Hl lc; Sec. GhsH d Gh H ͒ ͑ ͒ U Ϸ 0 + s 0 , ͑165͒ V , the deflection is given by 45 so that R2 d2 ␭ +2G p͑xЈ,yЈ͒dxЈdyЈ pl ͱ H͑x,y͒ = ͵ ϳ c , which has a minimum at d= hsR. Comparing d with lc ␲ ͑␭ ͒ ͱ͑ ͒2 ͑ ͒2 ϳͱ 4 G + G xЈ − x + yЈ − y G h0R, we see that the hydrodynamic pressure is localized Ͼ ͱ ͑158͒ if hs h0. For a localized force, d= hsR while for a nonlocalized force, d=R. The elastic energy of a localized defor- so that mation, Ul, and a nonlocalized deformation, Un, are given by Gh2H2 plc s 0 2 ͑ ͒ ␩ ϳ , ͑159͒ Ul = , Un = GhsH0. 166 Gh0 R ϳͱ ͑ ͒ The moment exerted by the hydrodynamic pressure on the the size of the contact zone is lc h0R, so that 155 and ͑159͒ yield spherical shell slider is ϳ 3 ͑ ͒ ␮2V2 R5/2 M plc , 167 ϳ ͑ ͒ Ls 5/2 . 160 G h0 which is independent of h0. The work done by the moment ͑167͒, which acts through an angle ⌬␪ϳH /d,is ͑d͒ For an incompressible layer ͑Sec. VI͒, we have two 0 H cases depending on the thickness of the substrate relative to ⌬␪ ϳ 3 0 ͑ ͒ ␨ M plc . 168 the contact zone characterized by the parameter =Hl /lc. For d ␨տ1, l ϳͱh R and ␩ϳ͓␮͑V−␻R͒/G͔͑R/h2͒ so that ͑155͒ c 0 0 Balancing the work done by the fluid ͑168͒ with the stored yields elastic energy ͑166͒ for both nonlocal and local deformations ␮2V2 R5/2 yields ϳ ͑ ͒ L␨տ1 5/2 . 161 G h ␮VR ␮VR5/2 0 ␩ ϳ ␩ ϳ ͑ ͒ n , l 5/2 , 169 For the case ␨Ӷ1, the proximity of the undeformed substrate Ghsh0 Ghs h0 substantially stiffens the layer. In sharp contrast to a com- so that ͑155͒ and ͑169͒ yield the lift force on the sphere for pressible layer, a thin incompressible layer will deform via the two cases ⌬ shear with an effective shear strain u/Hl. An incompress- ␮2V2 R4 h ١ ible solid must satisfy the continuity equation, ·u=0, L ϳ for s ӷ 1, ⌬ ϳ ⌬ l 5/2 3/2 which implies that u/lc H0 /Hl. Consequently, u/Hl G hs h0 h0 ϳ 2 ͐ ͑ 2͒2 lcH0 /Hl . Balancing the elastic energy G lcH0 /Hl dV ͑170͒ ϳG͑l H /H2͒2H l2 with the work done by the pressure pH l2 ␮2V2 R5/2 h c 0 l l c 0 c ϳ s Շ yields Ln 3/2 for 1. G hsh0 h0 p H3 ͑ ͒ ␩ ϳ l . ͑162͒ g For the ball and socket configurations, roughly the 2 ͑ ͒ G h0lc three-dimensional analog of the journal bearing Sec. IX , l ϳR and ␩ϳ͓␮␻R2H /͑2G+␭͒h3͔ so that the horizontal ϳͱ ͑ ͒ ͑ ͒ ͑ ͒ c l 0 Since lc h0R, 153 , 155 , and 162 yield force is given by ͑155͒: ␮ 3 ␮2 2 3 2 2 4 V Hl V Hl R ␮␻ R HlR ␩ ϳ , L␨Ӷ ϳ . ͑163͒ L ϳ . ͑171͒ 7/2 1/2 1 4 s ␭ 5 G h0 R G h0 2G + h0 ͑ ͒ ͑ ͒ ϳͱ e For a thin poroelastic layer Sec. VII , lc h0R and ␩ϳ͓␮͑ ␻ ͒ ␭͔͑ 1/2 5/2͒ ͑ ͒ V− R /2G+ HlR /h0 , so that 155 yields XI. DISCUSSION ␮2V2 H R2 L ϳ C͑␥͒ l , ͑164͒ s 2G + ␭ h3 The various combinations of geometry and material 0 properties in this paper yield some simple results of great where ␥ is the ratio of the poroelastic time scale to the time generality: the elastohydrodynamic interaction between soft scale of the motion. surfaces immersed in a viscous fluid leads generically to a ͑f͒ For a spherical shell slider ͑Sec. VIII͒ there are two coupling between tangential and normal forces regardless of cases: the thickness of the shell hs is smaller than the gap specific material properties or geometrical configurations. Ӷ thickness, i.e., hs h0 and all the elastic energy is stored in The fluid pressure deforms soft solid, thereby giving rise to a տ ␩ stretching; or hs h0 and bending and stretching energies are normal force. For small surface deformations, of the same order of magnitude.33 For a localized force, the =surface displacement/characteristic gap thicknessӶ1, the deformation will be restricted to a region of area d2. The dimensionless normal force is linear in ␩. Increasing ␩ ͑i.e., 2 2 ͒ stretching energy per unit area scales as GhsH0 /R , while the softening the material increases the asymmetry but de- 3 2 4 bending energy scales as Ghs H0 /d . The total elastic energy creases the magnitude of the pressure. The competition be- U of the deformation is then given by tween symmetry breaking, which dominates for small ␩, and

