Low Reynolds number oscillatory flow past a slowly rotating sphere

Abstract

In this paper the flow of a Newtonian viscous fluid, oscillating with angular velocityλ and amplitudeV∞, past a rigid sphere of radiusa which is slowly rotated with angular velocity ω is studied. Denoting the kinematic viscosity of the fluid byv, three parameters are involved in the analysis, namely: the Reynolds numberR =V∞ α/v and the Taylor numbersτ=a2ω/v andσ=a2gl/v. Asymptotic approximation is employed assuming thatR andτ are small whileσ is arbitrary. In the leading approximation, the swell velocity represents a double roll system while the oscillatory radial and polar velocities are reminiscent of Stokes (1851) celebrated solutions. In the terms of orderτ2/R approximation, the swell velocity is suppressed while the remaining two velocities are purely steady and independent ofσ. When the approximation is of orderR, the swell velocity is now purely oscillatory while the radial and polar velocities exhibit steady streaming components on which are superimposed oscillatory components. A quantitative study reveals that the steady streaming velocities are larger whenσ is small than when it is large.