Abstract
We examine the prescribed time-dependent motion of a rigid particle (a sphere or a cylinder) moving in a viscous fluid close to a deformable wall. The fluid motion is described by a nonlinear evolution equation, derived using lubrication theory, which is solved using numerical and asymptotic methods; a local linear pressure–displacement model describes the wall. When the particle moves from rest towards the wall, fluid trapping beneath the particle leads to an overshoot in the normal force on the particle; a similarity solution is used to describe trapping at early times and a multiregion asymptotic structure describes fluid draining at late times. When the particle is pulled from rest away from the wall, a peeling process (described by a quasisteady travelling wave) determines the rate at which fluid can enter the growing gap between the particle and the wall, leading to a transient adhesive normal force. When a cylinder moves from rest transversely over the wall, transient peeling motion is again observed (especially when the wall is initially indented), giving rise to an overshoot in the transverse drag. Simulations for a translating sphere show highly nonlinear wall deformations characterized by a localized crescent-shaped ridge. Despite generating sharp transient deformations, we found no numerical evidence of finite-time choking events.
1 Introduction
The motion of a smooth rigid particle in a viscous fluid near a plane wall may be described using lubrication theory, which captures the dominant hydrodynamic forces in the narrow gap between the particle and the wall. Within this framework, it is readily demonstrated that a particle moving under a constant force towards a rigid wall will not touch it in finite time, and that a sphere or cylinder moving parallel to a plane experiences zero normal force. There are numerous small-scale effects that alter these predictions (for example, sharp asperities on either surface can allow contact, and compliance of either surface can induce a lift force). By breaking the usual reversibility of Stokes flow, such effects can have profound consequences at macroscopic scales, influencing the rheology of concentrated suspensions of rough or deformable particles, for example (1; 2). These effects are of particular importance in biological situations involving interfaces with exotic mechanical properties. Our aim here is to use a simple lubrication-theory-based model to examine particle–wall interactions (with one or both interfaces compliant), focusing in particular on transient effects in two or three dimensions.
Elastohydrodynamic lubrication arises in diverse applications which have motivated a variety of previous theoretical studies. Lighthill (3) treated the problem of a tight-fitting particle moving through a tube under an imposed pressure gradient. This was one of the earliest models of a red blood cell in a capillary (following subsequent advances in cell mechanics, his study is now recognised as having more relevance to situations such as a rigid kidney stone in a flexible ureter). Lighthill demonstrated that low flow-induced pressures at the rear of the particle can draw the particle and wall close together. While he found that the interfaces do not touch under steady conditions, we will explore below the possibility that an unsteady ‘choking’ instability (in which the surfaces touch in finite time, a feature of some collapsible-tube flows (4)) might occur transiently. One motivation for the present study concerns the motion of inhaled particles trapped in the liquid lining of lung airways, which have been demonstrated experimentally to be pushed into the underlying epithelium by surface forces (5, 6). Knowing the resistance to transverse displacement of an indenting particle is important in understanding mechanisms of particle clearance from the lung via the mucociliary escalator, for example (a model for which is presented in (7)). Related ideas also appear in roll coating, an industrial process in which a thin liquid film is applied to a moving surface. The effects of allowing one roll to be deformable were examined by Coyle (8), who determined the flux through the ‘nip’ between two rolls as a function of the roll compliance (see also (9, 10)); Yin and Kumar (11) examined the effect of non-uniform surface topography in this context. Similar physical effects may be important in the motion of car and bicycle tyres on wet surfaces when hydroplaning (12).
