Constructing solutions of the Navier–Stokes equations for a fluid layer between moving parallel plates at low and moderate Reynolds numbers

In this paper, we study exact solutions of the Navier–Stokes equations for a layer between parallel plates, the distance between which varies according to an arbitrary power law and whose boundary has a no-slip condition, are under study. A solution in the form of a power series of the Reynolds number is obtained. Comparison with the exact solutions is performed, and high accuracy of expansions for Reynolds numbers $\mathrm{Re}=1-10$ is shown. An accurate estimate of the error of the Reynolds thin layer approximation is obtained.