We were discussing the basic
concept of Types
of fluid flow, Discharge
or flow rate, Continuity
equation in three dimensions, continuity
equation in cylindrical polar coordinates and
total acceleration, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to understand the
basic concept of velocity potential function and stream function, in the field
of fluid mechanics, with the help of this post.

Let us consider that V is the resultant velocity of a fluid particle at a point in a flow filed. Let us assume that u, v and w are the components of the resultant velocity V in x, y and z direction respectively.

We can define the components of
resultant velocity V as a function of space and time as mentioned here.

###
**Velocity
potential function**

Velocity potential function is basically
defined as a scalar function of space and time such that it’s negative
derivative with respect to any direction will provide us the velocity of the
fluid particle in that direction.

Velocity potential function will be represented
by the symbol ϕ i.e. phi.

Velocity components in cylindrical polar-coordinates
in terms of velocity potential function will be given as mentioned here.

Where,

u

_{r}is the velocity component in radial direction and u_{θ}is the velocity component in tangential direction.
We can write here the continuity
equation for incompressible steady flow in terms of velocity potential function
as mentioned here.

###
**Stream
function**

Stream function is basically defined as a scalar function of space and time such that it’s partial derivative with respect to any direction will provide us the velocity component at perpendicular to that direction.

Stream function will be represented by Ψ
i.e. psi. It is defined only for two dimensional flow.

Velocity components in cylindrical polar-coordinates
in terms of stream function will be given as mentioned here.

Where,

u

_{r}is the velocity component in radial direction and u_{θ}is the velocity component in tangential direction.
Let us use the value of u and v in
continuity equation; we will have following equation as mentioned here.

Therefore existence of stream function (Ψ)
indicates a possible case of fluid flow. Flow might be rotational or
irrotational.

If stream function (Ψ) satisfies the
Laplace equation, it will be a possible case of an irrotational flow.

We will discuss another term i.e. “Equipotential line and streamline” in fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

###
**Reference:**

Fluid mechanics, By R. K. Bansal

Image Courtesy: Google