Question

The stream function for a two-dimensional flow is given by ψ = 8xy, calculate the velocity at the point p(4, 5). Find the velocity potential function ϕ.

Answer 1

In velocity we can write 

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

Given $\psi=8xy$

$$u=\dfrac{\partial\psi}{\partial y}=8x, v=-\dfrac{\partial\psi}{\partial x}=-8y$$

Calculate the velocity at the point p(4, 5)

$$u=32, v=-40$$ $$|velocity|=\sqrt{u^2+v^2 }=\sqrt{(32)^2+(-40)^2}=8\sqrt{41}$$ $$u=\dfrac{\partial\varphi}{\partial x}, v=\frac{\partial\varphi}{\partial y}$$ $$u=\dfrac{\partial\varphi}{\partial x}=8x$$

Integrate with respect to $x$

$$\varphi=4x^2+g(y)$$ $$\frac{\partial\varphi}{\partial y}=\frac{dg}{d y}=-8y$$

Integrate with respect to $y$

$$g(y)=-4y^2+C$$

Then

$$\varphi(x,y)=4x^2-4y^2+C$$