Flow of Viscous Fluid Through Circular Pipe

When fluid flows through pipes there are frictional losses. These losses are due to the resistance due to which some energy of the fluid is lost. 

There are two types of energy loss:
Minor loss: due to friction
Major loss: due to sudden expansion or contraction or bend in the pipe.

Major loss

It can be calculated with the help of the following formula 

h_f = Loss of head due to friction

L = length of pipe

v = mean velocity of flow

d = diameter of the pipe

f = coefficient of friction in the viscous flow

Flow of Viscous fluid through circular pipe

Let us assume a circular cross section pipe of radius R .
In the above figure, we have taken a fluid element of radius ‘r’ and length dx.
Shear force on the element is given by  = τ*(2πr)*dx
The pressure at section ab is p then pressure at section cd will be (p+∂p/∂x  * dx)

Neglecting the acceleration of fluid element , by balancing net force = 0 , we get 

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Shear Stress distribution as per the above derived equation will be given by 
By this equation, we can obtain velocity distribution as we know,
τ = μ du/dy

But we are measuring y from the surface in the above equation, let’s correlate r & y 

r+y = R

y = R-r

dy = 0-dr

 τ = -μ du/dr

Now, let’s calculate discharge flowing in the small thickness dr

Discharge  dQ = velocity at radius r * Area of the ring

Average velocity is given by 

u_avg = Q/Area of cross section

 

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