VELOCITY POTENTIAL FUNCTION AND STREAM FUNCTION

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

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