Question

Q5.Cheak if ϕ = X2– Y2+ Y represents the velocity potential for 2 dimensional

Irrotational flow. If it does than determine than determine stream function Ψ.

Answer 1

The condition for irrotational motion

$$\dfrac{\partial u}{\partial y}-\dfrac{\partial v}{\partial x}=0$$

From the definition of stream function $ψ ,$ we get $u =\dfrac{\partial \phi}{\partial x}, v=\dfrac{\partial\phi}{\partial y}$

Thus, the velocity components become

$$u =\dfrac{\partial \phi}{\partial x}=2x, v=\dfrac{\partial\phi}{\partial y}=-2y+1$$

$$\dfrac{\partial u}{\partial y}=0, \dfrac{\partial v}{\partial x}=0$$

Since

$$\dfrac{\partial u}{\partial y}-\dfrac{\partial v}{\partial x}=0-0=0$$

then  $ϕ = x^2– y^2+y$ represents the velocity potential for 2 dimensional Irrotational flow.

$$\dfrac{\partial \psi}{\partial y}=u=2x, -\dfrac{\partial \psi}{\partial x}=v=-2y+1$$

Then

$$\psi=2xy+g(x)$$

$$\dfrac{\partial \psi}{\partial x}=2y+g'(x)=-v=2y-1$$

$$g'(x)=-1$$

$$g(x)=-x+C$$

Stream function is

$$\psi=2xy-x+C$$