Class 7 - Constitutive Relations

7/27/2019 Class 7 - Constitutive Relations

1/23

AOE 5104 Class 7

Online presentations for next class:

Kinematics 1

Homework 3 Class next Tuesday will be given by Dr.

Aurelien Borgoltz

daVinci (Aaron Marcus, Justin Ratcliff)


2/23

Claude-Louis Navier

(February 10, 1785 in Dijon - August 21, 1836 in Paris)


3/23

Pump flow

VisEng L td

http://www.viseng.com/consult/flowvis.html
http://www.viseng.com/index.htmlhttp://www.viseng.com/consult/flowvis.htmlhttp://www.viseng.com/consult/flowvis.htmlhttp://www.viseng.com/index.html


4/23

Equations for Changes Seen From

a Lagrangian Perspective

0=dDt

D

R

S

zyx

SRR

dS).(+).(+).(+dSp-d=dDt

DknjninnfV

dST).k(+dS.++p-+d.=d)2

V+(e

Dt

D

SS

zyx

R

2

R

nVknjninnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V.

Dt

D

kjifV

).().().( zyxpDt

D

).().().().().(.)(

2

21

TkwvupDt

VeDzyx

VVf


5/23

O

Pump flow

VisEng L td

http://www.viseng.com/consult/flowvis.html
http://www.viseng.com/index.htmlhttp://www.viseng.com/consult/flowvis.htmlhttp://www.viseng.com/consult/flowvis.htmlhttp://www.viseng.com/index.html


6/23

Conversion from Lagrangian to

Eulerian rate of change - Derivative

x

y

z(x(t),y(t),z(t),t)

.Vt

zw

yv

xu

tt

z

zt

y

yt

x

xt

t part

Time Derivative Convective Derivative

.V

tDt

D

The Substantial Derivative


7/23

Conversion from Lagrangian to

Eulerian rate of change - Integral

x

y

z

The Reynolds

Transport

Theorem

SR

R

R

R

R

RRRsys

dSdt

dt

dt

ddDt

D

Dt

Ddd

Dt

D

Dt

dDd

Dt

D=d

t

nV

V

VV

V

.

).(

..

.

.V

tDt

D

Volume R

Surface S

Apply

Divergence

Theorem

SRR

dSdt

=dDt

DnV.


8/23


a Lagrangian Perspective

0=dDt

D

R

S

zyx

SRR

dS).(+).(+).(+dSp-d=dDt

DknjninnfV

dST).k(+dS.++p-+d.=d)2

V+(e

Dt

D

SS

zyx

R

2

R

nVknjninnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V.

Dt

D

kjifV

).().().( zyxp

Dt

D

).().().().().(.)(

2

21

TkwvupDt

VeDzyx

VVf

parttDt

D

SRR

dSdt

=dDt

DnV.

.V

tDt

D


9/23


an Eulerian Perspective

Differential Form (for a fixed point in space)

Integral Form (for a fixed control volume)0=dSd

t SR

nV.

S

zyx

SRR

dS).(+).(+).(+dSp-d=dSdt

knjninnfnVVV

).(

dST).k(+dS.++p-+d.=dSV+ed)t

V+e

SS

zyx

RS

22

R

nVknjninnfVnV ).().().(.)()(

212

1

V.

Dt

D

kjifV

).().().( zyxp

Dt

D

).().().().().(.)(

2

21

TkwvupDt

VeDzyx

VVf

.V

tDt

D


10/23

Equivalence of Integral and

Differential Forms

0=dSdt

SR

nV.

d=dS RS VnV ..

0.

d

tRV

0.

V

t0..

VV

t

V.

Dt

D

Cons. of mass

(Integral form)

Divergence

Theorem

Conservation of

mass for any

volume R

Then we get or

Cons. of mass

(Differential form)


11/23


12/23

Constitutive Relations - Closing

the Equations of Motion

Could we solve, in principle, the equationswe have derived for a particular flow?


13/23


an Eulerian Perspective

Differential Form (for a fixed point in space)

Integral Form (for a fixed control volume)0=dSd

t SR

nV.

S

zyx

SRR

dS).(+).(+).(+dSp-d=dSdt

knjninnfnVVV

).(

dST).k(+dS.++p-+d.=dSV+ed)t

V+e

SS

zyx

RS

22

R

nVknjninnfVnV ).().().(.)()(

212

1

V.

Dt

D

kjifV

).().().( zyxp

Dt

D

).().().().().(.)(

2

21

TkwvupDt

VeDzyx

VVf

.V

tDt

D


14/23

Constitutive Relations

Equations of motion

5 eqns: Mass (1), Momentum (3), Energy (1)

13 unknowns: p, , u, v, w, T, 6 , e

Need 8 more equations!

