Question

For the velocity components given as : u = ay sin xy, v = ax sin xy. Obtain an expression for the velocity potential function.

Answer 1

If $\overrightarrow{v}$ denotes the flow velocity, then for a velocity potential function $\Phi$, $\overrightarrow{v}$ can be represented as the gradient of a scalar function $\Phi$ :

${\displaystyle \overrightarrow{v} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y}}\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.}$

In our case, $\frac {\partial \Phi }{\partial x}=u=ay\sin xy,\ \ \frac {\partial \Phi }{\partial y}=v=ax\sin xy.$

Then $\Phi(x,y)=\int ay\sin xy dx=-a\cos xy+C(y)$, and therefore,

$\frac {\partial \Phi }{\partial y}=ax\sin xy+C'(y)$.

It follows that $ax\sin xy+C'(y)=ax\sin xy$, and consequently, $C'(y)=0$. Then $C(y)=C$, and we conclude that

$\Phi(x,y)=-a\cos xy+C.$