2014new fluids lecture 08-09fmorriso/cm310/2014Lecture8-9.pdf · Fluids Lecture 8 Morrison CM3110 10/15/2014 3 © Faith A. Morrison, Michigan Tech U. To solve for complex flow fields:

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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Image from: gkdot.blogspot.com

Complex Flows


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


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Can we apply this modeling method to more complex problems?

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Image from: gkdot.blogspot.com Image fro

m: P

randtlan

d

Tietjens(1929); p

141 of o

ur textJet engine model.

Image from: www.stanford.edu


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)


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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



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?




Complex flow fields

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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


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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•


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)


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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)


) Newtonian Constitutive Equation(all types of flow fields)

Both expressions give the link between:

• Deformation (change of shape)• and Stress

Complex flow fields

Image from: www.labspaces.net


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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


We used here:

Newtonian Constitutive Equation


)

2

2

2

Gives the link between deformation and stress.


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Newtonian Constitutive Equation


2

2

2

)

00

⇒

Newton’s law of viscosity is a special case of the Newtonian Constitutive equation.

(Unidirectional flow)


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


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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 )


)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


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1. No slip2. Symmetry 3. Extrema4. Matching velocity, stress between fluids

Boundary Conditions


Boundary Conditions(Ex 6.5, p464)


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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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(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.


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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


ˆsurface

S

Q n v dS


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.


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Common surface shapes:

2

:

:

:

: ( ) 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?


``

V

nS

b

dS

Write the force on a small piece of surface

, and sum over the entire surface.


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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.


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.


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z‐component of force on the wall

ˆ ˆz z

S at surface

F e n pI dS



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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?


``

V

nS

b

dS

Write the torque on a small piece of surface , and sum over the entire surface.

Again, we use Π.


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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?


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


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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



4. Express torques on surfaces due to fluids

ˆS at surface


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)



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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 ?




Torque in Couette FlowSolution:

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1. Solve for velocity field (microscopic momentum bal)2. Calculate stress tensor3. Formulate equation for torque (an integral)4. Integrate5. Apply boundary conditions


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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


on a surfaceT

What is lever arm, R?

Etc…Chapter 2 ReduxChapter 2 ReduxChapter 2

ˆsurface

Sz

S

n v dSQ

vSdS

average velocity


ˆsurface

S

Q n v dS

force on a surface

ˆS at surface

F n pI dS



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Complex flow fields

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3. Calculations of quantities of interest• ????

Three questions remain: How do we handle the following:



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:


2014new fluids lecture 08-09fmorriso/cm310/2014Lecture8-9.pdf · Fluids Lecture 8 Morrison CM3110 10/15/2014 3 © Faith A. Morrison, Michigan Tech U. To solve for complex flow fields:

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