Development of a Compact Fusion Device based on the Flow Z ...
1/49 Chapter 6 Equations of Continuity and...
Transcript of 1/49 Chapter 6 Equations of Continuity and...
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Chapter 6 Equations of Continuity and Motion
Session 6-2 Equation of motion
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Chapter 6 Equations of Continuity and Motion
Contents
6.1 Continuity Equation
6.2 Stream Function in 2-D, Incompressible Flows
6.3 Rotational and Irrotational Motion
6.4 Equations of Motion
6.5 Examples of Laminar Motion
6.6 Irrotational Motion
6.7 Frictionless Flow
6.8 Vortex Motion
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6.4 Equations of Motion
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• Apply Newton's 2nd law of motion
• External forces = surface force + body force
〮 Surface force:
~ normal force + tangential force
〮 Body forces:
~ due to gravitational or electromagnetic fields, no contact
~ act at the centroid of the element → centroidal force
6.4 Equations of Motion
(A)F ma=∑
x xF ma∆ = ∆
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Consider only gravitational force
6.4 Equations of Motion
x y zg ig jg kg= + +
LHS of (A):
( )x xF gx y zρ∆ = ∆ ∆ ∆
xx xy z y zxx
σσ σ ∂ − ∆ ∆ + ∆ ∆+ ∆ ∂
yxyx yxx z x zyy
ττ τ
∂ − ∆ ∆ + ∆ ∆+ ∆ ∂
zxzx zxx y x yzz
ττ τ ∂ − ∆ ∆ + ∆ ∆+ ∆ ∂
body force (B)
normal force
tangential force
tangential force
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6.4 Equations of Motion
Divide (B) by volume of element
yxx x zxx
F gx y z x y z
τσ τρ∂∆ ∂ ∂
= + + +∆ ∆ ∆ ∂ ∂ ∂
(C)
RHS of (A):
xx
ma ax y z
ρ∆ =∆ ∆ ∆
(D)
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6.4 Equations of Motion
Combine (C) and (D)
yxx zxx xg ax y z
τσ τρ ρ∂∂ ∂
+ + + =∂ ∂ ∂
xy y zyy yg ax y z
τ σ τρ ρ
∂ ∂ ∂+ + + =
∂ ∂ ∂
yzxz zz zg ax y z
ττ σρ ρ∂∂ ∂
+ + + =∂ ∂ ∂ (6.21)
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6.4 Equations of Motion
6.4.1 Navier-Stokes equations– Eq (6.21) ~ general equation of motion
– For Newtonian fluids (with single viscosity coeff.), use stress-strain
relation given in (5.29) and (5.30)
→ Navier-Stokes equations
Eq. (5.29):
( )223
pressure normal stress due to fluid deformation and viscosity
xup qx
σ µ µ∂ = − + − ∇ ⋅ ∂
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6.4 Equations of Motion
( )223y
vp qy
σ µ µ∂ = − + − ∇ ⋅ ∂
( )223z
wp qz
σ µ µ∂ = − + − ∇ ⋅ ∂
Eq. (5.30):
yx xyv ux y
τ τ µ∂ ∂ += = ∂ ∂
yz zyw vy z
τ τ µ∂ ∂ += = ∂ ∂
zx xzu wz x
τ τ µ ∂ ∂ = = + ∂ ∂
(6.23)
(6.22)
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Substitute Eqs. (5.29) & (5.30) into (6.21)
Assume constant viscosity (neglect effect of pressure and temperature on
viscosity variation)
6.4 Equations of Motion
( )223x x
v up u u wg aq x yx x y zx z xρ µ µ ρµ µ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +− + + + =− +∇ ⋅ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂
( )223x x
v up u u wg aq x yx x y zx z xρ µ µ µ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +− + + + =− +∇ ⋅ ∂ ∂∂ ∂ ∂ ∂∂ ∂ ∂
u v wx y z
∂ ∂ ∂+ +
∂ ∂ ∂
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6.4 Equations of Motion
Expand and simplify
u v wx y zx
∂ ∂ ∂∂ + + ∂ ∂ ∂∂
2 2 2 2 2 2 2 2
2 2 2 2
2. . 23x
