Hagen Poiseuille Flow Problem

1. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
Hagen Poiseuille Flow Problem
Prof. Dr.-Ing. Naseem Uddin
Mechanical Engineering Department
NED University of Engineering & Technology
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

2. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
1 Problem Definition
2 Assumptions
3 Governing Equations
4 Boundary Condition
5 Solution
Mean Velocity
Centerline Velocity
Shear Stress and Reynolds Number
Skin friction Coefficient
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

3. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
Problem Definition
Prove that
under Fully developed
flow conditions
velocity profile is
Fluid in a long pipe is
moving at steady rate with u = um (1 − r∗2 )
fully developed conditions
under applied constant where r∗ = r/rw
pressure gradient. and at the center of
pipe
−dP 2
rw
um = umax =
dx 4µ
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

4. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
Assumptions
1 Fully Developed Flow
2 No body forces, fb = 0
3 Non-porous walls, v = 0
3 Steady Flow
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

5. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
x momentum equation
We analyze the problem in (r,θ,x) coordinates with x as
streamwise coordinate. x momentum equation:
∂ 2 u 1 ∂u 1 ∂P
2
+ = (1)
∂r r ∂r µ ∂x
1 ∂ ∂u 1 ∂P
(r ) = (2)
r ∂r ∂r µ ∂x
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

6. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
∂u 1 dP r2 C1
= + (3)
∂r rµ dx 2 r
dP r2
u= + C1 ln(r) + C2 (4)
dx 4µ
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

7. Outline
Problem Definition
Assumptions
Governing Equations
Boundary Condition
Solution
Boundary/Initial Condition
No Slip Condition u(rw , t) = 0
Centerline Condition (∂u/∂r)r=0 = 0
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

8. Outline
Problem Definition Mean Velocity
Assumptions Centerline Velocity
Governing Equations Shear Stress and Reynolds Number
Boundary Condition Skin friction Coefficient
Solution
Solution
At Centerline gradient of velocity profile is zero so from
equation 3 C1 =0 At Walls velocity is zero so from equation 4
2
dP rw
C2 = − (5)
dx 4µ
dP r2 2
dP rw
u= − (6)
dx 4µ dx 4µ
dP 1 2 2 dP 1 2
u= r − rw = − r 1 − (r/rw )2 (7)
dx 4µ dx 4µ w
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

9. Outline
Problem Definition Mean Velocity
Assumptions Centerline Velocity
Governing Equations Shear Stress and Reynolds Number
Boundary Condition Skin friction Coefficient
Solution
Mean Velocity
1
u= udA (8)
A A
rw
1
u= 2
2uπrdr (9)
πrw 0
rw
dP 1 2 1
u=− r 2 1 − (r/rw )2 πrdr (10)
dx 4µ w πrw 0
2
rw
β β
u= 2 1 − (r/rw )2 πrdr = (11)
πrw 0 4
where,
dP 1 2
β=− r (12)
dx 2µ w
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

10. Outline
Problem Definition Mean Velocity
Assumptions Centerline Velocity
Governing Equations Shear Stress and Reynolds Number
Boundary Condition Skin friction Coefficient
Solution
Centerline Velocity
β β
uc = 1 − (r/rw )2 = (13)
2 2
Ratio of Centerline velocity to Mean velocity is
uc β/2
= =2 (14)
u β/4
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

11. Outline
Problem Definition Mean Velocity
Assumptions Centerline Velocity
Governing Equations Shear Stress and Reynolds Number
Boundary Condition Skin friction Coefficient
Solution
Shear Stress is calculated from the Newton’s law of viscosity
τw = µ(du/dr)w
dP 1 2
|τw | = µ( r )(1/rw ) = µβ/rw (15)
dx 2µ w
Reynolds number is defined as uD/ν where D is the diameter of
pipe and ν is kinematic viscosity
uDρ βDρ
ReD = = (16)
µ 4µ
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem

12. Outline
Problem Definition Mean Velocity
Assumptions Centerline Velocity
Governing Equations Shear Stress and Reynolds Number
Boundary Condition Skin friction Coefficient
Solution
Fanning Friction Coefficient is defined as
|τw | 32µ
Cf = 1 2
= (17)
2 ρu
ρβrw
32µ ρDβ 8D
Cf ReD = = = 16 (18)
ρβrw 4µ rw
Dary friction coefficent is defined as
64
f = 4Cf = (19)
ReD
Prof. Dr.-Ing. Naseem UddinMechanical Engineering Department NED University of Engineering
Hagen Poiseuille Flow Problem