Question

X-component of velocity in a 2D incompressible flow is given by \( u = x^2 + y^2 \). If Y-component of the velocity is equal to zero in y=0, then the equation for Y is given by

• A. \( 2y^2 \)
• B. \( -2y^2 \)
• C. \( 4y \)
• D. \( 2x^2 \)

Hint:

In incompressible fluid flow, always apply the continuity equation to solve for unknown velocity components.

Answer 1

The Correct Option is B

In a 2D incompressible flow, the continuity equation states that: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \] Given \( u = x^2 + y^2 \), taking the partial derivative with respect to x: \[ \frac{\partial u}{\partial x} = 2x \] Thus, \( \frac{\partial v}{\partial y} = -2x \). When \( x = 0 \), \( v = -2y^2 \).