Question

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

Answer 1

$\phi=x^2-y^2$

Point (4,5)

We know that

$u_x=\frac{d\phi}{dt}=2x$

$u_y=\frac{d\phi}{dt}=-2y$

$u_x=2\times2=4$

$u_y=-2\times5=-10$

Velocity

$$v=\sqrt{u_x^2+u_y^2}=\sqrt{4^2+(-10)^2}=\sqrt{16+100}=\sqrt{116}$$

$|v|=10.77$