We were discussing the concept of laminar
and turbulent flow, Reynolds
experiment, frictional loss in pipes and also the derivation of expression for loss of head due to friction in pipes, in the subject of fluid mechanics,
in our recent posts.

###

Now we will go ahead to find out the expression for
the coefficient of friction in terms of shear stress, in the subject of fluid
mechanics, with the help of this post.

###
**Expression
for coefficient of friction in terms of shear stress **

We will determine here the expression for coefficient
of friction in terms of shear stress.

Let us consider that fluid is flowing through a
uniform horizontal pipe with steady flow as displayed here in following figure.

Now we will assume two sections of pipe i.e. section
1-1 and section 2-2.

Let us consider the following terms to derive the
required expression of coefficient of friction in terms of shear stress.

P

_{1}= Pressure intensity at section 1-1
V

_{1}= Velocity of flow at section 1-1
P

_{2}= Pressure intensity at section 2-2
V

_{2}= Velocity of flow at section 2-2
L = Length of pipe between section 1-1 and section
2-2

h

_{f}= Loss of head due to friction
d = Diameter of the pipe

A = Area of pipe = (П /4) x d

^{2 }
τ

_{0}= Shear stress
F

_{1}= Force due to shear stress (τ_{0})
F

_{1 }= τ_{0 }x П d x L
Now we will apply the Bernoulli’s equations between
section 1-1 and section 2-2.

Because,

Pipe is horizontal and hence, Z

_{1}= Z_{2 }
Diameter of uniform pipe is same at both sections
and hence, V

_{1}= V_{2 }
Where h

_{f }is the Darcy-Weisbach equation which is commonly used to determine the loss of head due to friction in pipes.
Now we will write here the equation of equilibrium of
forces

P

_{1}A – P_{2}A – F_{1}=0
(P

_{1}– P_{2})A = F_{1}
(P

_{1}– P_{2})A = Force due to shear stress (τ_{0})
(P

_{1}– P_{2}) (П /4) x d^{2}= τ_{0 }x П d x L
(P

_{1}– P_{2}) = 4 τ_{0 }x L/d
Now we will equate the expression for (P

_{1}– P_{2}) and we will have following equation as mentioned here.
Above equation is the expression for the coefficient
of friction in terms of shear stress.

Further we will go ahead to derive the expression of shear stress in turbulent flow, in the subject of fluid mechanics, with the
help of 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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