2.25 Advanced Fluid Mechanics
October 15th 2008
Couette & Poiseuille Flows . 1 Some of the fundamental solutions for fully developed
viscous flow are shown next. The flow can be pressure or viscosity driven, or a combination of
both. We consider a fluid, with viscosity µ and density ρ. (Note: W is the depth into the page.)
• a) PLANE Wall-Driven Flow (Couette Flow)
Parallel flow: u(y) = u(y)x̂, flow between parallel plates at y = 0 and y = H, wall-driven,
and resisted by fluid viscosity.
Umax = max(Utop , Ubottom )
Umin = min(Utop , Ubottom )
Uavg = 0.5 ∗ (Utop + Ubottom )
u(y)
H
ŷ
Q′ = Q/W
Q′ =
H
2
(Utop
− Ubottom ) (signed)
τw,top = −µ
(Utop −Ubottom )
H
x̂
u(y) = (Utop − Ubottom )(y/H) + Ubottom
• b) PLANE Pressure-Driven Flow (Poiseuille Flow) (Stationary walls)
Parallel flow: u(y) = u(y)x̂, flow between parallel plates at y = 0 and y = H,
pressure driven.
Umax = − dP
dx
H2
8µ
Uavg = 23 Umax
Q′ = − dP
dx
τw =
>0
u(y )
H
y
H3
12µ
H
dP
2 (− dx )
(τw on either wall in the x direction)
1
−dP
dx
2.25 Advanced Fluid Mechanics Problem
x
u(y) =
U H 2 ) dP y
2µ (− dx )( H (1 −
y
H ))
• c) TUBE Pressure Driven Flow (Poiseuille Flow) (Stationary walls)
Parallel flow: u(r) = u(r)x̂, flow along a tube.
−dP
dx
Umax =
Uavg =
>0
U R2 ) dP
4µ (− dx )
U R2 )U
8µ
r̂
dP
dx
−
)
4
u(r)
2R = D
x̂
πR
Q = − dP
dx 8µ
τw (2πRL) = πR2 L(− dP
dx )
(τw in the x direction)
f=
ΔP
L
1
ρU 2 D
2
=
u(r) =
16
ReD
2
U R2 )
4µ
r 2
(− dP
dx )(1 − ( R ) )
MIT OpenCourseWare
http://ocw.mit.edu
2.25 Advanced Fluid Mechanics
Fall 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.