Question

The y-component of velocity for a certain 2-D flow field is given as u = 3xy + x2 . Determine the x-component of velocity if the flow is incompressible.

Answer 1

Answer:

$-\frac{3x^{2}}{2}$

Explanation:

It is given that y component is

v = 3xy + $x^{2}$

$\Rightarrow \frac{\partial v}{\partial y}= 3 x$

For an incompressible flow, the continuity equation is written in differential form as

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

$\Rightarrow \frac{\partial u}{\partial x}= -\frac{\partial v}{\partial y}$

$\Rightarrow \frac{\partial u}{\partial x}= - 3x$

Now solving for x component of velocity is

u = - $\int 3x.dx$

  = - $\frac{3x^{2}}{2}$

Therefore, x component of velocity is - $\frac{3x^{2}}{2}$