p u u v w v u u wL H S gx x x x y x z x y y z x z
ρ µ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
= − + − ++ + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2 22 2 2
22 2 2
13x
u v wp u u ugx x y x zx x y z
ρ µ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + += − + ++ + ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂
( )2 2 2
2 2 2
13x
p u u ug qx xx y z
ρ µ µ ∂ ∂∂ ∂ ∂
= − + ++ + ∇ ⋅ ∂ ∂∂ ∂ ∂
normal stress shear stress
normal stress + shear stress
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6.4 Equations of Motion
( )2 2 2
2 2 2
13x x
p u u ug aqx xx y z
ρ µ µ ρ ∂ ∂∂ ∂ ∂
− + + =+ + ∇ ⋅ ∂ ∂∂ ∂ ∂
( )2 2 2
2 2 2
13y y
p v v vg aqy yx y z
ρ µ µ ρ ∂ ∂∂ ∂ ∂
− + + =+ + ∇ ⋅ ∂ ∂∂ ∂ ∂
( )2 2 2
2 2 2
13z z
p w w wg aqz zx y z
ρ µ µ ρ ∂ ∂∂ ∂ ∂
− + + =+ + ∇ ⋅ ∂ ∂∂ ∂ ∂
→ Navier-Stokes equation for compressible fluids with constant viscosity
(6.24)
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1) For inviscid (ideal) fluid flow, (µ = 0) → viscous forces are neglected.
6.4 Equations of Motion
♦ Vector form
( ) ( )23
qg p q qq qt
µρ µ ρ ρ∂− ∇ + ∇ + ∇ = +∇ ⋅ ⋅∇∂
( )dq qa qqdt t
∂= = + ⋅∇
∂
where --- Eq. (2.5)
( )qg p qqt
ρ ρ ρ∂− ∇ = + ⋅∇∂
→ Euler equations for ideal fluid
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2) For incompressible fluids, (Continuity Eq.)
Define acceleration due to gravity as
6.4 Equations of Motion
0q∇ ⋅ =
( )2 qg p q qqt
ρ µ ρ ρ∂− ∇ + ∇ = + ⋅∇∂
(6.25)
xhg gx
∂= −
∂
yhg gy
∂= −
∂g g h= − ∇
zhg gz
∂= −
∂
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where h = vertical direction measured positive upwardFor Cartesian axes oriented so that h and z coincide
→ minus sign indicates that acceleration due to gravity is in the negative
h directionThen, N-S equation for incompressible fluids and isothermal flows are
6.4 Equations of Motion
(6.26)0 , 1x yhg gz
∂= = =
∂
zg g= − (6.27)
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6.4 Equations of Motion
2 2 2
2 2 2
1u u u u h p u u uu v w gt x y z x x x y z
µρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2 2
2 2 2
1v v v v h p v v vu v w gt x y z y y x y z
µρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2 2
2 2 2
1w w w w h p w w wu v w gt x y z z z x y z
µρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
(6.28)
Local acceleration
Convective acceleration
Body force per mass
Pressure force per mass
Viscosity force per mass
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Eq. (6.28): unknowns – u, v, w, p→ We need one more equation to obtain a solution when the boundary
conditions are specified.
→ Eq. of continuity for incompressible fluid
♦ Boundary conditions
1) kinematic BC: velocity normal to any rigid boundary (wall) equal the
boundary velocity (velocity = 0 for stationary boundary)
2) physical BC: no slip condition (continuum stick to a rigid boundary)
→ tangential velocity relative to the wall vanish at the wall surface
6.4 Equations of Motion
0u v wx y z
∂ ∂ ∂+ + =
∂ ∂ ∂
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6.4 Equations of Motion
♦ General solutions for Navier-Stocks equations are not available because
of the nonlinear, 2nd-order nature of the partial differential equations.
→ Only particular solutions may be obtained by simplifications.
