February | 2015 | Snapshots in Mathematics!

Let us give a few examples of boundary layer solutions to the Prandtl problem, derived in the last lecture. In 2D, we recall the Prandtl layer problem:

$\displaystyle \left \{ \begin{aligned} u_t + u u_x + v u_z &= \mu u_{zz} - p_x, \qquad v = -\int_0^z u_x(x,\theta)\; d\theta \\ u_{\vert_{t=0}} &= u_0(x,z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,x,z) = u^E(t,x), \end{aligned} \right. \ \ \ \ \ (1)$

with the pressure gradient: ${p_x = -u_t^E- u^Eu^E_x(t,x)}$ , where ${u^E}$ denotes the tangential component of Euler flow on the boundary ${y=0}$ . Here, the tangential velocity component ${u}$ is an (only) unknown scalar function, and the normal velocity component ${v}$ is defined through the divergence-free condition. A simplest example: in the case ${u^E}$ is independent of ${x}$ and ${t}$ , any solution to the following heat problem