Vortex Flow (Kinematics and Dynamics of Flow Part-4)

Rotating mass of fluid is known as vortex flow.

Vortex flow is of two types 
1. Forced vortex flow
2. Free vortex flow

Forced Vortex Flow

When there is external torque is applied to the flow to rotate the flow then it is called as a forced vortex flow.

ω = constant

or

v/r = constant

Example: Fluid inside the centrifugal pump or fluid is in a container and container is being rotated around its axis  

Free Vortex Flow

In this kind of flow external torque is zero i.e. no external torque is applied.
Example:Flow in a circular pipe or A whirlpool in river.

First of all, let’s take a general case where the fluid is rotating about some axis and we will find an expression for the pressure at some point.

Let us consider a small element of fluid element rotating at uniform velocity in horizontal plane about an axis perpendicular to out of plane.

∂P/∂r is a pressure gradient.
If it is positive then pressure increases with radiaus and if it is negative pressure decreseas with radius

But we already know that pressure variation in vertical plane is given by hydrostatic law, as 

∂P/∂h = ρg

Hence, it can be said that pressure at any point is function of h and r

P = f(r,h)

Equation of free vortex flow

In this case vr = constant putting in the derived equation
It shows that Bernoulli’s equation is valid for free vortex flow and can be applied.

Equation of forced vortex flow

In above derived equation for forced vortex flow ω is constant.
ω = v/r → v = rω
Putting we get,

The pressure difference between two points can be calculated with the help of the integrating function.

Let’s discuss few important notes from the above expression.
1. Bernoulli’s equation is not valid for forced vortex flow and can’t be applied.
2. If any point lies on the axis of rotation then r= 0 →v = ω * 0 = 0
3. If both points 1 & 2 lies on the free surface the P1 = P2
4. If both case 2& case 3 are present then 
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