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).

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:

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