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The wall speed setsUsing $x$ as the local streamwise length giveswhich is large when $x\gg \sqrt{\nu /\alpha }$. The boundary-layer scalingthen givesso the stretching-sheet layer has constant thickness to leading order.

Let $U(x)$ and $\Delta (x)$ denote the characteristic vertical speed and width. Integrating the result of part (a), with $b(x)=B{x}^{-1/5}$, gives the scalingThe boundary-layer scaling in the momentum equation balances vertical inertia against transverse viscosity:Substituting this into the momentum-flux relation givesand thereforeReturning to the viscous-inertial balance then givesThese are the similarity scaling of a two-dimensional laminar plume.

Boundary-layer theory assumes steady incompressible high-Reynolds-number flow, $\delta /x\ll 1$, weak streamwise viscous diffusion, nearly constant pressure across the layer, and outer pressure gradient $-{\rho }^{-1}{p}_x=U{U}^{\prime }$.

For $\varphi =-A{r}^k\cos (k\theta )/k$,At $\theta =\pm \pi /k$ the normal velocity vanishes; along the upper wall $U(x)=A{x}^{k-1}$. Boundary-layer scaling givesSet $m=k-1$. The Falkner-Skan equation becomesso $\alpha =k-1$, $\beta =-k/2$, $\gamma =1-k$, with
$f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. Finallyrequires $x\gg (\nu /A{)}^{1/k}$; thus $a=1/k$ and the tip region violates the approximation.