Question

Which of the following velocity fields satisfies the condition of steady incompressible flow?

1. $u=3y$, $v=3x$

2. $u=3xt$, $v=3yt$

3. $u=xy+y^{2}t$, $v=xy+x^{4}t$

4. $u=4x^2{y}^3$, $v=-2x{y}^4$

• A. 1 and 2
• B. 2 and 3
• C. 3 and 4
• D. 1 and 4

Answer 1

The correct option is D 1 and 4

For steady incompressible flow, the continuity equation is given by

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

1. $\frac{\partial u}{\partial x}=0$ $and$ $\frac{\partial v}{\partial y}=0$
$\therefore$ It does satisfy continuity

2. $\frac{\partial u}{\partial x}=3t$ $and$ $\frac{\partial v}{\partial y}=3t$
$\therefore$ It does not satisfy continuity

3. $\frac{\partial u}{\partial x}=y$ $and$ $\frac{\partial v}{\partial y}=x$
$\therefore$ It does not satisfy continuity

4. $\frac{\partial u}{\partial x}=8x{y}^3$ $and$ $\frac{\partial v}{\partial y}=-8x{y}^3$
$\therefore$ It does satisfy continuity

$\therefore$ Velocity field (1) and velocity field (4) shows steady incompressible flow.