Feeds:
Posts
Comments

Archive for February, 2015

We study the linearization of 2D Navier-Stokes around a boundary layer {\vec v_\mathrm{app} =[\bar u, \sqrt \nu \bar v](t', x', y'/\sqrt \nu)}, in which {(t',x',y')} denotes the original coordinates for Navier-Stokes. Materials in this lecture are drawn from the joint paper(s) with Grenier and Guo (here and here).

(more…)

Read Full Post »

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

\displaystyle u_t = \mu u_{zz} , \qquad u_{\vert_{t=0}} = u_0(z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,z) = u^E

gives a boundary layer solution of (1) in the form {[u,0]}. We call a solution of this form to be a shear flow.

(more…)

Read Full Post »