Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / b

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.

With zero imposed pressure gradient, the two-dimensional boundary-layer equations areIntroduce the stream function byThe boundary conditions areThe additive constant in $\psi$ may be chosen so that $\psi (x,0)=0$.

PutThenSubstitution in the boundary-layer momentum equation, using $\nu /{\delta }^2=\alpha$, givesThe wall and far-field conditions becomeThese are the similarity equations for the boundary layer over a linearly stretching sheet.

Set $f=A+B{e}^{-C\eta }$ with $C>0$. The differential equation reduces toso $A=C$. The two wall conditions giveHence $C^2=1$, and positivity of $C$ yieldsor

As $\eta \to \infty$,Thus the tangential velocity matches the quiescent outer fluid, but the normal velocity tends toThe stretching sheet entrains fluid toward itself at a constant leading-order speed. Consequently a globally stationary exterior flow is impossible: the outer flow must provide this normal influx and complete the mass balance away from the local boundary-layer approximation.