Creeping motion of a solid particle inside a spherical elastic cavity

Abstract

On the basis of the linear hydrodynamic equations, we present an analytical theory for the low-Reynolds-number motion of a solid particle moving inside a larger spherical elastic cavity which can be seen as a model system for a fluid vesicle. In the particular situation where the particle is concentric with the cavity, we use the stream function technique to find exact analytical solutions of the fluid motion equations on both sides of the elastic cavity. In this particular situation, we find that the solution of the hydrodynamic equations is solely determined by membrane shear properties and that bending does not play a role. For an arbitrary position of the solid particle within the spherical cavity, we employ the image solution technique to compute the axisymmetric flow field induced by a point force (Stokeslet). We then obtain analytical expressions of the leading-order mobility function describing the fluid-mediated hydrodynamic interactions between the particle and the confining elastic cavity. In the quasi-steady limit of vanishing frequency, we find that the particle self-mobility function is higher than that predicted inside a rigid no-slip cavity. Considering the cavity motion, we find that the pair-mobility function is determined only by membrane shear properties. Our analytical predictions are supplemented and validated by fully resolved boundary integral simulations where a very good agreement is obtained over the whole range of applied forcing frequencies.

Towards an analytical description of active microswimmers in clean and in surfactant-covered drops

“…Our approach is based on the method originally introduced by Fuentes et al [178,179], who derived the solution for a Stokeslet acting outside a clean viscous drop. An analogous approach was employed by some of us to derive the Stokeslet solution near [180,181] or inside [182,183] a spherical elastic object, and outside a surfactant-covered drop [184]. We find that the presence of the surfactant alters the swimming behavior of the encaged microswimmer by enhancing its rate of rotation.…”

“…The details of derivation have previously been reported by some of us in ref. [182], and will thus be omitted here. As shown there, the free-space Stokeslet for an axisymmetric point force F = F d can be expanded in terms of an infinite series of harmonics centered at x 1 via the Legendre expansion as…”

“…In addition, the image solution inside the drop can readily be determined from Lamb's general solution [191,194], and can conveniently be expressed in terms of interior harmonics based at x 1 as [182]…”

“…Next, the solution of the flow problem outside the drop can likewise be obtained using Lamb's general solution, and can be expressed in terms of exterior harmonics based at x 1 as [182]…”

Asymmetric Stokes flow induced by a transverse point-force acting near a finite-sized elastic membrane

“…Notably, the in-plane traction jumps f r and f φ are solely determined by the shear elasticity of the membrane whereas the out-of-plane traction jump f z is determined by bending resistance only. It is worth noting that this behavior is in stark contrast to curved membranes where coupling between shear and bending occurs [74][75][76][77] .…”