Stream & Velocity Potential Function (Kinematics and Dynamics of Flow Part-3)

Stream Function: A stream function is an arbitrary function whose derivatives give velocity components of a particular flow situation.

The partial derivative of stream function with respect to any direction gives the velocity component at right angles to that direction. It is denoted by ψ

According to the above definition

Continuity Equation for flow is given by 

** As ψ satisfies the coninuity equation hence, if ψ exists then it is possible case of fluid flow. **

Now let us see the rotational components of fluid particles 

Here u,v,w are the velocities in x, y, z-direction respectively.

Putting values of u & v in terms of ψ, we get

This is a Laplace equation for stream function (ψ)

Conclusion:

1. If stream function (ψ) exists, it is a possible case of fluid flow. But we can’t decide whether the flow is rotational or irrotational.

2. If stream function satisfies Laplace equation then, it is a possible case of irrotational flow.

Velocity Potential Function: 

It is a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. 
It is denoted by Φ
Continuity Equation for flow is given by 

This is a Laplace equation for velocity potential function (Φ)

**If Φ satisfies above equation then it is a possible case of fluid flow.**

Now let us see the rotational components of fluid particles 

Here u,v,w are the velocities in x, y, z-direction respectively.

Above equation shows that Φ exists then, flow will be irrotational.

Stream Function Ψ

If stream function (ψ) exists, it is a possible case of fluid flow.
If stream function satisfies Laplace equation then, it is irrotational flow.

Velocity Potential Φ

If Velocity Potential Function (Φ) exists then, flow is irrotational.
If Φ satisfies the Laplace equation then, a possible case of fluid flow.

Relation between stream function and velocity potential

Comparing both stream function (Ψ) and velocity potential function (Φ)

Stream Line

It is the line along which stream function (ψ) remains constant.

Equipotential Line

It is the line along which Velocity potential (Φremains constant.

**From above equations, it can be concluded that stream line and equipotential line are perpendicular to each other.**

Because product of their slope of line is -1
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