→ Numerical solutions are usually sought.
r - component:2
r r r rr z
v vv v v vv vt r r r z
θ θρθ
∂ ∂ ∂ ∂+ + − +
∂ ∂ ∂ ∂
( ){ } 2 22 2 2 21 21 r rr rp v v vg rvr r r r zr r θρ µ θ θ ∂ ∂ ∂ ∂ ∂∂= − + + − + ∂ ∂ ∂ ∂ ∂∂
♦ Navier-Stocks equations in cylindrical coordinates for constant density
and viscosity
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θ - component:
z - component:
6.4 Equations of Motion
rr z
v v v v v v vv vt r r r zθ θ θ θ θ θρ
θ∂ ∂ ∂ ∂ + + − + ∂ ∂ ∂ ∂
( ){ } 2 22 2 2 21 1 21 rp v v vg rvr r r r zr r θ θθ θρ µθ θ θ ∂ ∂ ∂ ∂ ∂∂= − + + + + ∂ ∂ ∂ ∂ ∂∂ z z z z
r zvv v v vv v
t r r zθρ
θ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂
2 2
2 2 21 1 z zz
zp v vvg rz r r r zr
ρ µθ
∂ ∂ ∂ ∂∂ = − + + + ∂ ∂ ∂ ∂∂ (6.29)
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6.4 Equations of Motion
Continuity eq. for incompressible fluid
( ) ( ) ( )1 1 0r zvrv vr r r zθθ∂ ∂ ∂
+ + =∂ ∂ ∂
(6.30)
Normal & shear stresses for constant density and viscosity
2 rrvpr
σ µ ∂= − +∂
12 rv vpr r
θθσ µ θ
∂ = − + + ∂
2 zzvpz
σ µ ∂= − +∂
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6.4 Equations of Motion
1 rr
vvrr rr
θθτ µ θ
∂ ∂ = + ∂ ∂
1 zz
v vz rθ
θτ µ θ∂ ∂ = + ∂ ∂
r zzr
v vz r
τ µ ∂ ∂ = + ∂ ∂
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6.5 Examples of Laminar Motion
- N-S equations are important in viscous flow problems.
♦ Laminar motion
~ orderly state of flow in which macroscopic fluid particles move in layers
~ viscosity effect is dominant
♦ Laminar flow through a tube (pipe) of constant diameter
~ instantaneous velocity at any point is always unidirectional (along the
axis of the tube)
~ no-slip condition @ boundary wall
~ apply concept of the Newtonian viscosity
~ velocity gradient gives rise to viscous force within the fluid
~ low Re
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6.5 Examples of Laminar Motion
[Re] Reynolds number = inertial force / viscous force = destabilizing force /
stabilizing force
♦ Viscous force
~ dissipative
~ have a stabilizing or damping effect on the motion
~ use Reynolds number
[Cf] Turbulent flow
~ unstable flow
~ instantaneous velocity is no longer unidirectional
~ destabilizing force > stabilizing force
~ high Re
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2-D flow (x, z)
steady flow
parallel flow
z-axis coincides with h
6.5.1 Laminar flow between parallel platesConsider the two-dimensional, steady, laminar flow between parallel plates
in which either of two surfaces is moving at constant velocity and there is
also an external pressure gradient.
♦ Assumptions:
6.5 Examples of Laminar Motion
( )0 ; 0vy
∂→ = =
∂( ) 0t
∂→ =
∂
( )0 ; 0ww ∂→ = =
∂
0 ; 1h h hx y z
∂ ∂ ∂→ = = =
∂ ∂ ∂
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6.5 Examples of Laminar Motion
a
y
zaz/a
0px
∂<
∂0p
x∂
=∂0
px
∂<
∂
p1 p2
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6.5 Examples of Laminar Motion
♦ External pressure gradient
1 2p p>
0px
∂< →
∂i) pressure gradient assists the viscously induced motion to
overcome the shear force at the lower surface
0px
∂> →
∂ii) pressure gradient resists the motion which is induced by the
motion of the upper surface
1 2p p
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6.5 Examples of Laminar Motion
Continuity eq. for two-dimensional, parallel flow:
0u wx z
∂ ∂+ =
∂ ∂
( )
2
2
only
0ux
u f z
∂=→ ∂
=
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6.5 Examples of Laminar Motion
N-S Eq.:
. : ux dirt
∂−
∂uux
∂+
∂uvy
∂+
∂uwz
∂+
∂
hgx
∂= −
∂
2
2
1 p ux x
µρ ρ
∂ ∂− +
∂ ∂
2
2u
y∂
+∂
2
2u
z ∂
+ ∂
2
2
10 p ux z
µρ ρ
∂ ∂∴ = − + ∂ ∂
Steady flowContinuity eq. for incompressible fluid
2D flow
parallel flow
(6.31a)
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6.5 Examples of Laminar Motion
. : wz dirt
∂−
∂wux
∂+
∂wvy
∂+
∂wwz
∂+
∂
2
2
1h p wgz z x
µρ ρ
∂ ∂ ∂= − − +
∂ ∂ ∂
2
2w
y∂
+∂
2
2w
z∂
+∂
10 pgzρ
∂∴ = − −
∂ (6.31b)
(6.31b): p gz
ρ γ∂ = − = −∂
( )p z f xγ∴ = − + (6.32)
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→ hydrostatic pressure distribution normal to flow
→ For any orientation of z -axis. in case of a parallel flow, pressure is distributed hydrostatically in a direction normal to the flow.
