Capillary lubrication of a spherical particle near a fluid interface

TL;DR

Capillary-lubrication theory is used to analyze a sphere translating near a deformable fluid interface, where the interfacial tension and gravity couple to generate lift at zero Reynolds number. The authors perform a perturbation expansion in the small capillary compliance $\kappa$ and solve for the zeroth-order pressure $P_0$ and the first-order interfacial deflection $\Delta_1$ (split into $\Delta_{10}$, $\Delta_{11}$) and pressure $P_1$ ($P_{10}$, $P_{11}$) to obtain scaling laws and numerical results across the Bond number $Bo$. They derive explicit expressions for the first-order vertical, horizontal forces and the torque, with $\alpha_k$ and $\beta_k$ prefactors that depend on $Bo$ and interpolate between capillary-dominated and Winkler-like responses; the crossover occurs near $Bo\sim 1$. This work provides quantitative predictions for soft-lubrication effects near tensile interfaces, with potential relevance to microbial motility at interfaces, levitating Leidenfrost droplets, and oil-droplet lifetimes near walls in soft matter.

Abstract

The lubricated motion of an object near a deformable boundary presents striking subtleties arising from the coupling between the elasticity of the boundary and lubricated flow, including but not limited to the emergence of a lift force acting on the object despite the zero Reynolds number. In this study, we characterize the hydrodynamic forces and torques felt by a sphere translating in close proximity to a fluid interface, separating the viscous medium of the sphere's motion from an infinitely-more-viscous medium. We employ lubrication theory and perform a perturbation analysis in capillary compliance. The dominant response of the interface owing to surface tension results in a long-ranged interface deformation, which leads to a modification of the forces and torques with respect to the rigid reference case, that we characterise in details with scaling arguments and numerical integrations.

Figures (7)

• Figure 1: Schematic of the system. A sphere of radius $a$ immersed in a viscous fluid of viscosity $\eta$ and density $\rho$ moves near a fluid interface. The undeformed gap profile is noted $h(r,t)$, with $r$ the horizontal radial coordinate and $t$ the time. The origin of coordinates is located at the undeformed fluid interface ($z = 0$) in line with the center of mass of the sphere ($r = 0$). The interface separates the top fluid from a secondary fluid of viscosity $\eta_{\textrm{sl}}$, with $\eta_{\textrm{sl}}\gg \eta$, and density $\rho_{\textrm{sl}}$ at the bottom, i.e.$\rho_{\textrm{sl}}>\rho$. The sphere has prescribed horizontal velocity $u$ and vertical velocity $\dot{d}$, where $d=h(0,t)$ denotes the instantaneous distance between the sphere bottom and the undeformed fluid interface. The interface deflection field is denoted as $\delta(r,t)$, the acceleration of gravity is denoted as $g$, and the surface tension is denoted as $\sigma$.
• Figure 2: Isotropic component $\Delta_{10}$ of the amplitude of the first-order interface deflection as a function of the radial coordinate $R$ (solid black line), as calculated from Eq. (\ref{['eq:delta10_soln']}), for $\textrm{Bo} = 0.01$, $D = 1$ and $\dot{D} = 1$. For comparison, the inner solution (red solid line), the outer solution (blue solid line), and the matched crossover expression (symbols), from Eqs. (\ref{['eq:delta_10_inner_soln']}), Eq. (\ref{['eq:delta_10_outer_soln']}) and (\ref{['eq:delta_10_inner_soln_matched']}) respectively, are also shown.
• Figure 3: Isotropic component $\Delta_{10}$ (a) and anisotropic component $\Delta_{11}$ (b) of the amplitude of the first-order interface deflection as a function of the radial coordinate $R$, as calculated from Eqs. (\ref{['eq:delta10_soln']}) and (\ref{['eq:delta11_soln']}), for $D = 1$, various $\textrm{Bo}$ as indicated, and for either $\dot{D} = 1$ (a) or $U=1$ (b). The black solid line denotes the limiting profile for $\textrm{Bo} = 0$ in the anisotropic case, as given in Eq. (\ref{['eq:delta_11_lim']}).
• Figure 4: Auxiliary functions $\phi_k$ (see Eq. (\ref{['eq:p10_decomposition']})) of the isotropic component $P_{10}$ of the first-order magnitude $P_1$ of the excess pressure field, as functions of the radial coordinate $R$, for various Bond numbers $\textrm{Bo}$, as obtained by numerically solving Eq. (\ref{['eq:governing_eqn_P_10']}) with $D = 1$.
• Figure 5: Auxiliary functions $\phi_k$ (see Eq. (\ref{['eq:p11_decomposition']})) of the anisotropic component $P_{11}$ of the first-order magnitude $P_1$ of the excess pressure field, as functions of the radial coordinate $R$, for various Bond numbers $\textrm{Bo}$, as obtained by numerically solving Eq. (\ref{['eq:governing_eqn_P_11']}) with $D = 1$.