The asymptotic expansions at large Reynolds numbers for steady motion between non-coaxial rotating cylinders

Abstract

The motion of a viscous fluid contained between two rotating, circular cylinders whose axes are set slightly apart is considered. The equations of viscous motion are linearized by expanding the stream function in the form $\sum\limits^\infty_{n=0}\psi _n \gamma^n$, where γ is a parameter which depends on the distance between the cylinder axes. The ensuing analysis appears to hold for all values of the fluid viscosity v, and in particular for small values of v.

The asymptotic behaviour of the solutions for small v is examined, attention being mainly confined to the first order stream function ψ₁ and the corresponding component of vorticity ζ₁. Outside the boundary layers, where, for small v, ψ₁ may be expanded asymptotically as $\sum \limits ^\infty _{n=0} \psi^{(n)}_1 v^{n|2$, the terms ζ⁽ⁿ⁾₁ of the corresponding expansions for the vorticity are shown to be uniform throughout the fluid. It is noted that the asymptotic expansions of ψ₁ for the region of the boundary layers and for the region outside the boundary layers may be combined in a single expansion which holds in both regions. The leading terms of this expansion are calculated by boundary layer methods.