(6.31a): ~ independent of z
6.5 Examples of Laminar Motion
p dpx dx
∂→
∂
(A)2
2dp udx z
µ ∂∴ =∂
Pressuredrop
Energy loss due toviscosity
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Integrate (A) twice w.r.t. z
Use the boundary conditions,
6.5 Examples of Laminar Motion
2
2dp udzdz dzdzdx z
µ ∂=∂∫∫ ∫∫
1dp uz dz dz C dzdx z
µ ∂= +∂∫ ∫ ∫
2
1 22dp z u C z Cdx
µ= + + (6.33)
i) ( ) 2 20, 0 0 00dpz u C Cdx
µ= = → × = + ∴ =
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6.5 Examples of Laminar Motion
ii) 2
1, 2dp az a u U U C adx
µ= = → = +
2
11
2dp aC U
a dxµ
∴ = −
∴ (6.33) becomes
2 212 2
dp z dp au zUdx a dx
µ µ = + −
2
2 2z dp az zu Ua dx
µ µ ∴ = − −
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6.5 Examples of Laminar Motion
( ) 12
U a dp zu z u z za dx aµ
= = − −
(6.34)
Velocitydriven
Pressuredriven
i) If 0dpdx
= Couette flow (plane Couette flow)→
Uu za
= (6.35)
→ driving mechanism = (velocity)U
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6.5 Examples of Laminar Motion
ii) If 0U = 2-D Poiseuille flow (plane Poiseuille flow)→
( ) parabolic1 ~2
dpu z a zdxµ
= − (6.36)
→ driving mechanism = external pressure gradient, dpdx
max @ 2au z =
2
max 8a dpu
dxµ= − (6.37)
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6.5 Examples of Laminar Motion
V = average velocity
(6.38)2
max23 12
Q a dpuA dxµ
= = = −
[Re] detail
( )2 30 0
1 12 12
a a dp dpQ u dz z az dz adx dxµ µ
= = − = −∫ ∫2
max21
12 3Q a dpA a V uA dxµ
= × ∴ = = − =
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6.5 Examples of Laminar Motion
[Re] Dimensionless form
2
2
12
2
1
u z a dp z zU a U dx a a
a dpPU dx
u z z zPU a a a
µ
µ
= − −
= −
= + −
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6.5 Examples of Laminar Motion
[Cf] Couette flow in the narrow gap of a journal bearing
Flow between closely spaced concentric cylinders in which one
cylinder is fixed and the other cylinder rotates with a constant
angular velocity, ω
i
o i
U ra r r
Ua
ω
τ µ
=
= −
≈
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→ Hagen-Poiseuille flow
→ Poiseuille flow: steady laminar flow due to pressure drop along a tube
Assumptions:
– use cylindrical coordinates
6.5 Examples of Laminar Motion
6.5.2 Laminar flow in a circular tube of constant diameter0p
x∂
<∂
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6.5 Examples of Laminar Motion
parallel flow →
Continuity eq. →
paraboloid →
steady flow →
00
rvvθ
==
0zvz
∂=
∂
0zvθ
∂=
∂
0zvt
∂=
∂
0zv ≠
0zvr
∂≠
∂
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Eq. (6.29c) becomes
By the way,
independent of r
6.5 Examples of Laminar Motion
10 zzp vg rz r r r
ρ µ∂ ∂ ∂ = − + + ∂ ∂ ∂ (A)
( ) ( )zp dg p h p hz z dz
ρ γ γ∂ ∂
− + = − = −+ +∂ ∂
zhg gz
ρ ρ ∂ = − ∂
( )
comp. Eq.
