Answer to Question #149106 in Vector Calculus for Usman

Question

If for a two dimensional potential flow, the velocity potential is given by : φ = 4x(3y - 4), determine the velocity at the point (2, 3). Determine also the value of stream function ψ at the point (2, 3).

Answer 1

In velocity we can write 

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

Given $\varphi=4x(3y-4)$

$$u=\dfrac{\partial\varphi}{\partial x}=12y-16, v=\dfrac{\partial\varphi}{\partial y}=12x$$

Calculate the velocity at the point (2, 3)

$$u=20, v=24$$

$$|velocity|=\sqrt{u^2+v^2 }=\sqrt{(20)^2+(24)^2}=4\sqrt{61}$$

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

$$u=\dfrac{\partial\psi}{\partial y}=12y-16$$

Integrate with respect to $y$

$$\psi=6y^2-16y+g(x)$$

$$-\frac{\partial\psi}{\partial x}=-\frac{dg}{d x}=12x$$

Integrate with respect to $x$

$$g(x)=-6x^2+C$$

Then

$$\psi(x,y)=-6x^2+6y^2-16y+C$$