Question

For the velocity potential function, φ = x^2 - y^2 . Find the velocity components at the point (4, 5).

Answer 1

In velocity we can write 

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

Given $\varphi=x^2-y^2$

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

Calculate the velocity at the point (4, 5)

$$u=8, v=-10$$

$$|velocity|=\sqrt{u^2+v^2 }=\sqrt{(8)^2+(-10)^2}=2\sqrt{41}$$