On the Motion of Two Microspheres in a Stokes Flow Driven by an External Oscillator Field

Mohammed M Al-Hatmi,Anton Purnama

doi:10.1155/2021/9211272

Mohammed M Al-Hatmi, Anton Purnama

Open Access

https://doi.org/10.1155/2021/9211272

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Abstract

Hydrodynamic interactions of a two-solid microspheres system in a viscous incompressible fluid at low Reynolds number is investigated analytically. One of the spheres is conducting and assumed to be actively in motion under the action of an external oscillator field, and as the result, the other nonconducting sphere moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the two-sphere system is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of each sphere is formulated. The trajectory of the sphere was found to be inversely proportional to the frequency of the external field. The results demonstrated that no collisions occur between the spheres as the system moves in a circular motion with a fixed separation distance.

Highlights

Dynamics of microparticles in a viscous fluid play important roles in many applications in medicine and technology, such as minimising surgical invasion and controlling drug delivery, see, e.g., [1] and [2]. e study of the motion of small particles in suspension has been of interest to scientists for many years and is still an active area of research, see, e.g., [3,4,5,6,7,8]. e movement of a particle due to the oscillatory external forces in a viscous fluid represents a classical problem of fluid dynamics. is motion represents a model problem for the use of the dynamical approach in fluid dynamics and for the studies of turbulence
We studied analytically the motion of a system of two microspheres in a Stokes flow driven by an external oscillator field
Our choice of slow time s εt and fast time τ t/ε led to a result that agrees with the experimental studies of an oscillating sphere in a viscous fluid, see, e.g., [23] and [21]

Summary

Introduction

Dynamics of microparticles in a viscous fluid play important roles in many applications in medicine and technology, such as minimising surgical invasion and controlling drug delivery, see, e.g., [1] and [2]. e study of the motion of small particles in suspension has been of interest to scientists for many years and is still an active area of research, see, e.g., [3,4,5,6,7,8]. e movement of a particle due to the oscillatory external forces in a viscous fluid represents a classical problem of fluid dynamics. is motion represents a model problem for the use of the dynamical approach in fluid dynamics and for the studies of turbulence. A conducting or active microsphere suspended in an external oscillatory force tends to accelerate in the direction of the applied external field. This type of interaction is classical in character, there are certain features that do not seem to be widely understood. Interactions of a system of two microspheres are two-fold: first, the active sphere moves in the direction of the external field, and second, the other nonconducting sphere moves due to the local surrounding fluid velocity generated by the motion of the active sphere. E asymptotic formulation of the problem leads us to study the motion of the spheres with a timeperiodic external force, as described by the Stokes equation, where the fluid inertial effects are neglected. Our analytical treatments are simple, but it can be considered as the basis for the development of a full theory of particle suspension

Formulation of Problem

Two-Timing Method and Asymptotic Procedure

Conclusion