Downloaded 09 Sep 2005 to 128.103.60.225. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp 092101-21 Soft lubrication Phys. Fluids 17, 092101 ͑2005͒ decreasing pressure, which dominates for large ␩, produces a ͱ2␮VR1/2 maximum in the lift force as a function of ␩, the material’s ⌫ = ⌫Ј, ͑A2͒ h1/2 softness. 0 Additional complications such as nonlinearities and an- so that the dimensionless version of ͑A1͒ is isotropy in both the ﬂuid and the solid, streaming potentials 1 40,41 ⌫ = ͵ n ϫ ␴ · n dx, ͑A3͒ and current generated stresses would clearly change ␧ f some of our conclusions. However, the robust nature of the coupling between the tangential and the normal forces illus- where trated in this paper should persist and suggest both experi- n = ͑2␧x,− 1͒, ͑A4͒ ments and design principles for soft lubrication. and we have dropped the prime. Equation ͑A3͒ is evaluated using ͑8͒ yielding ACKNOWLEDGMENTS

͒ ͑ 2 ͉ ץ͉ ͵ ⌫ The authors thank Mederic Argentina for assistance in = zvx z=1+x dx. A5 using the AUTO2000 software package, and both Howard ␻ ͒ ͑ ͉ ͉ ץ Stone and Tim Pedley for their thoughtful comments. We We find z vx z=1+x2 by solving 7 with Ј=0, so that ץ acknowledge support via the Norwegian Research Council 1 h xp ͑J.M.S.͒, the U.S. Office of Naval Research Young Investi- ⌫ = ͵ͩ + ͪdx, ͑A6͒ h 2 gator Program ͑L.M.͒, and the U.S. National Institutes of Health ͑L.M.͒. where positive ⌫ corresponds to a torque pointing out of the page in Fig. 1. We drop the prime and solve ͑7͒ to find p ץh 1 v ͉ dx = ͵ͩ + x ͪdx. ͑A7͒ ץ͉ ͵ = ⌫ APPENDIX: THE ELASTOHYDRODYNAMIC z x z=h h 2 RECIPROCAL THEOREM To calculate the torque for small deflections we need to in- Jeffrey and Onishi42 calculated the flow around a cylin- clude terms of order ␩2, so that der near a wall in a viscous fluid and found that translation p = p͑0͒ + ␩p͑1͒ + ␩2p͑2͒ + O͑␩3͒. ͑A8͒ produces no torque on the cylinder. Similarly, a rotating cyl- ͑ ͒ inder experiences no net force. We now consider how elas- For p 2 ͑9͒, ͑10͒, and ͑25͒ yield a system: 0͒͒3͑ ͑ ץ2͒ ͑ 1͒͒͑ 0͒͑ ͑ ץ2͒2 ͑ 1͒͑ ͓ ץ ␩2 tohydrodynamics will alter this force balance. First, we con- :0 = x 6p +3 1+x x p p + 1+x x p sider the torque on a translating cylinder, 2͔͒͑ ץ2͒3 ͑ + 1+x xp , ͑ ͒ ⌫ = ͵ Rn ϫ ␴ · n dx, ͑A1͒ A9 f 0=p͑2͒͑− ϱ͒ = p͑2͒͑ϱ͒. where n=͑x/R,−1͒. Based on the scalings used in Sec. II, The order ␩2 pressure is found using MATHEMATICA5.0 ͑Ref. we write the dimensionless torque as 43͒ to be