We examine here the motion of a rigid sphere or cylinder moving close to a deformable wall. The indendation of the wall is assumed sufficiently small to allow the use of lubrication theory. Treating the wall as an incompressible elastic half-space would require the pressure displacement to be non-local (see, for example, (13, 10)). To avoid this complication, we assume the displacement of the wall is linear in the local pressure (section 2), treating the wall either as a spring-backed membrane (8), or (as justified in (14 to 17)) a thin compressible elastic layer bonded to a plane surface. This simple model is consistent with the biological application motivating the study; furthermore, Carvalho and Scriven showed that such a model provides an acceptable first approximation of deformation of a compressible half-space (9) (a more detailed comparison is provided in (17)). We assume throughout that the particle moves with prescribed displacement and compute the resulting force that it experiences; motion under prescribed force is described in (7). We first consider motion of a cylindrical particle normal to the wall (section 3). We show how squeezing of fluid out of the gap between the particle and wall leads to a transient overshoot in the force. A transient overshoot in force also arises if the particle is lifted rapidly off the wall (section 4), because of the delay in the peeling of the wall away from the base of particle. Analogous pushing and peeling motions for a drop or cell near a wall, in which tension rather than a linear spring force is dominant, have been described in (18, 19). We then examine the transverse motion of a particle over a wall, focusing on the transient response as it starts from rest. The motion ultimately reaches a steady configuration described in the case of a cylinder by numerous previous authors (8, 3, 16, 17) (and revisited in the Appendix). For a cylinder (section 5) and a sphere (section 6), we show numerically that the wall forms a sharp transient corner (in two dimensions) or a ridge (in three dimensions) but, at least according to our simulations, never chokes in finite time.
2 The model
2.1 Motion of a sphere
We consider a sphere of radius R* a distance H* ≪ R* from a deformable wall. The sphere is immersed in fluid of constant viscosity μ*. For times t* > 0 the sphere moves in a prescribed fashion either normal to the undeformed wall over a timescale T*, or parallel to it with speed V*. The motion is assumed slow enough that viscous forces dominate inertia. In equilibrium the wall is assumed to be flat; wall displacements are assumed to be linear in the applied normal stress with an effective spring constant κ*.
Sketch showing the particle and the deformable interface
2.2 Motion of a cylinder
2.3 Numerical method
For pushing and pulling motion of a cylinder (sections 3, 4 below), (2.9a) was solved using a second-order finite-difference method and the method of lines in the domain [0, L] (exploiting the symmetry of the problem), applying (2.9b) at x = L. In the simulations we used 1001 meshpoints with L = 10. For transverse motion (section 5 below), (2.9a) was integrated over [−L, L] using 3001 meshpoints and L = 10. In both cases the accuracy of the results was checked by varying the number of meshpoints and the domain length and by checking conservation of fluid volume (accounting for the flux passing across the ends of the domain).
For translation of a sphere (section 6), we set h = h0 + (x · x)/2 + H in (2.6) and solved for H (which vanishes in the far field) using an alternating-direction-implicit (ADI) method to integrate forward in time. We integrated over a rectangular domain, – 10 ≤ x ≤ 10, – 10 ≤ y ≤ 10, with a grid of 200 × 200 meshpoints, and from 0 ≤ t ≤ 20 with 106 timesteps. Accuracy was checked by varying the number of meshpoints and timesteps, and results were validated using asymptotics (see below).
We now describe the motion arising when the cylinder is pushed normally towards the wall (section 3), pulled normally away from it (section 4) and translated across it (section 5).
3 Cylinder: pushing motion
Figure 2(a) shows the evolution of the gap thickness h when Xc = 5 and T = 10. At large times the system approaches its equilibrium position, although fluid remains trapped beneath the particle for all time. At early times the pressure rises beneath the cylinder (Fig. 2(b)); the elevated pressure drives fluid out of the thin gap, which thins slowly with time at large times (Fig. 2(c)) with h(0, t) ∝ t−1/2. A corner develops in the membrane near x = Xc over a diminishing lengthscale of width O(t−1/2) at large times (Fig. 2(d)). The corresponding force on the cylinder is shown in Fig. 3(a), illustrating how trapped fluid increases the force on the particle above the equilibrium value that would arise in the absence of fluid. The force relaxes to its equilibrium level as fluid is slowly squeezed out of the gap between the particle and the wall. There is a significant overshoot in the force when t ≈ T, particularly when the particle is pushed down rapidly (with T = O(1)).
