= Turbulent mixing-layer similarity equation
{title2=$F'''+(\xi+F)F''=0$}
For a <planar turbulent mixing layer>, let $U_c=(U_a+U_b)/2$, $\Delta U=U_a-U_b$, $\nu_T=C\Delta U\delta$, $\delta=S\Delta U x/U_c$, and $\xi=\sqrt{S/C}\,y/\delta$. The <stream function> $\psi=U_c[y+\delta\sqrt{C/S}F(\xi)]$ enforces mean <incompressibility>. Substitution into $UU_x+VU_y=\nu_TU_{yy}$ gives the displayed <ordinary differential equation>, with $F'(\pm\infty)=\pm\Delta U/(2U_c)$. When $\Delta U/U_c$ is small, its linear approximation is $F'''+\xi F''=0$, so $F''$ is a <Gaussian function> and $F'$ an <error function>.
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