Question

Which differential equation would be obtained, if the boundary layer transition from a boundary of zero thickness and the lateral inflow is zero in canal?

• A. \(\frac{d^2y}{dx^2} + b^2y = 0\)
• B. \(\frac{dy}{dx} + b^2y = 0\)
• C. \(\frac{d^2y}{dx^2} - b^2y = 0\)
• D. \(\frac{dy}{dx} - b^2y = 0\)

Hint:

Boundary layer equations in canals:
Second-order homogeneous differential equations.
Solutions involve exponential functions (e^{bx} and e^{-bx}).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests the derivation of differential equations for boundary layer in open channel flow.

Step 2: Detailed Explanation:
When analyzing the boundary layer transition from zero thickness with no lateral inflow, the governing equation takes the form of a second-order homogeneous differential equation.
For these conditions, the differential equation is:
\[ \frac{d^2y}{dx^2} - b^2y = 0 \] where:
- y represents the boundary layer thickness or some related parameter.
- b is a constant.
The equation \(\frac{d^2y}{dx^2} - b^2y = 0\) has the solution:
\[ y = C_1 e^{bx} + C_2 e^{-bx} \] Other equations:
- \(\frac{d^2y}{dx^2} + b^2y = 0\) (A): For oscillatory solutions.
- \(\frac{dy}{dx} + b^2y = 0\) (B): First-order decay equation.
- \(\frac{dy}{dx} - b^2y = 0\) (D): First-order growth equation.
Thus, the correct equation is \(\frac{d^2y}{dx^2} - b^2y = 0\).

Step 3: Final Answer:
Thus, the differential equation obtained is \(\frac{d^2y}{dx^2} - b^2y = 0\).
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