Exact solution for the slow motion of a spherical particle in the presence of an interface with slip regime

Abstract

An analytical and numerical study for the creeping flow caused by a solid spherical particle with a slip-flow surface is considered in the presence of a fluid–fluid plane interface. The particle rotating about or translating along an axis perpendicular to the interface. The motion is investigated in the limit of low capillary number where in this situation the interface is of negligible deformation. Using a bipolar coordinate system, the stream functions are constructed for both fluid phases as Reynolds number tends to zero. The novelty of this work is allowing the slip on the surface of the particle. The matching boundary conditions at the plane interface and the slip boundary condition on the particle’s surface are applied to the truncated solutions to specify the unknown coefficients. A comparison is made between the results of the analytical solution and the results obtained from a boundary collocation method. The torque and drag force exerted on the particle are calculated using both techniques, which are found in perfect agreement. In addition to compression with collocation techniques, we also studied the predicted changes in the drag force and torque due to the presence of the plane interface and the slippage at the surface of the particle. Our results of the drag force and torque are compared with the available data in the literature for the special cases. The work is motivated by its possible application as an analytical tool in the study of locomotion of microswimmers near an interface such as synthetic swimmers and microorganisms.

Appendix

$$\begin{aligned} \alpha _{n}= & {} \left( n-1\right) \left( \sinh n_{2} \alpha +\lambda \cosh n_{2} \alpha \right) , \end{aligned}$$

(A.1)

$$\begin{aligned} \beta _{n}= & {} -\left( 2\beta ^{*} \sqrt{L^{-2} -1} \, +3\sinh \alpha \right) \left( \cosh n_{1} \alpha +\lambda \sinh n_{1} \alpha \right) \nonumber \\&-\left( 2n+1\right) \cosh \alpha \left( \sinh n_{1} \alpha +\lambda \cosh n_{1} \alpha \right) , \end{aligned}$$

(A.2)

$$\begin{aligned} \gamma _{n}= & {} \left( n+2\right) \left( \sinh n_{3} \alpha +\lambda \cosh n_{3} \alpha \right) , \end{aligned}$$

(A.3)

$$\begin{aligned} f_{n}= & {} -4\sqrt{2} \beta ^{*} \left( L^{-2} -1\right) \, \Omega ae^{-n_{1} \alpha }. \end{aligned}$$

(A.4)

$$\begin{aligned} C_{1n} \left( \zeta \right)= & {} \left( \cosh n_{1} \alpha +\lambda \sinh n_{1} \alpha \right) P_{n}^{1} \left( \zeta \right) \nonumber \\&+{\textstyle \frac{1}{2}} n_{1} w\left( \sinh n_{1} \alpha +\lambda \cosh n_{1} \alpha \right) P_{n}^{1} \left( \zeta \right) . \end{aligned}$$

(A.5)

$$\begin{aligned} h_{1n}= & {} \left( {\textstyle \frac{1}{8}} \lambda n_{2} \left( \cosh n_{3} \alpha -\cosh n_{2} \alpha \right) +\sinh n_{2} \alpha \right) \, G_{n+1} \left( \zeta \right) , \end{aligned}$$

(A.6)

$$\begin{aligned} h_{2n}= & {} \left( {\textstyle \frac{1}{8}} \lambda n_{3} \left( \cosh n_{3} \alpha -\cosh n_{2} \alpha \right) +\sinh n_{3} \alpha \right) \, G_{n+1} \left( \zeta \right) , \end{aligned}$$

(A.7)

$$\begin{aligned} h_{3n}= & {} \left[ g_{n} \left( {\textstyle \frac{1}{4}} \lambda n_{2} \left( \cosh n_{3} \alpha -\cosh n_{2} \alpha \right) +\sinh n_{2} \alpha \right) \right. \nonumber \\&-\sqrt{L^{-2} -1} \, \beta ^{*} w^{{\textstyle \frac{3}{2}} } n_{2} \left( {\textstyle \frac{1}{2}} \lambda \left( n_{3} \sinh n_{3} \alpha -\sinh n_{2} \alpha \right) +\cosh n_{2} \alpha \right) \nonumber \\&\quad \left. \, -w^{{\textstyle \frac{5}{2}} } n_{2} \left( {\textstyle \frac{1}{2}} \lambda \left( n_{3}^{2} \cosh n_{3} \alpha -n_{2}^{2} \cosh n_{2} \alpha \right) +n_{2} \sinh n_{2} \alpha \right) \right] G_{n+1} \left( \zeta \right) , \nonumber \\ \end{aligned}$$

(A.8)

$$\begin{aligned} h_{4n}= & {} \left[ g_{n} \left( {\textstyle \frac{1}{2}} \lambda \left( n_{3} \cosh n_{3} \alpha -n_{3} \cosh n_{2} \alpha \right) +\sinh n_{3} \alpha \right) \right. \nonumber \\&-\sqrt{L^{-2} -1} \, \beta ^{*} w^{{\textstyle \frac{3}{2}} } n_{3} \left( {\textstyle \frac{1}{2}} \lambda \left( n_{3} \sinh n_{3} \alpha -n_{2} \sinh n_{2} \alpha \right) +\cosh n_{3} \alpha \right) \nonumber \\&\left. -w^{{\textstyle \frac{5}{2}} } n_{3} \left( {\textstyle \frac{1}{2}} \lambda \left( n_{3}^{2} \cosh n_{3} \alpha -n_{2}^{2} \cosh n_{2} \alpha \right) +n_{3} \sinh n_{3} \alpha \right) \right] G_{n+1} \left( \zeta \right) , \nonumber \\ \end{aligned}$$

(A.9)

where $g_{n}$ which appears in $h_{3n}$ and $h_{4n}$ is given by

$$\begin{aligned} g _{n} ={\textstyle \frac{3}{4}} w^{{\textstyle \frac{1}{2}} } \left( \sinh ^{2} \alpha +2\sqrt{L^{-2} -1} \, \beta ^{*} \sinh \alpha -\sin ^{2} \xi \right) \, +{\textstyle \frac{3}{2}} w^{{\textstyle \frac{3}{2}} } \cosh \alpha -n\left( n+1\right) w^{{\textstyle \frac{5}{2}} }. \nonumber \\ \end{aligned}$$

(A.10)