Question

A boundary layer develops due to a two-dimensional steady flow over a horizontal flat plate. Consider a vertical line away from the leading edge which extends from the wall to the edge of the boundary layer. Which of the following quantity/quantities is/are not constant along the vertical line? \( u \) and \( v \) represent the components of velocity in the direction along the plate and normal to it, respectively, and \( x \) is taken along the length of the plate while \( p \) is the pressure.

• A. ( u \)
• B. ( \frac{\partial u}{\partial x} \)
• C. ( v \)
• D. ( p \)

Hint:

In boundary layer flow, velocity components \( u \) and \( v \) vary with the vertical position, while pressure remains constant along the vertical line for steady flow.

Answer 1

The Correct Option is A, B, C

In boundary layer theory, the flow is steady and two-dimensional, with the velocity components \( u \) and \( v \) representing the flow along the plate and normal to the plate, respectively. The boundary layer is formed when the flow velocity near the plate changes from zero to the freestream velocity. We need to analyze the constancy of different quantities along a vertical line extending from the wall to the edge of the boundary layer.
Step 1: Analyze \( u \).
The velocity component \( u \), which represents the flow along the length of the plate, is not constant along the vertical line. The velocity changes from zero at the wall to a maximum value at the edge of the boundary layer, so \( u \) is not constant. Hence, statement (A) is true.
Step 2: Analyze \( \frac{\partial u}{\partial x} \).
The derivative \( \frac{\partial u}{\partial x} \) represents the rate of change of velocity in the direction along the plate. Since the flow is steady and the velocity varies with both \( x \) and the vertical direction, this derivative is not constant along the vertical line. Hence, statement (B) is true.
Step 3: Analyze \( v \).
The velocity component \( v \) represents the flow in the direction normal to the plate. In the boundary layer, \( v \) changes across the vertical line, from zero at the wall to a non-zero value at the edge of the boundary layer. Thus, \( v \) is not constant along the vertical line. Hence, statement (C) is true.
Step 4: Analyze \( p \).
The pressure \( p \) is assumed to be constant along the vertical line for steady flow, as there is no change in pressure in the direction normal to the plate in the boundary layer. Hence, statement (D) is false.
Step 5: Conclusion.
The correct answers are (A), (B), and (C).