2014new fluids lecture 08-09fmorriso/cm310/2014Lecture8-9.pdf · Fluids Lecture 8 Morrison CM3110...
Transcript of 2014new fluids lecture 08-09fmorriso/cm310/2014Lecture8-9.pdf · Fluids Lecture 8 Morrison CM3110...
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
CM3110 Transport IPart I: Fluid Mechanics
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
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Complex Flows
© Faith A. Morrison, Michigan Tech U.
CM3110
Transport Processes and Unit Operations I
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Fluid MechanicsMicroscopic Momentum Balances
Let’s take stock
gvPvvt
v
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1. Control volumes2. Coordinate systems3. Continuity equation (microscopic mass balance)4. Navier‐Stokes (microscopic momentum balance)5. Newton’s law of viscosity6. Boundary conditions7. Solving differential equations8. Calculate quantities of interest
vvt
Lecture 8
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Can we apply this modeling method to more complex problems?
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m: P
randtlan
d
Tietjens(1929); p
141 of o
ur textJet engine model.
Image from: www.stanford.edu
© Faith A. Morrison, Michigan Tech U.
To solve for complex flow fields:
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gvPvvt
v
2
1. Sketch and simplify if possible2. Use convenient coordinate system
• Match system symmetries• Minimize the number of components of velocity
3. Continuity equation (microscopic mass balance)• Use table for components
4. Navier‐Stokes (microscopic momentum balance)• Use table for components
5. Newton’s law of viscosity• ????
6. Boundary conditions• ????
7. Solve differential equations• Use advanced methods• Use computers
8. Calculate quantities of interest• ????
vvt
Continuity equation(microscopic mass balance)
Navier‐Stokes equation(microscopic momentum balance)
(let’s try)
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
To solve for complex flow fields:
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gvPvvt
v
2
1. Sketch and simplify if possible2. Use convenient coordinate system
• Match system symmetries• Minimize the number of components of velocity
3. Continuity equation (microscopic mass balance)• Use table for components
4. Navier‐Stokes (microscopic momentum balance)• Use table for components
5. Newton’s law of viscosity• ????
6. Boundary conditions• ????
7. Solve differential equations• Use advanced methods• Use computers
8. Calculate quantities of interest• ????
vvt
Continuity equation(microscopic mass balance)
Navier‐Stokes equation(microscopic momentum balance)
(let’s try)
What should we use in the complex case?
What should we use in the complex case?
What should we use in the complex case?
© Faith A. Morrison, Michigan Tech U.
Complex flow fields
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1. Newton’s law of viscosity• ????
2. Boundary conditions• ????
3. Calculations of quantities of interest• ????
Three questions remain: How do we handle the following:
• Flow rate, • Average velocity, • Forces due to fluids• Torques due to fluids
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Complex flow fields
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1. Newton’s law of viscosity
zyz
dv
dy
Newton’s Law ofViscosity
(Scalar relationship; one coordinate system)
• Works for unidirectional flow•
© Faith A. Morrison, Michigan Tech U.
Complex flow fields
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1. Newton’s law of viscosity
zyz
dv
dy
Newton’s Law ofViscosity
(Scalar relationship; one coordinate system)
• Works for unidirectional flow
(that’s it)
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.9
In flows other than unidirectional flow, we need the more general relationship: the Newtonian Constitutive Equation
(Tensor relationship; all coordinate systems)
Complex flow fields
In general, there are 9 components of stress at every
location in a fluid
)Newtonian Constitutive Equation
zyz
dv
dy
Newton’s Law of Viscosity(unidirectional flow)
© Faith A. Morrison, Michigan Tech U.10
) Newtonian Constitutive Equation(all types of flow fields)
Both expressions give the link between:
• Deformation (change of shape)• and Stress
Complex flow fields
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Fluids Lecture 8 Morrison CM3110 10/15/2014
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Equation of Motion
V
ndSS
microscopic momentumbalance written on an arbitrarily shaped volume, V, enclosed by a surface, S
vv v P g
t
Gibbs notation: general fluid
gvPvvt
v
2Gibbs notation:
Newtonian fluid
Navier‐Stokes Equation
© Faith A. Morrison, Michigan Tech U.11
We used here:
Newtonian Constitutive Equation
© Faith A. Morrison, Michigan Tech U.12
)
2
2
2
Gives the link between deformation and stress.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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Newtonian Constitutive Equation
© Faith A. Morrison, Michigan Tech U.13
2
2
2
)
00
⇒
Newton’s law of viscosity is a special case of the Newtonian Constitutive equation.
