Topic: Fluid Mechanics
The velocity components in the x and y directions are given by $u=\lambda xy^{3}-x^{2}y$; $v=xy^{2}-\frac{3}{4}y^{4}$. The value of $\lambda$ for a possible flow field involving an incompressible fluid is
- A. $-\frac{3}{4}$
- B. 3
- C. $\frac{4}{3}$
- D. $-\frac{4}{3}$
Answer: B. 3
Explanation: For a possible incompressible flow, continuity equation must be satisfied: $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$. Given $u=\lambda xy^{3}-x^{2}y$, $\frac{\partial u}{\partial x}=\lambda y^{3}-2xy$. Given $v=xy^{2}-\frac{3}{4}y^{4}$, $\frac{\partial v}{\partial y}=2xy-3y^{3}$. Substituting into the continuity equation: $\lambda y^{3}-2xy+2xy-3y^{3}=0 \implies y^{3}(\lambda-3)=0$. Since $y\ne0$, $\lambda=3$.