Question

Consider steady incompressible flow over a flat plate, where the dashed line represents the edge of the boundary layer, as shown in the figure. Which one among the following statements is true?

• A. Bernoulli’s equation can be applied in Region I between any two arbitrary points.
• B. Bernoulli’s equation can be applied in Region I only along a streamline.
• C. Bernoulli’s equation cannot be applied in Region II.
• D. Bernoulli’s equation cannot be applied in Region I.

Hint:

Boundary layer $\Rightarrow$ viscous $\Rightarrow$ \textbf{no Bernoulli}. Outside the boundary layer (inviscid/irrotational region) $\Rightarrow$ Bernoulli holds (even between any two points).

Answer 1

The Correct Option is D

Step 1: Identify the regions.
Region I (below the dashed line) is the boundary layer: viscous effects and shear stresses are significant. Region II (outside the dashed line) is the inviscid/free stream: viscous effects are negligible; the flow is essentially irrotational for external flow over a smooth flat plate.
Step 2: Recall when Bernoulli’s equation applies.
The steady-flow Bernoulli equation holds \[ \frac{p}{\rho}+\frac{V^2}{2}+gz=\text{const} \] (i) along a streamline for {inviscid} flows; and (ii) between any two points in the domain if the flow is additionally {irrotational}. It does {not} hold where viscous dissipation is important (finite shear stresses).
Step 3: Assess each statement.
(A) Region I is viscous $\Rightarrow$ Bernoulli cannot be used between arbitrary points there. False.
(B) Even along a streamline in Region I, viscosity invalidates Bernoulli. False.
(C) Region II is inviscid/irrotational $\Rightarrow$ Bernoulli is applicable (indeed between any two points). False.
(D) In Region I (boundary layer), Bernoulli {cannot} be applied. True. Final Answer: \[ \boxed{\text{(D)}} \]