![(a) Gap thickness h(x, t) as t increases from t = 0 to t = 18 in unit intervals, with T = 10, Xc = 5 and h0(0) = 5. (b) Pressure as time increases from t = 0 to t = 0·02 in uniform intervals when T = 1. (c) log h(0, t) versus log t with T = 10 (dashed) and \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\sqrt{6}t^{{-}1/2}\) \end{document} (solid). (d) Rescaled solutions of (2.9a) in the neighbourhood of x = Xc for 10 ≤ t ≤ 150 (solid), and the solution of (3.7) (dashed)](https://cdn.statically.io/img/oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/qjmam/59/2/10.1093/qjmam/hbl002/2/m_qjmamjhbl002f02_lw.gif?Expires=1731669295&Signature=aTLiaa14Zcio1zBWER66BF0HfZUdefKf1s0wMkWo6QL-J6SYi4gLa7eoNpGxrMPd9BRUJWX15jgsAbyQ32SQs3ktkJQ0YsHz99a3g--OxPjl~z4r2qAPlZDoIt7xpsI2PEtxZ1LvJXxQiwMf9zLV7d7~yFU4UtYsEsacC5Sq6DK3Q5uozhzAkZi0zWhQe-M0LhA45Ra8oBIxqcU7jtMvbd6rUAyz8I8y7Pgw70lEMJbqvfgdlOAh9EnC3U8EH-Q8s1S6kcTA2ipRPpicOYPjW61Ay0oViSIos2dc275afju9b2pJ9Cq2Rx4Q46VoKx0bjoOsFBGBLEC1DpNq-8w78A__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Gap thickness h(x, t) as t increases from t = 0 to t = 18 in unit intervals, with T = 10, Xc = 5 and h0(0) = 5. (b) Pressure as time increases from t = 0 to t = 0·02 in uniform intervals when T = 1. (c) log h(0, t) versus log t with T = 10 (dashed) and
(a) Normal force per unit length versus time for T = 2, 5, 10, 20, 25 (solid). Dashed lines show the corresponding equilibrium curves. (b) Early-time force when T = 100 (solid) versus quasisteady prediction (3.4) (dashed). (c) Solution H(ζ) of (3.5). (d) log(Fn/2) versus log t for T = 1 (solid) and approximation (3.6) with A = 78·16
We can understand this behaviour in more detail by examining the early- and late-time behaviour asymptotically.
3.1 Early-time behaviour
In contrast, when t ≪ T = O(1), the motion of the particle is sufficiently rapid for the quasisteady assumption to fail. Instead, the particle moves so quickly that fluid is trapped within the gap, having insufficient time to drain away as the particle moves. This is illustrated by the early-time pressure field in Fig. 2(b); the region of elevated uniform pressure corresponds to trapped fluid. We estimate the amount of fluid trapping as follows. For
3.2 Late-time behaviour
4 Cylinder: pulling motion
Figure 4(a) shows the gap thickness as t increases when T = 10, Xc = 5 and h0(T) = 5. There is a well-defined corner that moves towards x = 0 before the particle then lifts off the wall. We define X0(t) to be the maximum value of x for which h = δ; this characterizes the location of the corner. Figure 4(b) shows how X0(t) changes with time when the particle is pulled off at different speeds. As the particle begins to move there is a ‘waiting time’ during which Xc remains stationary. The waiting time increases as T increases. Figure 4(c) shows how the force acting on the particle depends on the speed with which the particle is pulled off the wall. As the particle is pulled away, the upward force due to the compression of the wall decreases as the wall relaxes. Later on, the wall is pulled upwards behind the particle, exerting a downward (negative) force on the particle (for example, Fig. 5(d)). As the wall relaxes to its undeformed state the force on the particle returns to zero. For particles pulled off quickly (for example, T = 1), the suction force on the particle is much greater than for particles pulled off more slowly (for example, T = 20) and the membrane is more sharply deformed in the corner region.
(a) h(x, t) versus x for t increasing from 0 to 42 in unit intervals, with T = 10, Xc = 5 and h0(T) = 5. Inset shows asymptotic regions I to IV. (b) Corner position X0(t), as particle is pulled off the wall with T = 5 and 10, obtained from numerical simulations (solid) and (4.7) (dashed). (c) Force versus time for T = 1, 3, 10 and 20. (d) Fn/2, as the particle is pulled away from the wall with T = 10 calculated by numerical simulations (dashed), compared with approximation (4.8) (solid)
Particle lifted off with T = 1, where t = (a) 0, (b) 0·4, (c) 0·7, (d) 1, (e) 1·3, (f) 2
We now explore this behaviour by examining the asymptotic structure of the corner region.
4.1 The corner region
In region II we assume h = δH(η) and ξ = (δ3/w)η, so that (4.3) becomes
5 Cylinder: transverse motion
We now use (2.9a) with V > 0 to determine the evolution of the wall and the force on the particle when the particle moves sideways. Initially the particle either starts above the wall (with h0 > 0 held constant) or else we use a similar initial condition to that used for pulling simulations (with the particle indented into the wall and then moved sideways with h0 < 0 held constant).
