This paper considers the dynamics of cylindrically arranged parallel layers of smectic A liquid crystal subjected to Couette flow. Governing equations are constructed using a recently developed dynamic theory for smectic A (Stewart 2007 Contin. Mech. Thermodyn. 18 343–60). These equations are solved to provide analytical solutions for the smectic layer undulations and velocity profiles. Results show the dependence of the response time of the smectic layers upon the permeation constant and the layer compression modulus. The relaxation times for the flow profiles are shown to depend upon two viscosities; estimates for these times are shown to be shorter than that for a typical approximation to the relaxation time of the smectic layer undulations.