(Unidirectional flow)
© Faith A. Morrison, Michigan Tech U.14
For other coordinate systems, use the handout
Π I )Total stress
2
2
2
yx x x z
xx xy xz
y y yx zyx yy yz
zx zy zz xyzyx z z z
xyz
vv v v vp
x y x z x
v v vv vp
y x y z y
vv v v vp
z x z y z
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.15
For other coordinate systems, use the handout
Total stress
Pressure only acts as a normal (perpendicular) push
2
2
2
yx x x z
xx xy xz
y y yx zyx yy yz
zx zy zz xyzyx z z z
xyz
vv v v vp
x y x z x
v v vv vp
y x y z y
vv v v vp
z x z y z
Π I )
© Faith A. Morrison, Michigan Tech U.16
)Newtonian Constitutive Equation
• The viscous stresses are due to molecular forces• How deformation and stress are linked depends on the
molecules• Some molecules do not follow the Newtonian
Constitutive Equation• Rheology! (Non‐Newtonian Fluid Mechanics)
Notes:
(CM4650 Polymer Rheology; spring)
Gives the link between:
• Deformation (change of shape)• and Stress
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.17
1. No slip2. Symmetry 3. Extrema4. Matching velocity, stress between fluids
Boundary Conditions
© Faith A. Morrison, Michigan Tech U.18
Boundary Conditions(Ex 6.5, p464)
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.19
How do we calculate quantities of interest?
1. Calculate flow rate
2. Calculate average velocity
3. Express forces on surfaces due to fluids
4. Express torques on surfaces due to fluids
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© Faith A. Morrison, Michigan Tech U.
(The expressions are different in different coordinate systems)
Engineering Quantities of Interest
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0 0
0 0
W H
z
z W H
v dx dy
v
dx dy
average velocity
volumetric flow rate
0 0
W H
x zQ v dx dy WH v z‐component of force on the wall
0 0
L W
z xz x HF dy dz
H is the height of the film
xz
fluid
xvz
air
Our strategy has been to develop the equation for each special case.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Engineering Quantities of Interest(tube flow)
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cross-section A:A
r z
r
z
L vz(r)
Rfluid
Our strategy has been to develop the equation for each special case.
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0 0
2
0 0R
R
z
z
drdr
drdrv
vaverage velocity
volumetric flow rate z
R
z vRdrdrvQ 22
0 0
z‐component of force on the wall dzRdF
L
Rrrzz
0
2
0
ˆsurface
Sz
S
n v dS
vdS
average velocity
volumetric flow rate
ˆsurface
S
Q n v dS
© Faith A. Morrison, Michigan Tech U.
Engineering Quantities of Interest(any flow)
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Instead, we can use general expressions that work in all cases.
Using the general formulas will help prevent errors.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Common surface shapes:
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:
:
:
: ( ) sin sin
rectangular dS dxdy
circular top dS r drd
surface of cylinder dS Rd dz
sphere dS Rd r d R d d
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(Fore more areas, see inside back cover)
Note: spherical coordinate system in use by fluid mechanics community
uses 0 as the angle from the =axis to the point.
What is the general expression for fluid force on a surface?
© Faith A. Morrison, Michigan Tech U.
``
V
nS
b
dS
Write the force on a small piece of surface
, and sum over the entire surface.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
We can show:(any flow, small surface)
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Π ≡
Force on the surface
⋅ ΠV
nS
b
dS
This is the power of the stress tensor: It allows us to calculate fluid forces on any surface.
Total stress tensor, Π:The stress tensor was invented to make this calculation easier.