Figure 6 shows four examples of transient motion. Panels (a) and (b) illustrate the effect of reducing h0 > 0 with V fixed. The corresponding wall deformation in case (b) is shown in Fig. 7. A region of high pressure arises in front of the particle, pushing the wall downwards. Low pressure behind the particle pulls the wall upwards forming a sharp corner. The gap thickness between the particle and the wall never falls appreciably below its initial minimum value h0. As t increases, a pressure-gradient-driven flow through the gap reduces the pressure difference between front and back, the corner becomes less sharp and a steady configuration is approached. The corner is sharpened, albeit transiently, both by reducing h0 and by increasing V (cf. Fig. 6(a, c)).
Gap thickness h versus x at times as indicated for (a) V = 1, h0 = 0·6; (b) V = 1, h0 = 0·2; (c) V = 3, h0 = 0·6; (d) V = 1, h0 = – 3·125 with a nearly-uniform initial film thickness h ≈ 0·2417. Arrows indicate increasing times, which are: (a) 0, 0·5, 1·5, 2·5, 3·5, 5·5, 6·5; (b) 0, 0·5, 1·5, 2·5, 4·5, 5·5, 6·5; (c) 0, 0·5, 1, 1·5, 2, 2·5, 3·5; (d) 0, 1, 3, 5, 7, 9, 12, 15
Wall deformation at times indicated for V = 1, h0 = 0·2. In each case the parabola denotes the base of the particle
When the particle is indented into the wall before moving sideways (Figs 6(d) and 8) there is a well-defined peeling process in which the high-pressure region expands along the initially uniform film, reminiscent of Fig. 4(a). At large times, there is an increase in the minimum gap thickness (for this particular choice of parameters) as the system approaches steady state. In no instance did we find an example where h → 0 in finite time.
Wall deformation as t increases for the case V = 1, h0 = – 3·125 with a nearly-uniform initial film thickness h ≈ 0·2417
Figure 9(a to c) illustrates the evolution of the transverse and normal stress distributions (the integrands in (2.10)) on the particle. The large spike in the transverse stress distribution is associated with the transient sharp corner of the wall. The large pressure gradients across the gap ultimately relax to a steady state dependent on V (Fig. 9(c)). Figure 9(d, e) shows the corresponding transverse and normal forces acting on the particle when h0 > 0. There is a transient overshoot in the transverse force, although the normal force increases monotonically in magnitude. Both force components increase as V increases or h0 decreases. There is a net lift force on the particle arising from its transverse motion. We now examine transient peeling motion (Figs 6(d) and 8) and the ultimate steady state asymptotically (Appendix).
(a, b) Transverse and (c) normal stress distributions (τt and τn are the integrands in (12)), for V = 1, h0 = 0·6, at times (a) 0, 0·5, 0·75; (b) 3, 4·5, 15; (c) 1, 3, 15. Arrows indicate increasing time. (d) Transverse and (e) normal force on the particle for (i) V = 1, h0 = 0·6, (ii) V = 1, h0 = 0·2, (iii) V = 3, h0 = 0·6
5.1 Peeling asymptotics
For a particle which is initially indented into the wall and then pushed sideways, we observe a peeling motion (Figs 6(d) and 8). The motion of the wall (in the frame of the particle) drags a small fluid flux out through the downstream (left-hand) limit of the gap region at x = x− (see the inset to Fig. 10(a)). A region of low pressure where the surfaces separate causes the wall to be sucked upwards forming a sharp transient corner. The wall motion also drags a much larger amount of fluid into the upstream (right-hand) limit of the gap region, peeling the wall away from the particle. We wish to predict the time over which this transient behaviour persists.
(a) X0 versus t when V = 1, h0 = – 3·125 using (5.2) (solid curve); crosses mark values of X0(t) from numerical simulations. Inset shows moving corner X0(t) and the quasisteady boundary of the thin-gap region x−. (b) x− versus V for values of h0 indicated; crosses denote values from the numerical simulations
We exploit the similarity between the transverse peeling motion evident in Fig. 6(d) and the perpendicular pulling behaviour (Fig. 4). We see from Fig. 6(d) that h is quasisteady in the left-hand side of the domain and has uniform thickness, a result of the draining flow described in section 3.2. Draining is very slow on the timescale of peeling, so we take h ≈ δ in x− < x < X0, as illustrated in the inset to Fig. 10(a).