Information about the fluid is needed

Constitutive relations Thermodynamics: p, , T, e

Viscous stress relations

p = p(,T) and

e = e(,T)

Newtonian fluid


15/23

Newtonian (Isotropic) Fluid

Viscous Stress is Linearly Proportional to Strain Rate Relationship is isotropic (the same in all directions)

ij

zzzyzx

yxyyyx

xzxyxx

Stress, is a tensor

and so has some basic properties when we rotate the coordinate system used

to represent it like

Principal axes axis directions for

which all off shear stressesare zero

Tensor invariants combinations

of elements that dont

change with the axis directions

zz

yy

xx

00

00

00

)(

)(

)(3

1

3

1

3

1

CubictDeterminanIII

QuadraticII

LinearI

i j jiijjjii

i ii

But what is strain rate (or rate of deformation?)

(Symmetric so yx= xy,

yz= zx, xz= zx)


16/23

Distortion of a Particle in a Flow

M>1, accelerating,

expanding flow

Total change

= rotation

+ dilation

+ shear deformation

Physical ly

Rate of

deformation

orstrain

rate

Cauchy Stokes

Decomposition


17/23

Distortion of a Particle in a Flow

rrrV

rr

VV

d

z

v

y

w

x

w

z

u

z

v

y

w

y

u

x

v

x

w

z

u

y

u

x

v

d

z

w

y

vx

u

d

z

v

y

w

x

w

z

u

z

v

y

w

y

u

x

v

x

w

z

u

y

u

x

v

d

dd

d

dz

dy

dx

z

w

y

w

x

w

z

v

y

v

x

v

z

u

y

u

x

u

dw

dv

du

d

0

0

0

00

00

00

0

0

0

21

21

21

21

21

21

21

21

21

21

21

21

V+dV

V

Deformation isrepresented by

dVtime so rate of

deformation is

given by dV

Total

change= rotation + dilation + shear deformation

Mathematical ly

Rate of deformation orstrain rate


18/23

Newtonian (Isotropic) Fluid

z

w

z

v

y

w

x

w

z

u

z

v

y

w

y

v

y

u

x

v

xw

zu

yu

xv

xu

toalproportion

LLYISOTROPICAzzzyzx

yxyyyx

xzxyxx

21

21

21

21

21

21

z

w

y

vx

u

toalproportion

LLYISOTROPICAzz

yy

xx

00

00

00

00

00

00

V.22

x

u

z

w

y

v

x

u

x

uxx

So

So Each stress= Const.Corresponding strain+ Const. F irst invariant of

comp. rate component strain rate tensor

Or And likewise for

y andz.


19/23

Stokes Hypothesis

Stokes hypothesized that the total normal

viscous stress xx+yy+zz should be zero, so

that they cant behave like an extra pressure

(i.e. he wanted to simplify things so that the

total pressure felt anywhere in the flow wouldbe the same as the pressure used in the

thermodynamic relations).

This implies =- and remains controversial

With this, and in general (non-principal) axes,

we finally have

y

u

x

v

x

u

xy

xx

V.232 and likewise foryyand zz

and likewise foryzand xz

V.22

x

u

z

w

y

v

x

u

x

uxx


20/23

The Equations of MotionDifferential Form (for a fixed volume element)

V.

Dt

D

).(2)()(f

)().(2)(f

)()().(2f

31

31

31

V

V

V

z

w

zy

w

z

v

yx

w

z

u

xz

p

Dt

Dw

y

w

z

v

zy

v

yy

u

x

v

xy

p

Dt

Dv

xw

zu

zyu

xv

yxu

xxp

DtDu

z

y

x

).(2)()()().(2)(

)()().(2).()(.)(

31

31

31

2

21

VV

VVVf

z

ww

y

w

z

vv

x

w

z

uu

zy

w

z

vw

y

vv

y

u

x

vu

y

x

w

z

uw

y

u

x

vv

x

uu

xTkp

Dt

VeD

The Continuity equation

The Navier Stokes equations

The Viscous Flow Energy Equation

These form a closed set when two

thermodynamic relations are

specified


21/23

Leonhard Euler

1707-1783


22/23

Assumptions made / Info encodedAssumption/Law Mass NS VFEE

Conservation of massConservation of momentum

Conservation of energy

ContinuumNewtonian fluid

Isotropic viscosity

Stokes Hypothesis

Fouriers Law of Heat conduction

No heat addition except by

conduction


23/23

Summary

Conservations laws

Lagrangian and Eulerian perspectives

Equations of motion dervied from a Lagrangian

perspective The Substantial Derivative and the Reynolds

Transport Theorem connect Lagrangian withEulerian

Constitutive Relations provide information aboutthe fluid material

Assumptions

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Class 7 - Constitutive Relations

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