The elastohydrodynamic force on a sphere near a soft wall

Abstract

The influence of soft boundaries on the forces experienced by a small sphere undergoing slow translation and rotation near a wall is investigated using asymptotic and numerical methods. The clearance between the sphere and the wall is assumed to be small, so that the lubrication approximation holds in the gap. The forces induced by boundary deformation break the symmetry of the Stokes equations, leading to irreversibility of the motion of the sphere and yielding a nonzero lift force. A general formulation, applicable to any constitutive equation of the wall, is presented, and an asymptotic analysis for slightly soft boundaries is developed and applied to a thin compressible elastic layer coating a rigid surface. Expressions are derived for the elastohydrodynamic lift exerted on a sphere moving parallel to the wall, which include the influence of the sphere rotation in the direction of its motion. The results extend and correct previous work, and are different from those for the elastohydrodynamic motion of a cylinder near a wall, in that the geometry eliminates the occurrence of maximum lift and optimum choice of the material properties of the boundary.

Rotation of a submerged finite cylinder moving down a soft incline

“…which measures the scale of the substrate deformation relative to the fluid gap thickness [45]. In this framework, previous theoretical studies [11][12][13][14][15][16] have shown that for Λ 1 and for given u c and h f the motion of an infinite cylinder is accompanied by the emergence of an elastohydrodynamic lift force F ∼ Λµu c 2 L/h 2 f , which was confirmed experimentally [35]. Since the cylinder also rotates with negligible inertia, the sum of torques due to elastohydrodynamics (induced by the substrate's deformation due to sliding) τ s and viscous damping of the rotational motion τ Ω vanish [36]: τ Ω +τ s = 0.…”

“…Often, this elastohydrodynamic coupling is seen in the presence of confined flow where pressure gradients are likely to be large. Previous theoretical works have studied confined flows in the soft lubrication approximation and accounted for the roles of elasticity [11][12][13][14][15][16], fluid compressibility [17], the inertia of the fluid and the elastic medium [18], and viscoelasticity of the substrate [19]. More recent works have focused on elastohydrodynamic effects for liquids confined at the micro and nano scales [20][21][22], which has important consequences for surface mechanical characterization [23,24].…”

On the hydrodynamic interaction between a particle and a permeable surface

“…This result is relaxed when the interacting surfaces are rough, the fluid is compressible, or when either or both surfaces are compliant. [2][3][4][5] A finite force at contact has also been shown to occur when the solid…”

Similarity theory of lubricated Hertzian contacts

“…1,5,6 In the limiting case of small elastic deformations, the problem can be solved perturbatively and reduces to a set of ordinary differential equations. 2,3,7 For the opposite case of strong deformations, the contact area becomes flat as in a classical Hertzian contact, 8 except for the effect of the thin lubrication layer. [9][10][11] Here we show that at the edge of the contact area, the film thickness is described by a similarity solution, whose shape is governed by an integro-differential equation, as opposed to the ordinary differential equations encountered in most singular fluid problems.…”