Unidirectional flow ($\mathbf{u}=u(y){\mathbf{e}}_x$)

A unidirectional flow has velocity parallel to a single fixed direction, for example $\mathbf{u}=u(y){\mathbf{e}}_x$. If the speed is independent of the coordinate along that direction, the convective acceleration $(\mathbf{u}\cdot \nabla )\mathbf{u}$ vanishes. The steady Navier-Stokes equation then reduces to a linear balance of the pressure gradient and viscosity, as in Couette flow and Poiseuille flow.