This preview shows page 1. Sign up to view the full content.
Unformatted text preview: P + ρg + η∇ 2 U + ∇(∇ ⋅ U )
Dt
3
DU
= −∇P + ρg + η∇ 2 U
Dt EE C245 ρ 17 U. Srinivasan © Cartesian Coordinates
ρ DU
= −∇P + ρg + η∇ 2 U
Dt x direction ∂U x
∂U x
∂U x
∂U x + Ux
+U y
+ Uz
=
∂x
∂y
∂z ∂t EE C245 ρ U. Srinivasan © ∂ 2U x ∂ 2U x ∂ 2U x ∂Px
−
+ ρg x + η 2 + 2 + ∂x
∂y
∂z 2 ∂x
18 9 Dimensional Analysis
DU
∇P
η∇ 2 U
=−
+g+
ρ
ρ
Dt • Each term has dimension L/t2 so ratio of any two gives
dimensionless group EE C245 • inertia/viscous
• pressure /inertia
• flow v/sound v =
=
= P/ρU2 =
=
= Reynolds number
Euler number
Mach number • In geometrically similar systems, if dimensionless numbers are
equal, systems are dynamically similar U. Srinivasan © 19 Two Laminar Flow Cases U
h τw τw EE C245 y U. Srinivasan © τw
τw High h
P Low
P y
Ux
U Ux
Umax 20 10 Couette Flow
• Couette flow is steady viscous flow
between parallel plates, where top plate is
moving parallel to bottom plate
• Noslip boundary conditions at plates ∇ ⋅ U = 0 U = Uxix
U = U x ( y )i x DU x
η ∂ 2U x ∂ 2U x 1 ∂P
=−
+ gx + + ∂y 2 Dt
ρ ∂x
ρ ∂x 2
U x = 0 at y = 0
∂ 2U x
= 0, and
U x = U at y = h
∂y 2
EE C245 U h τw τw y Ux = Ux
U U. Srinivasan © 21 Couette Flow
• Shear stress acting on plate due to motion, , is dissipative
• Couette flow is analogous to resistor with power dissipation
corresponding to Joule heating τ w = −η
U τw EE C245 h U. Srinivasan © τw RCouette = ∂U x
∂y =
y =h τ w A ηA
U = h PCouette = RU 2
22 11 Poiseuille Flow
• Poiseuille flow is a pressuredriven flow between stationary
parallel plates
• Noslip boundary conditions at plates ∂ 2U x ∂ 2U x DU x
1 ∂P
=−
+ gx +η
+ ∂x 2
∂y 2 Dt
ρ ∂x ∂P
∆P
=−
L
∂x ∂ 2U x
∆P
=−
,
2
∂y
Lη U x = 0 at y = 0, h
Ux = EE C245 y τw
τw Highh
P Low
P U max = Ux
Umax U. Srinivasan © 23 Poiseuille Flow
• Volumetric flow rate Q ~ [h3] Q = W ∫0 U x dy =
h • Shear stress on plates, , is
dissipative
• Force balance
• Net force on fluid and plates is zero
since they are not accelerating
• Fluid pressure force ∆PWh (+x)
balanced by shear force at the walls
of 2 WL (x) EE C245 • Wall...
View
Full
Document
This note was uploaded on 04/01/2014 for the course CHBE 6200 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

Click to edit the document details