Past exam of the theoretical mechanics course of the University of Oxford / 2011 / A / 2

Take the tangential length and velocity scales as order one. In the boundary layer, continuity reads $u_x+({\delta }_2/{\delta }_1)V_Y=0$, so a nontrivial balance requires ${\delta }_2\sim {\delta }_1$. Tangential inertia is order one, and the normal viscous term has size ${\mathrm{Re}}^{-1}{\delta }_1^{-2}$. Balancing them givesup to constants absorbable in the scaled variables. Streamwise diffusion is then smaller by ${\mathrm{Re}}^{-1}$. The normal momentum equation has a pressure term of size ${\delta }_1^{-1}{p}_Y$ and inertial/viscous terms of size ${\delta }_1$, hence $p_Y=0$ at leading order. The remaining Prandtl boundary-layer equations areThe boundary conditions are no slip and no penetration at the plate, and matching to the outer tangential velocity:The outer inviscid tangential momentum equation supplies $p_x=-U_s{U}_s^{\prime }$; pressure is shared between the boundary layer and outer flow because it has no leading normal variation.

Continuity allows a stream function with $u={\Psi }_Y$ and $V=-{\Psi }_x$. Substituting these into the tangential equation givesThe outer inviscid equation gives $-p_x=U_s{U}_s^{\prime }$, while $u_{YY}={\Psi }_{YYY}$. ThusThe wall no-penetration condition makes $\Psi$ constant along the wall; its additive freedom may be used to set that constant to zero.

For a boundary layer with exponentially varying outer velocity, the ansatz has $u={\Psi }_Y=e^{(\beta +\gamma )x}{f}^{\prime }(\eta )$. Matching it to $A{e}^{\alpha x}$ requires $\beta +\gamma =\alpha$ and $f^{\prime }(\infty )=A$. The nonlinear terms in the stream-function equation scale as $e^{2(\beta +\gamma )x}$, whereas ${\Psi }_{YYY}$ scales as $e^{(\beta +3\gamma )x}$. Matching exponents gives $\beta =\gamma$, henceMore explicitly,Canceling the common factor givesTogether with $f(0)=0$, the other two conditions areThey represent wall no slip and outer-flow matching, respectively.

Use autonomous third-order reduction using the dependent variable. On a monotone branch where $f$ can serve as the independent variable, the chain rule givesTherefore the reduced second-order equation isThe wall and matching conditions becomeFor the usual positive outer flow this is the limit $f\to +\infty$. The reduced equation is singular at $G=0$: if $f^{\prime \prime }(0)=b\ne 0$, then $f\sim b{\eta }^2/2$ and $G\sim b\eta$, so $G$ has square-root behavior as a function of $f$ near the wall. A finite value for $G_f(0)$ should not be imposed as an extra boundary condition.

An attached similarity profile must approach $f^{\prime }\to A$ as $\eta \to \infty$. Linearizing about $f\sim A\eta +\text{constant}$ gives, for the small velocity correction $g=f^{\prime }-A$,When $\alpha A>0$, one independent far-field correction decays with a Gaussian factor; when $\alpha A<0$, the two nontrivial asymptotic modes grow, one algebraically and one exponentially, so a nonconstant wall profile cannot decay into the prescribed outer flow. Thus the attached exponential similarity construction requiresEquivalently the outer speed increases along the direction of motion, giving a favorable pressure gradient. In the intended flow away from the leading edge along positive $x$, this means $A>0$ and $\alpha >0$. The sign-reversed choice $A<0,\alpha <0$ describes locally favorable flow in the opposite direction; it is not a layer developing downstream from the stated leading edge.

The asymptotic approximation also requires a geometrically thin layer, away from the leading-edge region. Comparing its thickness with the streamwise variation length $1/|\alpha |$ gives the local condition $\mathrm{Re}|A|e^{\alpha x}/|\alpha |\gg 1$. An exponentially decaying outer speed eventually violates this condition on an infinite downstream interval. These scale and orientation requirements distinguish formal similarity reduction from a uniformly valid attached boundary layer.