01
0
r
rp gr
p rr
ρ
γ
→
= −
−∂
+∂
∂→ + =
∂
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6.5 Examples of Laminar Motion
Then (A) becomes
( ) 1 zd vp h rdz r r r
µγ∂ ∂ =+ ∂ ∂
( )1 zd vrp h rdz r r
γµ
∂ ∂ =+ ∂ ∂ (B)
Integrate (B) twice w.r.t. r
( )2
11
2zd r vr Cp h
dz rγ
µ∂
= ++∂
(C)
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6.5 Examples of Laminar Motion
( ) 112
zd v Crp hdz r r
γµ
∂= ++
∂
( )2
1 21 ln
2 2 zd r v C r Cp hdz
γµ
= + ++ (D)
Using BCs
→ (C) :
→ (D) :
max0 , z zr v v= = 1 0C =
0 , 0zr r v= = ( )2
02
12 2
rdC p hdz
γµ
= + (D1)
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6.5 Examples of Laminar Motion
Then, substitute (D1) into (D) to obtain νzpiezometricpressure
( ) ( )2 201
4zdv r rp hdz
γµ
∴ = − −+
( )22
0
0
14zrd rv p h
dz rγ
µ = − + −
(6.39)
→ equation of a paraboloid of revolution
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(1) maximum velocity,
(2) mean velocity, Vz
6.5 Examples of Laminar Motion
maxzv
(6.40)max @ 0zv r =
( )max
20
4zd rv p hdz
γµ
= − + (6.41)
zdQ v dA=
( ) ( )2 201 2
4d rdrr rp hdz
πγµ
= − −+
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6.5 Examples of Laminar Motion
( ) ( )0 2 2001 2
4r dQ rdrr rp h
dzπγ
µ = − −+ ∫
( )02 4
202 2 4
r
r
d r rp h rdz
πγ
µ = − + −
( )4
0
8r d p h
dzπ
γµ
= − +
( ) max2
02
0 8 2z
z
vQ Q r dV p hA r dz
γπ µ
∴ = = = =− + (E)
[Cf] For 2 - D Poiseuille flow max23
V u=
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6.5 Examples of Laminar Motion
(3) Head loss per unit length of pipe
Total head = piezometric head + velocity head
Here, velocity head is constant.
Thus, total head loss = piezometric head change
( ) 2 20
1 8 32f z zh V Vd p hL r Ddz
µ µγ
γ γ γ ≡ = =− +
(E)where 0 diameter2D r= =
(6.42)
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6.5 Examples of Laminar Motion
[Re] Consider Darcy-Weisbach Eq.
(F)21
2f z
h VfL D g
=
head loss due to friction= fh
friction factorf =
Combine (6.42) and (F)
2
232 1
2z zV Vf
D D gµ
γ= (6.43)
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Differentiate (6.39) w.r.t. r
6.5 Examples of Laminar Motion
64 64 64/ Rez z
fV D V D
νν
= = = → For laminar flow (6.44)
(4) Shear stress
rzr
vz
τ µ ∂=∂
zz vvrr
µ ∂∂ =+ ∂∂
(G)
( ) 12
zv d rp hr dz
γµ
∂= +
∂ (H)
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6.5 Examples of Laminar Motion
Combine (G) and (H)
( )12zr
d rp hdz
τ γ= +
Linear profile
(6.45)
At center and walls
0 , 0zrr τ= =
( )max0 0
1,2zr zr
dr r rp hdz
τ τγ= = =+
Chapter 6 Equations of Continuity and MotionChapter 6 Equations of Continuity and Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.4 Equations of Motion6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion 6.5 Examples of Laminar Motion