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Use the same stream function convention and denote differentiation with respect to the dimensionless variable $\eta =y/\delta$ by a prime. The proposed similarity solution givesIts continuity equation holds identically. Substitution in the tangential Prandtl boundary-layer equation givesChoose the constant scale $\delta =\sqrt{\nu /A}$. The Hiemenz flow equation and its boundary conditions areThe first two conditions impose the no-penetration boundary condition and no-slip boundary condition. The last matches the tangential outer velocity. One also has $f^{\prime \prime }\to 0$ and $f(\eta )/\eta \to 1$. It is too strong to impose $f-\eta \to 0$: the displacement thickness generally permits a constant offset, $f\sim \eta -B$.

The normal Navier-Stokes equation makes the approximation in part (c) explicit. With this similarity solution, it givesAt the wall this differs from $(p_{\infty }{)}_y=-\rho g$ by $-\rho A^2\delta f^{\prime \prime }(0)$. This is a concrete next-order counterexample to interpreting the printed normal pressure gradient equality exactly; the tangential Hiemenz flow equation remains valid at its stated order.

The Hiemenz flow profile changes on $\eta =O(1)$, so its boundary layer thickness isA specified edge convention, such as $u/(Ax)=0.99$, introduces an order-one numerical multiplier requiring the profile to be solved. Scaling alone does not determine that multiplier. The result has no separate dependence on $x$, $y$, $\rho$ or $p_0$ at fixed kinematic viscosity. At fixed dynamic viscosity, it is $\sqrt{\eta /(\rho A)}$.

The viscous diffusion time across a thickness $\delta$ is ${\delta }^2/\nu$; the imposed strain has time scale $A^{-1}$. Their balance sets the thickness. Increasing kinematic viscosity spreads the layer, while increasing the imposed velocity gradient presses the layer toward the wall. The outward tangential velocity increases as $Ax$, so the decreasing travel time per unit distance offsets the usual downstream broadening.

For a conventional estimate of transition to turbulence, form the local horizontal Reynolds number:If transition is assumed to occur at an empirical value ${\mathrm{Re}}_{\mathrm{crit}}$, thenA rough borrowed threshold of order $10^5$–$10^6$ would give $l$ of order $10^2$–$10^3\sqrt{\nu /A}$. This is a heuristic estimate, not a universal stability prediction: the accelerating outer flow and favorable pressure gradient can delay transition, and disturbance amplitudes and wall roughness matter. Simply setting ${\mathrm{Re}}_x\sim 1$ locates failure of the thin-layer approximation near the stagnation point, not onset of turbulence.