Consider the two-dimensional velocity field given by $\to V = (5 + a_{1} x + b_{1} y)^i + (4 + a_{2} x + b_{2} y)^j$ . where $a_{1}, b_{1}, a_{2}$ and $b_{2}$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

a_{1} + b_{1} = 0

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a_{1} + b_{2} = 0

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a_{2} + b_{1} = 0

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a_{2} + b_{2} = 0

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Solution

The correct option is B $a_{1} + b_{2} = 0$
$\to V = (5 + a_{1} x + b_{1} y)^i + (4 + a_{2} x + b_{2} y)^j$
$= u^i + v^j$
For, incompressible flow $d i v \to V = 0 \Rightarrow ▽ . \to V = 0$
or $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$
$a_{1} + b_{2} = 0$

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