The main purpose of this paper is to present some analytical solutions for the Taylor-Couette flow problem of vortex-cell formation in the gap bounded by two concentric rotating circular cylinders of finite length. The solutions are obtained by using the basic equations of the mechanical part of the theory of a Cosserat point with any number of directors, the general thermomechanical formulation of which has been given recently by Green & Naghdi (Q. Jl Mech. appl. Math 44, 335-355 (1991)). Several special solutions are given for a particular cylinder geometry and Reynolds number of flow to compare with the experimental results of Benjamin & Mullin (J. Fluid Mech. A 121, 219-230 (1982)). Flow stability and other possible types of solutions are also briefly discussed.