87 465x + 38 815x3 + 121 394x5 + 95 062x7 + 39 237x9 + 6699x11 p͑2͒ =− . ͑A10͒ 5600͑1+x2͒8

ץ ץ ץ Consequently, zp =0, xp = zzvx,

ץ ץ 21␲ xvx + zvz =0, ⌫ = ␩2 + O͑␩4͒, ͑A11͒ 128 p͑x → − ϱ,z͒ =0, p͑x → ϱ,z͒ =0, so that elastohydrodynamic effects cause a cylinder translat- ͑A12͒ ͑ ␩ ͒ ͑ 2͒ ing from left to right above a substrate to rotate clockwise. vx x,− H =0, vx x,1 + x =−1, When ␩=O͑1͒, we evaluate ͑A7͒ numerically and show ⌫͑␩͒ in Fig. 21. ͑ ␩ ͒ ͑ 2͒ vz x,− H =0, vz x,1 + x =−2x, We now consider a stationary cylinder rotating clockwise. In this case the governing dimensionless lubrication where we have used the scaling of Sec. II but with ͑V equations are −␻R͒→␻R. We solve ͑A12͒ and ﬁnd the familiar Reynolds

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with the velocity V and rate of rotation ␻ via a matrix of second rank tensors so that

F AB V ͩ ͪ = ͩ ͪͩ ͪ. ͑A19͒ ⌫ CD ␻

The reciprocal theorem for viscous ﬂow44 states that B=CT. Due to the nonlinear coupling between translational and rotational modes, a priori there is no reason to expect any elastohydrodynamic analog to the reciprocal theorem. Nev- ertheless, for pure translation and rotation ͑A18͒ shows us that we may write ⌫ FIG. 21. Dimensionless torque due to translation above a thin soft elastic ͑␩͒␻ ⌫ ͑␩͒ ͑ ͒ layer. Surface deformation is needed to couple rotation to translation. For Fx = B , = C V, A20 ␩Ӷ1, ⌫=͑21␲/128͒␩. A cylinder translating from left to right above a deformable substrate rotates clockwise. where, somewhat surprisingly,

