Low Reynolds number oscillatory flow past a slowly rotating sphere (Q1062844)

The paper presents analytical solutions to the equations that describe the viscous flow, oscillating with an angular velocity \(\lambda\) and an amplitude \(U_{\infty}\) past a sphere of radius a. The sphere is slowly rotating with an angular velocity \(\omega\) about an axis normal to the oscillatory flow direction. The problem involves three parameters, namely: the Reynolds number \(Re=U_{\infty}a/\nu\) and the Taylor numbers \(\tau =a^ 2\omega /\nu\) and \(\sigma =a^ 2\lambda /\nu\), where \(\nu\) is the kinematic viscosity of the fluid. Asymptotic approximations for the velocity components in terms free and dependent of \((\tau^ 2/Re\), Re,\(\tau)\) etc. with Re and \(\tau\) small parameters, while \(\sigma\) is arbitrary, are used. Closed form solutions for the leading terms of these series expressed in terms of Bessel functions are obtained. The author shows that for the leading terms, the swell velocity represents a double roll system while the oscillatory radial and polar velocities represent the well-known Stokes solution. Then, in the terms of order \(\tau^ 2/Re\) approximation, the swell velocity is suppressed while the other two velocities are purely steady and independent of \(\sigma\). It is further shown that when the approximation is of order Re, the swell velocity is purely oscillatory and the radial and polar velocities exhibit steady streaming components on which are superimposed oscillatory components. The author has solved a difficult problem of three dimensional flows in a very elegant way. This is a good paper for applied mathematicians to study the method for solving such difficult types of nonlinear partial differential equations.