Question

The velocity potential function ϕ is given by ϕ = x2 - y^2. Find the velocity components in x and y direction. Also, show that ϕ represents a possible case of fluid flow.

Answer 1

In velocity we can write 

$$u=\dfrac{\partial\varphi}{\partial x}, v=\frac{\partial\varphi}{\partial y}$$

Given $\varphi=x^2-y^2$

$$u=\dfrac{\partial\varphi}{\partial x}=2x, v=\dfrac{\partial\varphi}{\partial y}=-2y$$

$$\dfrac{\partial u}{\partial x}=\dfrac{\partial^2\varphi}{\partial x^2}=2$$ $$\dfrac{\partial v}{\partial y}=\dfrac{\partial^2\varphi}{\partial y^2}==2$$ $$\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}=0$$

The continuity equation is satisfied and the flow is irrotational.