Couette flow

Example

Consider a fluid of density $\rho > 0$ and dynamic viscosity $\mu > 0$ between two parallel plates $y = 0$ and $y = h$, with $h > 0$. The lower plate is at rest and the upper plate moves in the $x$-direction with constant speed $U$. Plane Couette flow is $$u = \Big( \frac{U y}{h}, 0, 0 \Big), \qquad p = p_0,$$ with $p_0$ a constant.

Check: $u$ depends only on $y$ and points in the $x$-direction, so $\nabla \cdot u = 0$ and $(u \cdot \nabla) u = (U y / h) \, \partial_x u = 0$. Also $\Delta u = 0$ since $u$ is linear, and $\nabla p = 0$. So the steady Navier–Stokes equations $\rho (u \cdot \nabla) u = -\nabla p + \mu \Delta u$ hold. At $y = 0$ the velocity is $0$ and at $y = h$ it is $(U, 0, 0)$, so the no-slip condition holds on both plates.