Question

A stream function is given by Ψ = 2x- 5y. Calculate the velocity components

and also magnitude and direction of the resultant velocity at any point

Answer 1

In velocity we can write

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

Given $\psi=2x-5y$

$$u=\dfrac{\partial\psi}{\partial y}=-5, v=-\dfrac{\partial\psi}{\partial x}=-2$$ $$u=-5, v=-2$$ $$|velocity|=\sqrt{u^2+v^2 }=\sqrt{(-5)^2+(-2)^2}=\sqrt{29}$$ $$\tan\theta=\dfrac{v}{u}=\dfrac{-2}{-5}=0.4$$

The vector of the resultant velocity at any point consists the angle $\tan^{-1}(0.4)\approx21.8\degree$ with the positive direction of the x-axis.