At the point of laminar boundary layer separation on a smooth aerodynamic surface, which of the following physical and mathematical conditions must be satisfied?

(A) The wall shear stress vanishes $\left(\tau_w = \mu \left.\frac{\partial u}{\partial y}\right|_{y=0} = 0\right)$ in the presence of an adverse pressure gradient $\left(\frac{dp}{dx} > 0\right)$

(B) The velocity gradient at the wall becomes infinite

(C) The static pressure gradient must be zero ($dp/dx = 0$)

(D) The displacement thickness becomes identically zero

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In 2D steady boundary layer theory: - Flow against an **adverse pressure gradient** ($dp/dx > 0$) causes near-wall fluid to decelerate. - At the boundary layer separation point, the velocity profile $u(y)$ develops an inflection point, and the velocity slope at the solid wall drops to zero: $$\left.\frac{\partial u}{\partial y}\right|_{y=0} = 0 \implies \tau_w = \mu \left.\frac{\partial u}{\partial y}\right|_{y=0} = 0$$ Downstream of this separation point, reverse flow ($\\partial u / \partial y < 0$) develops. ### 2. Distractor Forensics - Zero displacement thickness is physically meaningless; in fact, displacement thickness grows very rapidly near separation. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Boundary layer separation strictly requires an *adverse* pressure gradient ($dp/dx > 0$). In favorable gradients ($dp/dx < 0$), separation can never occur. - **Standard Reference:** Schlichting, H., *Boundary-Layer Theory*, Chapter 8.

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