Von Mises boundary-layer transformation ($U_x=(U{U}_{\Psi }{)}_{\Psi }$)

In an attached two-dimensional boundary layer with zero pressure gradient, use the stream function $\Psi$ as transverse coordinate, assuming ${\Psi }_Y=u>0$ away from the wall. If $u=U(x,\Psi )$ and ${\Psi }_x=-V$, the chain rule gives $u_x=U_x-V{U}_{\Psi }$, $u_Y=U{U}_{\Psi }$ and $u_{YY}=U(U{U}_{\Psi }{)}_{\Psi }$. Substitution into $u{u}_x+V{u}_Y=u_{YY}$ proves the displayed degenerate nonlinear diffusion equation. Its flat-plate similarity form $U=g(\Psi /\sqrt{x})$ obeys $(g{g}^{\prime }{)}^{\prime }+\eta g^{\prime }/2=0$, $g(0)=0$, $g(\infty )=1$.

The wall coordinate is singular: finite physical shear corresponds to $g\sim C\sqrt{\eta }$, rather than a finite transformed derivative. A monotone profile exists by putting $q=g{g}^{\prime }$, $q(0)=a>0$. Then $q^{\prime }/q=-\eta /(2g)$ and $(g^2{)}^{\prime }=2q$. Since $q\le a$, $g\le \sqrt{2a\eta }$; this forces $q$ to decay at least as $\exp [-c{\eta }^{3/2}]$. Thus $g$ has a finite positive limit, which can be normalized to one by $g(\eta )\mapsto Ag(\eta /\sqrt{A})$.