1. HAGEN POISEUILLE EQUATION
GUIDED BY
PROF D V CHAUHAN
SUBMITTED BY

2. CONTENTS
INTRODUCTION
DERIVATION
REFERENCE

3. INTRODUCTION
In fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe.
It can be successfully applied to air flow in lung alveoli, for the flow through a drinking straw or through a hypodermic needle.
It was experimentally derived independently by Gotthilf Heinrich Ludwig Hagen in 1839 and Jean Léonard Marie Poiseuille in 1838, and published by Poiseuille in 1840 and 1846.

4. DERIVATIONS
Consider the steady flow of fluid of constant density in fully developed flow through a horizontal pipe. Visualize a disk shaped element, concentric wit the access of the tube, of radius r and length dL as shown in the figure.

5. βa upstream = βb downstream
Assumptions:
Fluid pressure acting on upward stream pa = p
Fluid pressure acting on downward stream pb = -(p + dp)
Fluid is possessing viscosity
Pipe is an horizontal pipe
za = zb
The flow is fully developed
(i.e) Va upstream = Vb downstream
βa upstream = βb downstream

6. According to momentum equation
= ΣF = 0
{that is resultant of all the forces acying on the fluid element would be zero}
ΣF = 0
= pasa – pbsb + f w – fg = (1)
[ where pasa = force acting upward stream
pbsb = force acting downward stream
f w =force acting on the stream due to wall
fg =force due to gravity]

7. Pressure force on upward stream pasa = pπr2
Pressure force on downward stream pbsb = (p + dp) πr2
force acting on the wall = - (shear stress on the rim of the element)
f w = - fs
As fs = (2πrdL)τ
f w = - (2πrdL)τ
Now substituting the values and dividing by dLπr2 eq (1) we get,
(2)

8. Now here pressure drop will occur and this is due to skin friction
therefore dp = ΔPs
Now if we take eq (2) for definite length dL the eq becomes
(3)

9. Now to calculate friction we use bernouliis equation
(i.e)
as the flow is fully developed
Va upstream = Vb downstream
βa upstream = βb downstream
za = zb
therefore the eq becomes
(4)

10. As we know
therefore
(5)
substituting (5) in (4) we get
(6)
therefore,
[putting eq (3)]

11. So the eq becomes
therefore we get
(7)
since the fluid is unsteady it will possess some velocity hence considering a thin ring with radius r and width dr, area is given by:
ds = 2πr dr

12. The velocity description is found using the definition of viscosity.
(8)
therefore
(9)
Integrating eq (8) with u = 0 and r = rw

13. Therefore
(10)
Now the average velocity is given by
(11)
Substituting the value of eq (8),(9),(10) in (11) we get
(12)

14. Substituting the value of (3) in eq (12) we get the following
now for practical calculation D = rw
so the equation becomes,
on further simplifying it becomes

15. So ,the below equation
is known as Hagen
Poiseuille Equation.
One of its uses is to measure the viscosity of the fluid given its pressure drop and volumetric flow rate of a tub eof given diameter and lenght.

16. REFERENCES
Unit Operation Of Chemical Engineering ,Warren L. McCabe , Julian C. Smith , Peter Harriot. (7th Edition 2005) ISBN

17. THANK YOU