Stream Function:

-It is defined as the scalar function of space and time, such that its potential derivative with respect to any direction gives the velocity component at right angles to that direction.

-It is denoted by $'\psi$ and defined for only 2-D flow.

Mathematically for steady flow

$\psi=f(x,y)$

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

Since $\psi=f(x,y)$

$\partial \psi=\frac{\partial \psi}{\partial x}dx+\frac{\partial \psi}{\partial y}dy$

$=-vdx+udy$

Also $\partial \psi =0$

$\therefore -vdx+udy=0$

$udy=vdx$

$\frac{dy}{dx}=\frac{v}{u}$

$\therefore \text{ Slope of stream line}=\frac{v}{u}$

The continuity equation for 2-D flow is $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

Consider $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=\frac{\partial }{\partial x}[\frac{\partial \psi}{\partial y}]+\frac{\partial }{\partial y}[-\frac{\partial \psi}{\partial x}]$

$=\frac{\partial ^2\psi}{\partial x\cdot \partial y}-\frac{\partial ^2\psi}{\partial x\cdot \partial y}$

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

Properties of stream function

1. If stream function $(\psi)$ exists, it is a possible case of fluid flow which may be rotational or irrotational.
2. If stream function $(\psi)$ satisfies the laplas equation, it is a possible case of an irrotational flow.

Velocity Potential Function

-It is defined as a scalar function of space & time such that its partial derivative in any direction gives the fluid velocity in that direction.

-It is denoted by $\phi$

$\therefore \text{ For steady flow} \phi=f(x,y,z)$

$\therefore u=-\frac{\partial \phi}{\partial x}; v=-\frac{\partial \phi}{\partial y}; w=-\frac{\partial \phi}{\partial z}$

-Negative sign satisfies that $\phi$ decreases with increase in values of x,y,z i.e., flow is always in the direction of decreasing $\phi$

$d \phi=\frac{\partial \phi}{\partial x}\cdot dx+\frac{\partial \phi}{\partial y}\cdot ydy$

$=-u\cdot dx-v\cdot dy$