Fluid kinematics (2-D incompressible flow): If the x-component of velocity is u(x,y) = y^2 + 4xy and the y-component v(x,y) = 0 at y = 0, determine the expression for v(x,y).
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A4y
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B2y^2
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C-2y^2
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D2xy
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E-4xy
Answer
Correct Answer: -2y^2
Explanation
Introduction / Context:For incompressible 2-D flow, continuity requires that the divergence of velocity is zero. Given one component, the other can be found (up to a function of the remaining variable) by integrating the continuity equation and applying a boundary condition.
Given Data / Assumptions:
- u(x,y) = y^2 + 4xy.
- Incompressible 2-D: du/dx + dv/dy = 0.
- Boundary condition: v(x,0) = 0 for all x.
Concept / Approach:
From continuity, dv/dy = −du/dx. Differentiate u with respect to x, integrate with respect to y to obtain v, and then apply the boundary condition to fix the integration “constant” (a function of x).
Step-by-Step Solution:
Compute du/dx = ∂(y^2 + 4xy)/∂x = 4y.From continuity: dv/dy = −4y.Integrate: v(x,y) = ∫(−4y) dy = −2y^2 + f(x).Apply v(x,0) = 0 ⇒ f(x) = 0.Thus v(x,y) = −2y^2.Verification / Alternative check:
Check: dv/dy = −4y and du/dx = 4y ⇒ du/dx + dv/dy = 0, satisfying incompressibility.
Why Other Options Are Wrong:
- 4y, 2xy: Wrong functional dependence and ignore boundary condition.
- 2y^2: Sign error; would violate continuity.
- −4xy: Not supported by integration of −4y.
Common Pitfalls:
- Forgetting the integration function f(x).
- Not applying the boundary condition properly.
Final Answer:
-2y^2