The subsequent steady state to which the system evolves is a problem that has been well studied in various forms (8, 3, 16, 17). To connect with and extend these studies, we present a brief analysis of this problem in the Appendix using phase-plane techniques. Table 1 summarizes limiting expressions for the normal and transverse forces acting on the cylinder when the flux passing between the cylinder and the wall takes extreme values. These expressions are consistent with, for example, Coyle's analysis of deformable roll coating (8) (although here we are able to determine coefficients exactly). Figure B in the Appendix illustrates force behaviour for intermediate fluxes.
Leading-order expressions for the magnitudes of the flux per unit length and horizontal and vertical forces per unit length, in dimensional variables, for a cylinder moving steadily near a deformable wall.
\(Q^{{\ast}}{\ll}{\kappa}^{{\ast}}{\vert}h_{0}^{{\ast}}{\vert}^{\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{<}0\) | \(Q^{{\ast}}{\gg}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{>}0\) | |
|---|---|---|
| Q* | \(\left(\frac{2}{81}\right)^{\frac{1}{4}}({\mu}^{{\ast}}V^{{\ast}3}/{\kappa}^{{\ast}})^{\frac{1}{2}}(R^{{\ast}}/{\vert}h_{0}^{{\ast}}{\vert})^{\frac{1}{4}}\) | \(\frac{2}{3}V^{{\ast}}h_{0}^{{\ast}}\) |
\(F_{\mathrm{t}}^{{\ast}}\) | \(0.2243\left(\frac{81}{2}{\vert}h_{0}^{{\ast}}{\vert}\right)^{\frac{3}{4}}\left({\mu}^{{\ast}}{\kappa}^{{\ast}}V^{{\ast}}R^{{\ast}\frac{1}{2}}\right)^{\frac{1}{2}}\) | \({\pi}{\mu}^{{\ast}}V^{{\ast}}(2R^{{\ast}}/h_{0}^{{\ast}})^{\frac{1}{2}}\) |
\(F_{\mathrm{n}}^{{\ast}}\) | \(\frac{4}{3}{\kappa}^{{\ast}}R^{{\ast}\frac{1}{2}}{\vert}2h_{0}^{{\ast}}{\vert}^{\frac{3}{2}}\) | \(3{\pi}R^{{\ast}\frac{3}{2}}({\mu}^{{\ast}}V^{{\ast}})^{2}/\left(2^{\frac{3}{2}}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}\right)\) |
\(Q^{{\ast}}{\ll}{\kappa}^{{\ast}}{\vert}h_{0}^{{\ast}}{\vert}^{\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{<}0\) | \(Q^{{\ast}}{\gg}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{>}0\) | |
|---|---|---|
| Q* | \(\left(\frac{2}{81}\right)^{\frac{1}{4}}({\mu}^{{\ast}}V^{{\ast}3}/{\kappa}^{{\ast}})^{\frac{1}{2}}(R^{{\ast}}/{\vert}h_{0}^{{\ast}}{\vert})^{\frac{1}{4}}\) | \(\frac{2}{3}V^{{\ast}}h_{0}^{{\ast}}\) |
\(F_{\mathrm{t}}^{{\ast}}\) | \(0.2243\left(\frac{81}{2}{\vert}h_{0}^{{\ast}}{\vert}\right)^{\frac{3}{4}}\left({\mu}^{{\ast}}{\kappa}^{{\ast}}V^{{\ast}}R^{{\ast}\frac{1}{2}}\right)^{\frac{1}{2}}\) | \({\pi}{\mu}^{{\ast}}V^{{\ast}}(2R^{{\ast}}/h_{0}^{{\ast}})^{\frac{1}{2}}\) |
\(F_{\mathrm{n}}^{{\ast}}\) | \(\frac{4}{3}{\kappa}^{{\ast}}R^{{\ast}\frac{1}{2}}{\vert}2h_{0}^{{\ast}}{\vert}^{\frac{3}{2}}\) | \(3{\pi}R^{{\ast}\frac{3}{2}}({\mu}^{{\ast}}V^{{\ast}})^{2}/\left(2^{\frac{3}{2}}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}\right)\) |
Leading-order expressions for the magnitudes of the flux per unit length and horizontal and vertical forces per unit length, in dimensional variables, for a cylinder moving steadily near a deformable wall.
