Velocity potential function and stream function are two scalar functions that help study whether the given fluid flow is rotational or irrotational. Both the functions provide a specific Laplace equation. The fluid flow can be rotational or irrotational flow based on whether it satisfies the Laplace equation or not.

The expression for velocity potential function and stream function, along with their relationship, is explained briefly in this article.

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## What is Velocity Potential Function?

Velocity potential function is a scalar function of space and time. If ‘phi’ is the representation of velocity potential function, then the velocity function for a steady fluid flow is given by the expression,

It is a scalar function, whose negative derivative, with respect to any direction, gives the velocity component in that direction.

Here,** u**, **v**, and **w** are the velocity components of the fluid flow along x, y, and z directions.

The velocity potential function is expressed in polar coordinates as,

Where u_{r }and u_{?} are called as the velocity components in the radial and the tangential direction i.e. along r and ‘theta’ direction.

We get the 3D Laplace equation in terms of velocity potential function by the given formula

The formula mentioned below gives the Laplace equation in terms of velocity potential function in 2D

If the velocity potential function satisfies the Laplace equation, then it corresponds to some case of fluid flow.

Also Read: Kinematics of Flow in Fluid Mechanics-Discharge and Continuity Equation

## Properties of Velocity Potential Function

The properties of the velocity potential function of a fluid is explained by understanding the rotational components along x, y, and z directions. It is given by :

By substituting the values of u, v and w from Eq.2 we get the rotational components as [Eq.6 as]

w_{x} = w_{y} = w_{z} = 0; Eq.7

We assume here that velocity potential function “phi” is a continuous function.

From the above explanation, it is concluded that,

- For the fluid flow to be irrotational, the rotational components are equal to zero.
- If there exists velocity potential, then the fluid flow is rotational.
- If the given velocity potential satisfies the Laplace equation (Eq.4), then the fluid flow is a representation of the steady incompressible irrotational flow.

Read More: Different Types of Fluid Flow in Kinematics

## What is Stream Function?

Stream function is a scalar function of space and time whose derivative with respect to any direction would give the velocity component at right angles to that direction. It is represented by ‘psi”, where

The stream function in cylindrical polar coordinated is given by,

Where, u_{r }and uo_{ }radial and tangential velocity.

### Properties of Stream Function

As it satisfies the continuity equation, the existence of a stream function proves a possible case of fluid flow. This flow can be either rotational or irrotational.

If the rotational component is given by the formula, w_{z} from the Eq.6, then we get the Laplace relation for Stream Function

The main properties of stream function are:

- The existence of stream function proves a possible case of fluid flow, that can be either rotational or irrotational in nature.
- A stream function of a fluid satisfying a Laplace equation is supposed to have an irrotational flow.

## Equipotential Lines and Stream Lines in Fluid Mechanics

### Equipotential Lines

The line along which the velocity potential function is constant is called as equipotential line. The slope of equipotential line is given by dy/dx = -u/v.

### Streamlines

Streamlines are defined as the lines along which the stream function is constant. In a flow field, a tangent drawn at any point on the streamline gives the direction of velocity. Hence the slope at any point on the streamline is given by dy/dx=v/u;

The above equation is the defining equation of streamline. With a constant stream function value, an infinite number of streamlines can be drawn. Given a family of streamlines, the flow patterns can be visualized very easily. Also, all the streamlines are parallel to each other.

The product of slope of equipotential line and streamline is obtained as -1. This means, both the lines are orthogonal to each other. Hence, knowing the value of stream function, the velocity potential value is determined or vice versa.

## Relationship Between Velocity Potential Function and Stream Function

From the expressions of velocity potential function and stream function,

By comparing both the equations, we get

Also Read: What is Velocity and Acceleration of a Fluid Flow?