© Faith A. Morrison, Michigan Tech U.26
Fluid force on the surface S
⋅
⋅
, , and evaluated at the surface
Π
To get the total force, we integrate over the entire surface of interest.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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z‐component of force on the wall
ˆ ˆz z
S at surface
F e n pI dS
© Faith A. Morrison, Michigan Tech U.
Engineering Quantities of Interest(any flow)
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ˆS at surface
F n pI dSforce on the wall
Using the general formulas will help prevent errors (like forgetting the pressure).
What is the general expression for fluid torque on an object?
© Faith A. Morrison, Michigan Tech U.
``
V
nS
b
dS
Write the torque on a small piece of surface , and sum over the entire surface.
Again, we use Π.
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
ˆS at surface
total fluid torqueR n dS
on a surfaceT
R lever arm
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pI total stress tensor
(Points from axis of rotation to position where torque is applied)
What is the general expression for fluid torque on an object?
© Faith A. Morrison, Michigan Tech U.
Example 4: In a liquid of density , what is the net fluid force on a submerged sphere (a ball or a balloon)? What is the direction of the force and how does the magnitude of the fluid force vary with fluid density?
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(p81)
H0f
air
x
z
Chapter 2 ReduxChapter 2 ReduxChapter 2
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Solution: We will be able to do this in this course (Ch4, p257).
2
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0 0
ˆ( cos ) sinrgR H R e d dF
From expression for force due to fluid, obtain (spherical coordinates):
We can do the math from here.
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ˆS at surface
total fluid forcen dS
on a surfaceF
Chapter 2 ReduxChapter 2 ReduxChapter 2
© Faith A. Morrison, Michigan Tech U.
4. Express torques on surfaces due to fluids
ˆS at surface
total fluid torqueR n dS
on a surfaceT
R lever arm
We will learn to write the stress tensor for our systems; then we can calculate stresses, torques.
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pI total stress tensor
(Points from axis of rotation to position where torque is applied)
Chapter 2 ReduxChapter 2 ReduxChapter 2
Fluids Lecture 8 Morrison CM3110 10/15/2014
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Example 5, Torque in Couette Flow: A cup‐and‐bob apparatus is widely used to measure viscosities for fluids. For the apparatus below, what is the torque needed to turn the inner cylinder (called the bob) at an angular speed of ?
© Faith A. Morrison, Michigan Tech U.33
Chapter 2 ReduxChapter 2 ReduxChapter 2
© Faith A. Morrison, Michigan Tech U.
Torque in Couette FlowSolution:
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Chapter 2 ReduxChapter 2 ReduxChapter 2
1. Solve for velocity field (microscopic momentum bal)2. Calculate stress tensor3. Formulate equation for torque (an integral)4. Integrate5. Apply boundary conditions
Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Torque in Couette FlowSolution:
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See problem 6.22 p487
2
2
0
1
0r z
R r Rv
R r
Velocity solution:
Tv v
pI
ˆS at surface
total fluid torqueR n dS
on a surfaceT
What is lever arm, R?
Etc…Chapter 2 ReduxChapter 2 ReduxChapter 2
ˆsurface
Sz
S
n v dSQ
vSdS
average velocity
volumetric flow rate
ˆsurface
S
Q n v dS
force on a surface
ˆS at surface
F n pI dS
© Faith A. Morrison, Michigan Tech U.
Engineering Quantities of Interest(any flow)
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Fluids Lecture 8 Morrison CM3110 10/15/2014
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© Faith A. Morrison, Michigan Tech U.
Complex flow fields
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1. Newton’s law of viscosity• ????
2. Boundary conditions• ????
3. Calculations of quantities of interest• ????
Three questions remain: How do we handle the following:
• Flow rate, • Average velocity, • Forces due to fluids• Torques due to fluids
© Faith A. Morrison, Michigan Tech U.
Complex flow fields -
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1. Newton’s law of viscosity →Use the Newtonian Constitutive Equation
2. Boundary conditions →Use vector relationships to write the boundary conditions for complex geometries
3. Calculations of quantities of interest → Use the general formulations (involve vector, matrix manipulations)
We handle these topics as follows:
• Flow rate, • Average velocity, • Forces due to fluids• Torques due to fluids