\(Q^{{\ast}}{\ll}{\kappa}^{{\ast}}{\vert}h_{0}^{{\ast}}{\vert}^{\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{<}0\) | \(Q^{{\ast}}{\gg}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{>}0\) | |
|---|---|---|
| Q* | \(\left(\frac{2}{81}\right)^{\frac{1}{4}}({\mu}^{{\ast}}V^{{\ast}3}/{\kappa}^{{\ast}})^{\frac{1}{2}}(R^{{\ast}}/{\vert}h_{0}^{{\ast}}{\vert})^{\frac{1}{4}}\) | \(\frac{2}{3}V^{{\ast}}h_{0}^{{\ast}}\) |
\(F_{\mathrm{t}}^{{\ast}}\) | \(0.2243\left(\frac{81}{2}{\vert}h_{0}^{{\ast}}{\vert}\right)^{\frac{3}{4}}\left({\mu}^{{\ast}}{\kappa}^{{\ast}}V^{{\ast}}R^{{\ast}\frac{1}{2}}\right)^{\frac{1}{2}}\) | \({\pi}{\mu}^{{\ast}}V^{{\ast}}(2R^{{\ast}}/h_{0}^{{\ast}})^{\frac{1}{2}}\) |
\(F_{\mathrm{n}}^{{\ast}}\) | \(\frac{4}{3}{\kappa}^{{\ast}}R^{{\ast}\frac{1}{2}}{\vert}2h_{0}^{{\ast}}{\vert}^{\frac{3}{2}}\) | \(3{\pi}R^{{\ast}\frac{3}{2}}({\mu}^{{\ast}}V^{{\ast}})^{2}/\left(2^{\frac{3}{2}}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}\right)\) |
\(Q^{{\ast}}{\ll}{\kappa}^{{\ast}}{\vert}h_{0}^{{\ast}}{\vert}^{\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{<}0\) | \(Q^{{\ast}}{\gg}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}R^{{\ast}{-}\frac{1}{2}}/{\mu}^{{\ast}},\) \(h_{0}^{{\ast}}{>}0\) | |
|---|---|---|
| Q* | \(\left(\frac{2}{81}\right)^{\frac{1}{4}}({\mu}^{{\ast}}V^{{\ast}3}/{\kappa}^{{\ast}})^{\frac{1}{2}}(R^{{\ast}}/{\vert}h_{0}^{{\ast}}{\vert})^{\frac{1}{4}}\) | \(\frac{2}{3}V^{{\ast}}h_{0}^{{\ast}}\) |
\(F_{\mathrm{t}}^{{\ast}}\) | \(0.2243\left(\frac{81}{2}{\vert}h_{0}^{{\ast}}{\vert}\right)^{\frac{3}{4}}\left({\mu}^{{\ast}}{\kappa}^{{\ast}}V^{{\ast}}R^{{\ast}\frac{1}{2}}\right)^{\frac{1}{2}}\) | \({\pi}{\mu}^{{\ast}}V^{{\ast}}(2R^{{\ast}}/h_{0}^{{\ast}})^{\frac{1}{2}}\) |
\(F_{\mathrm{n}}^{{\ast}}\) | \(\frac{4}{3}{\kappa}^{{\ast}}R^{{\ast}\frac{1}{2}}{\vert}2h_{0}^{{\ast}}{\vert}^{\frac{3}{2}}\) | \(3{\pi}R^{{\ast}\frac{3}{2}}({\mu}^{{\ast}}V^{{\ast}})^{2}/\left(2^{\frac{3}{2}}{\kappa}^{{\ast}}h_{0}^{{\ast}\frac{7}{2}}\right)\) |
6 Transverse motion of a sphere
(a) Steady downward wall deflection H along θ = 0 determined by simulation (solid) and (6.2) (dashed) for V = 0·01, h0 = 0·5. (b) Wall height – p(x, 0, t) (left-hand panels) and contours – p(x, y, t) (right) at increasing times for V = 1, h0 = 0·5 (contour values are chosen arbitrarily). (c) Transverse force, Ft, and normal force, Fn at steady state versus h0 for V = 1 and ϵ = 0·00125. Dashed lines show the leading-order values of the force in the stiff wall limit (6.3); symbols show the second-order correction (6.4)
As for a cylinder, high pressure in front of the sphere causes the wall to bulge downwards and low pressure behind pulls it upwards (Fig. 11(a)). For larger values of V, simulations show that nonlinear effects generate an extended ridge behind the particle (the analogue of the 2D corner). This is illustrated in Fig. 11(b), which shows how the ridge sharpens transiently during the initial stages of the sphere's motion. Further simulations (in (7)) show how the ridge sharpens at steady state as the particle speed increases. The ridge is illustrated more clearly in Fig. 12, which shows the steady-state film thickness arising after the particle is indented into the wall and then translated sideways. The ridge is an arc that is not quite circular. At no point did our simulations indicate that the gap thickness would reach zero in finite time.
