Question

For a flow, the velocity components are given by $u=(\lambda x{y}^2-x^3{y}^2)andv=(x^2{y}^3-3y^3)$. What is the value of $\lambda$ for the possible flow field which includes steady incompressible flow ?

• A. 3
• B. 5
• C. 7
• D. 9

Answer 1

The correct option is D 9

Data given : Velocity component :

$u=\lambda x{y}^2-x^3{y}^2$

$v=x^2{y}^3-3y^3$

For steady incompressible flow to exist, continuity equation must be satisfied .

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

(2-dimension continuity equation)

$\frac{\partial u}{\partial x}=\lambda y^2-3x^2{y}^2$

$\frac{\partial v}{\partial y}=x^2\left(3y^2\right)-3\times 3y^2=3x^2{y}^2-9y^2$

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=\lambda y^2-3x^2{y}^2+3x^2{y}^2-9y^2=0$

$\lambda =9$