Prandtl boundary layer equations

Remarks

Example: for a constant outer speed $U$ the function $u = U \operatorname{erf}\big( y / (2 \sqrt{\nu t}) \big)$, $v = 0$ is a solution, where $\operatorname{erf} = 1 - \operatorname{erfc}$. Indeed $u$ does not depend on $x$, so the equations reduce to $\partial_t u = \nu \, \partial_y^2 u$, which $u$ solves; and $u(0, t) = 0$, $u \to U$ as $y \to \infty$.

Non-example: the steady profile $u = U (1 - e^{-y})$, $v = 0$, with constant $U$, satisfies both boundary conditions but not the momentum equation, since the left side is $0$ and $\nu \, \partial_y^2 u = -\nu U e^{-y} \ne 0$.

The equations arise from the Navier–Stokes equations by rescaling $y$ and $v$ by $\mathrm{Re}^{-1/2}$ and letting the Reynolds number $\mathrm{Re}$ tend to infinity. The normal momentum equation becomes $\partial_y p = 0$, so the pressure is imposed by the outer flow of the Euler equations through Bernoulli's relation. For a steady flow the equations are parabolic in $x$, so one can march downstream from an initial profile. Their solutions may break down where the flow separates from the wall.