Contour plot showing wall deformation for the case V = 1, h0 = – 1 at t = 2. The initial film thickness in the indented region beneath the particle was approximately 0·2
The simulations also provide approximations for the transverse and normal forces Ft(t) and Fn(t) from (2.8) and (2.7). Figure 11(c) shows the steady-state forces versus h0 for V = 1 and ϵ = 0·00125. Each force increases as h0 is reduced, similar to Fig. 9(d). The transverse force shows a transient overshoot, the duration of which increases as h0 decreases (not shown; see (7)). The simulations converge to the leading-order force in the stiff wall limit (
7 Discussion
Using a simple model capturing the elastohydrodynamic interaction between a particle and a deformable wall, we have demonstrated how the motion of the particle normal or transverse to the wall is characterized by some striking transient effects. When the particle is pushed normally towards a wall, there is a transient overshoot in the normal force associated with trapping of fluid between the particle and the wall. This is evident even when the motion is slow ( T ≫ 1) and this overshoot persists at large times (Fig. 3a). We used an early-time similarity solution (Fig. 3c) to describe fluid trapping and large-time asymptotics to describe late-time fluid draining. If the particle is then pulled away from the wall, a quasisteady travelling wave (which exhibits waiting-time behaviour) determines the rate at which the wall peels away from the base of the particle. This peeling process causes the wall to ‘stick’ to the base of the particle, resulting in a transient adhesive force (so that Fn < 0 in Fig. 4(c)) which increases in magnitude with the speed of removal. Provided the initial film between the particle and the wall is sufficiently thin (δ ≪ 1 in (4.3)), the peeling speed is insensitive to δ. The wall forms a sharp corner during the peeling process (Fig. 5); clearly this places an upper bound on the speed under which the small-slope assumption of lubrication theory is valid. Our results in sections 3, 4 in a planar geometry for a cylindrical particle are readily extendible to an axisymmetric geometry for a spherical particle.
The steady motion of a cylinder moving transversely over a deformable substrate is a well-studied problem and the formation of a sharp corner towards the rear of the particle has been widely reported (8, 10, 3, 17). Using a phase-plane analysis and asymptotics for large and small fluxes, we recovered scalings identified previously by other authors (Table 1), providing new asymptotic predictions of coefficients in the small-flux limit. Of greater novelty are our predictions of the transient effects that precede steady motion. If the wall is initially flat, the corner that develops is transiently significantly sharper than at steady state (Fig. 7). This is also the case if the wall is initially indented; here, a quasisteady peeling process controls the time taken to reach steady state (Figs 6(d) and 8), which we were again able to capture using matched asymptotics. In both cases, the drag on the particle exhibits transient overshoot (Fig. 9d), while the elastohydrodynamic lift on the particle increases monotonically to steady state, at least for the cases we considered (Fig. 9e). Despite the development of very large transient gradients in gap thickness, under no conditions did we observe any evidence of ‘choking,’ whereby the gap thickness approaches zero in finite time, at least within the framework of the present model.
