Question

The velocity components in the x and y directions are given by $u=\lambda x{y}^3-x^{2}y,v=x{y}^2-\frac{3}{4}{y}^4.$

The value of $\lambda$ for a possible flow field involving an incompressible fluid is

• A. $-\frac{3}{4}$
  
• B. $-\frac{4}{3}$
• C. $\frac{4}{3}$
• D. 3

Answer 1

The correct option is D 3

$u=\lambda x{y}^3-x^{2}y$
$v=x{y}^2-\frac{3}{4}{y}^4$

Continuity equation for 2-D incompressible flow

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

where

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

$\frac{\partial v}{\partial y}=2xy-3y^3$

$\Rightarrow \lambda y^3-2xy+2xy-3y^3=0$

$\lambda y^3=3y^3$

or $\lambda =3$