Question

For the fluid flowing over the flat plate, the state of boundary layer separation occurs when

• A. Pressure gradient is zero
• B. Velocity gradient is zero
• C. Reynolds number is zero
• D. Flow is accelerating

Hint:

Boundary layer conditions at the solid wall ($y=0$): - $\left(\frac{\partial u}{\partial y}\right)_{y=0} > 0 \implies$ Stable flow (No separation). - $\left(\frac{\partial u}{\partial y}\right)_{y=0} = 0 \implies$ Onset of flow separation. - $\left(\frac{\partial u}{\partial y}\right)_{y=0} < 0 \implies$ Separated flow with backflow region.

Answer 1

The Correct Option is B

Concept: When a viscous fluid flows over a solid surface, such as a flat plate, shear forces create a boundary layer where fluid velocity increases from zero at the wall (due to the no-slip condition) to the free-stream velocity. Boundary layer separation occurs when fluid layers in immediate proximity to the solid wall slow down until they stop, causing the mainstream flow to detach from the surface. This behavior is governed by the velocity profile gradient evaluated directly at the solid boundary (\(y = 0\)). Let us analyze the velocity gradient \(\left(\frac{\partial u}{\partial y}\right)_{y=0}\) under different pressure gradient conditions along the plate:

Step 1: Favorable Pressure Gradient (\(\frac{\partial P}{\partial x} < 0\)).
When pressure drops in the direction of flow, the fluid accelerates. This external pressure force helps overcome viscous shear forces near the wall, keeping the velocity profile full and stable against separation. The velocity gradient at the wall remains strongly positive: \[ \left(\frac{\partial u}{\partial y}\right)_{y=0} > 0 \]

Step 2: Adverse Pressure Gradient (\(\frac{\partial P}{\partial x} > 0\)).
When pressure rises in the direction of flow, the fluid faces an opposing force. This adverse pressure gradient slows down the slow-moving fluid inside the boundary layer near the wall, reducing the velocity gradient \(\left(\frac{\partial u}{\partial y}\right)_{y=0}\).

Step 3: Identifying the point of separation.
As the adverse pressure gradient continues to retard the flow, the velocity profile reaches a critical point where the fluid velocity at an infinitesimal distance from the wall drops to zero. At this exact location, the shear stress at the wall (\(\tau_w = \mu \left(\frac{\partial u}{\partial y}\right)_{y=0}\)) vanishes entirely: \[ \left(\frac{\partial u}{\partial y}\right)_{y=0} = 0 \] This condition marks the onset of boundary layer separation. Downstream of this point, the velocity gradient becomes negative (\(\left(\frac{\partial u}{\partial y}\right)_{y=0} < 0\)), leading to localized flow reversal and vortex formation. Thus, Option (2) is the correct choice.