We also used spatially-two-dimensional simulations to examine the corresponding behaviour beneath a translating sphere. The corner beneath the cylinder appears here as a crescent-shaped ridge (Fig. 12): this too sharpens transiently as the sphere approaches steady state (Fig. 11b), and also as the speed of the sphere's motion increases. The expense of simulations hindered a comprehensive coverage of parameter space; while the large-flux limit is accessible to a regular perturbation expansion (6.2), the asymptotics of the small-flux limit remains a challenging open problem. In contrast to other numerical studies of related problems, we found the ridge to be more pronounced than that arising in an elastic half-space (2) (although it was not possible to compare equivalent parameters). More strikingly, Křupka et al. (22), using a lubricant with pressure-dependent viscosity, found both experimentally and theoretically that the maximum wall deformation can occur off the axis of symmetry, a feature not observed in our simulations. There are many other important features of practical application that we have not addressed here, including interfacial roughness (23), intermolecular forces causing film rupture and dewetting (12), surface slip (24), adsorbed polymer layers (25), and nonconforming geometries (17). Transient elastohydrodynamic effects are likely to be affected particularly by a poroelastic substrate having its own characteristic timescale (26, 17). These all merit further investigation.
APPENDIX
Steady transverse motion of a cylinder
![(a) Trajectories in the phase plane for \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{Q}}{=}1.\) \end{document} (b) Solutions of (A.1), (A.2) plotted in the \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(({\hat{x}},{\hat{h}})\) \end{document}-phase plane; (A.2) is applied at \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{x}}{=}{\pm}10,\) \end{document} with \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(0{\cdot}2{\leq}{\hat{Q}}{\leq}10.\) \end{document} (c) A trajectory in the small-flux limit (solid), showing the isocline \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{x}}{\hat{h}}^{3}{=}6{\hat{h}}{-}12{\hat{Q}}\) \end{document} (dashed) and the far-field condition \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{h}}{=}{\hat{h}}_{0}{+}\frac{1}{2}{\hat{x}}^{2}\) \end{document} (dot-dashed)](https://cdn.statically.io/img/oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/qjmam/59/2/10.1093/qjmam/hbl002/2/m_qjmamjhbl002a01_lw.gif?Expires=1731669295&Signature=Rf~BuvVvgtqPErk9-CBYi5ujgNJ5AFoPrYFdvzlpiUkBfvLmaFxVXr4lu7N4oWXtsAciNQM94Qt2Et~eF95Sg4CAmkMBLopSYUPrxM-sneKAjAWdpzqI7h7yghbdqV4rn2DTVgMW98kLubnrWacPw5gSUPUrhy~txReQGaiFobvTYtXoHjYnJuhI3k2WTulyBiHE5jcd-ATRP-DaBlfN1Uq21AwvSfznoo4CZHeaYur6nl~2y7QFy1tTdiPZf88OSB0yjirE7D0IZD7ktt85Z~ikpNKR7en3XIlJiiczfr4Lq8SRPSPNH8LtiKxyWxA-v2jjlt2WchCYd5yiVY3ybA__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Trajectories in the phase plane for
![(a) \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{h}}_{0}\) \end{document} versus \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\mathrm{log}{\hat{Q}}\) \end{document} from numerical simulations of (A.1), (A.2) (crosses) compared with small- and large- \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{Q}}\) \end{document}limits (A.5) (dotted) and (A.4) (dashed). (b) Normal force, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{F}}_{\mathrm{n}},\) \end{document} and transverse force, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{F}}_{\mathrm{t}},\) \end{document} versus \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{h}}_{0}\) \end{document} (solid). Dashed lines denote the asymptotic solutions for small and large \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \({\hat{h}}_{0}\) \end{document} (A.7a), (A.10), (A.11)](https://cdn.statically.io/img/oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/qjmam/59/2/10.1093/qjmam/hbl002/2/m_qjmamjhbl002a02_lw.gif?Expires=1731669295&Signature=LGs9mXZw-EgMtnRNCmnQ1f80FoDYoLz5Rk35uONn0BThZ92zfV9E0YPwM1WbAawUDF5kl3IoweRpKFK~15rKTo0AbNLHdDJIsBtU1nWF3Dyta-OEI5jOSnqrPg7hQ5kLjIw9F~YKlxhXRj~KGn7wgwh~X-0Dv7mVVFWm2iyaZJbJbvSnPtPw51gDya38GL1xdkLV14etBu1amb4foASpO9E4OjULQO2cuz41E-RdgkIk4YrHBaoFJCqkkvZ48-G5pT~aII7tzsS5z3~nOh-Q-cyfm~O1mboKS4oCdKXoLPU~kbUMy~3ZsICPJY9hNz64ORJVLg-IsopShhtAWCTJBQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a)
Present address: TRL Ltd, Crowthorne House, Nine Mile Ride, Wokingham RG40 3GA, UK.
This work was supported by the EPSRC.
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