Driven particles at fluid interfaces acting as capillary dipoles

Abstract

The dynamics of spherical particles driven along an interface between two immiscible fluids is investigated asymptotically. Under the assumptions of a pinned three-phase contact line and very different viscosities of the two fluids, a particle assumes a tilted orientation. As it moves, it causes a deformation of the fluid interface which is also computed. The case of two interacting driven particles is studied via the Linear Superposition Approximation. It is shown that the capillary interaction force resulting from the particle motion is dipolar in terms of the azimuthal angle and decays with the fifth power of the inter-particle separation, similar to a capillary quadrupole originating from undulations of the threephase contact line. The dipolar interaction is demonstrated to exceed the quadrupolar interaction at moderate particle velocities.

Drag and diffusion coefficients of a spherical particle attached to a fluid–fluid interface

“…According to the Stokes-Einstein relation (1.1), the diffusion coefficient D directly follows from the drag coefficient f . The work is based on a recent article Dörr & Hardt (2015) in which the flow field around a sphere attached to a fluid interface with a contact angle of 90 • was computed to obtain the deformation of the interface. These earlier results are complemented by a second perturbation expansion around a contact angle of 180 • .…”

“…This case has been studied by Zabarankin (2007), who provides drag coefficient values derived from a numerical solution of a Fredholm integral equation. Recently, Dörr & Hardt (2015) have derived the asymptotic expression f (Θ, 0) = 1 2 1 + 9 16 cos Θ + O(cos 2 Θ) .…”

“…Dörr & Hardt 2015) concerning necessary corrections to the method), which is based on spherical harmonics expansions and yields the velocity and pressure fields around a slightly deformed sphere. To this end, the particle shape (given by the pair of fused spheres of radius a in our case) needs to be parameterised in spherical coordinates (r, θ, ϕ), with the origin lying in the particle's symmetry plane, according to r = r p (θ , ϕ), (2.3) and subsequently expanded in a power series…”

“…The first-order problem has been solved by Dörr & Hardt (2015) in the particle's rest frame. Because the frame of reference only affects the zeroth-order flow u (0) by addition or subtraction of the velocity field U, the first-order flow field u (1) considered by Dörr & Hardt (2015) may be directly used in the present study. The supplementary material available at http://dx.doi.org/10.1017/jfm.2016.41 contains the complete set of expressions required to calculate the velocity field u (1) .…”

“…Then, using (2.12), (2.13), (2.4) and (2.5) in conjunction with the velocity field u (1) according to Dörr & Hardt (2015), the integral occurring in (2.19) can be evaluated, yielding…”