B͑␩͒ = C͑␩͒, ͑A21͒

ϱ͒ ͑ ͒ ץ3 ͑ ץ equation, 0= x 6h+h xp , p ± =0, as well as the shear at and a cylinder rotating clockwise will translate in the x di- the cylinder surface rection. ץ 1 h xp 1B. J. Hamrock and D. Dowson, Ball Bearing Lubrication: The Elastohy- v ͉ 2 =− + . ͑A13͒ ץ͉ z x z=1+x h 2 drodynamics of Elliptical Contacts ͑Wiley, New York, 1981͒. 2O. Reynolds, “On the theory of lubrication and its application to Mr. The Reynolds equation for the pressure is exactly the same Beauchamp Tower’s experiments, including an experimental determina- as for the translating cylinder and produces an identical pres- tion of the viscosity of olive oil,” Philos. Trans. R. Soc. London 177,157 ͑1886͒. sure field. The horizontal force on the cylinder is 3H. D. Conway and H. C. Lee, “The analysis of the lubrication of a flexible journal bearing,” ASME J. Lubr. Technol. 97, 599 ͑1975͒. 4 F = ͵ e · ␴ · ndx, ͑A14͒ D. Dowson and G. R. Higginson, “A numerical solution to the elasto- x x f hydrodynamic problem,” J. Mech. Eng. Sci. 1,7͑1959͒. 5B. J. Hamrock, Fundamentals of Fluid Film Lubrication ͑McGraw-Hill, where e =͑1,0͒. Based on the scalings in Sec. II, we write New York, 1994͒. x 6 the dimensional horizontal force as J. O’Donoghue, D. K. Brighton, and C. J. K. Hooke, “The effect of elastic distortions on journal bearing performance,” ASME J. Lubr. Technol. 89, ͑ ͒ ͱ2␮␻R3/2 409 1967 . Ј ͑ ͒ 7A. J. Grodzinsky, H. Lipshitz, and M. J. Glimcher, “Electromechanical Fx = 1/2 Fx . A15 h0 properties of articular cartilage during compression and stress relaxation,” Nature ͑London͒ 275, 448 ͑1978͒. After dropping primes ͑8͒, ͑A4͒, ͑A13͒, and ͑A14͒ yield 8V. C. Mow and X. E. Guo, “Mechano-electrochemical properties of articular cartilage: their inhomogeneities and anisotropies,” Annu. Rev. Biomed. 1 Eng. 4, 175 ͑2002͒. F = ͵ e · ␴ · n dx 9 x ␧ x f V. C. Mow, M. H. Holmes, and W. M. Lai, “Fluid transport and mechanical properties of articular cartilage: a review,” J. Biomech. 17,377 ͑1984͒. 10 E. R. Damiano, B. R. Duling, K. Ley, and T. C. Skalak, “Axisymmetric ͒ 2 ͉ ץ ͉͑ ͵ = − zux z=1+x −2xp dx pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer,” J. Fluid Mech. 314,163͑1996͒. p 11J. Feng and S. Weinbaum, “Lubrication theory in highly compressible ץh 1 = ͵ͩ − x −2xpͪdx. ͑A16͒ h 2 porous media: The mechanics of skiing, from red cells to humans,” J. Fluid Mech. 422, 281 ͑2000͒. 12 Using Eqs. ͑A7͒ and ͑A16͒, we find the difference between J. M. Fitz-Gerald, “Mechanics of red-cell motion through very narrow capillaries,” Proc. R. Soc. London, Ser. B 174, 193 ͑1969͒. the horizontal force on a rotating cylinder and the torque on 13M. J. Lighthill, “Pressure-forcing of tightly fitting pellets along fluid-filled a sliding cylinder in dimensionless terms, elastic tubes,” J. Fluid Mech. 34,113͑1968͒. 14T. W. Secomb, R. Hsu, and A. R. Pries, “A model for red blood cell motion in glycocalyx-lined capillaries,” Am. J. Physiol. 274, H1016 ͒ ͑ ͒ ץ ͑ ͵ ⌫ − Fx = h xp +2xp dx. A17 ͑1998͒. 15T. W. Secomb, R. Skalak, N. Özkaya, and J. F. Gross, “Flow of axisym- p metric red blood cells in narrow capillaries,” J. Fluid Mech. 163,405 ץh ␩ ץ h x2 ␩p x Since =1+ + , we have 2 = x − x and ͑1986͒. 16 ␩ H. Tözeren and R. Skalak, “The steady flow of closely fitting incompress- .2͉ϱ ible elastic spheres in a tube,” J. Fluid Mech. 87,1͑1978͒ ͉ ͒ ץ ␩ ץ ץ ͑ ͵ ⌫ − Fx = h xp + p xh − p xp dx = hp − p ϱ =0. 2 − 17W. Wang and K. H. Parker, “The effect of deformable porous surface ͑ ͒ layers on the motion of a sphere in a narrow cylindrical tube,” J. Fluid A18 Mech. 283, 287 ͑1995͒. 18S. Weinbaum, X. Zhang, Y. Han, H. Vink, and S. Cowin, “Mechanotrans- The linearity of Stokes’ equations for viscous flow equations duction and flow across the endothelial glycocalyx,” Proc. Natl. Acad. Sci. implies that we may linearly relate the force F and couple ⌫ U.S.A. 100, 7988 ͑2003͒.

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