Question

Consider a steady, two-dimensional, laminar, incompressible flow of water over a horizontal flat plate of length \(L\) with Reynolds number, \(Re_L = 10^4\). The flat plate is aligned with the incoming free-stream flow. The \(x\)-axis is along the length of the plate while the \(y\)-axis is normal to the plate. The respective velocity components are \(u\) and \(v\).

Which of the following statements is/are TRUE?

• A. For the flow within the boundary layer, \(u \gg v\).
• B. For the flow within the boundary layer, \(\dfrac{\partial u}{\partial y} \gg \dfrac{\partial u}{\partial x}\).
• C. The flow within the boundary layer is irrotational and inviscid.
• D. The thickness of the boundary layer is much less than the length of the flat plate.

Hint:

Use the thin-layer scaling of boundary layer theory to check each velocity-gradient and thickness claim.

Answer 1

The Correct Option is A, B, D

Step 1: Recall the boundary layer approximations.
Boundary layer theory works because the layer next to a wall is thin compared with the length of the surface. Flow inside it stays almost parallel to the wall, so gradients across the thin layer dominate over gradients along it.

Step 2: Check statement (A).
Inside a thin boundary layer the flow moves mostly along the plate. The streamwise velocity \(u\) is far larger than the tiny normal velocity \(v\) generated as the layer grows. This gives \(u \gg v\), so (A) is TRUE.

Step 3: Check statement (B).
Because the boundary layer is thin, \(u\) changes sharply over a very small distance \(y\), from zero at the wall to the free-stream value at the layer edge. It changes only gradually as \(x\) increases along the plate. So \(\partial u/\partial y \gg \partial u/\partial x\), and (B) is TRUE.

Step 4: Check statement (C).
This is wrong on both counts. The boundary layer exists precisely because viscosity is important there, so the flow cannot be called inviscid. The strong \(\partial u/\partial y\) gradient also means the flow carries vorticity, so it is rotational, not irrotational. So (C) is FALSE.

Step 5: Check statement (D).
Boundary layer thickness scales as \(\delta/L \sim 5/\sqrt{Re_L}\). With \(Re_L = 10^4\), this gives \(\delta/L \sim 5/100 = 0.05\). So \(\delta\) is about 5 percent of \(L\), much smaller than \(L\). So (D) is TRUE.

Final Answer:
Statements A, B and D describe genuine boundary layer behaviour; C contradicts the idea of a viscous boundary layer. \[ \boxed{\text{A, B, D}} \]