One-dimensional flow  is valid when both plates are infinitely long in the streamwise () and spanwise () directions. When the spanwise length is finite, the flow becomes two-dimensional and  is a function of both  and . However, the infinite length in the streamwise direction must be retained in order to ensure the unidirectional nature of the flow.
As an example, consider an infinitely long rectangular channel with transverse height  and spanwise width , subject to the condition that the top wall moves with a constant velocity . Without an imposed pressure gradient, the Navier–Stokes equations reduce to
with boundary conditions
Using separation of variables, the solution is given by
When , the planar Couette flow is recovered, as shown in the figure.