lecture 7 method of moment, 2nd law of thermodynamics,...

GIAN Course on

Rarefied & Microscale Gases and Viscoelastic Fluids:

a Unified Framework

Lecture 7

Method of Moment, 2nd Law of

Thermodynamics, Cumulant Expansion,

and Balanced Closure

R. S. Myong

Gyeongsang National University

South Korea

Feb. 23rd ~ March 2nd, 2017

GIAN Lecture 7-1 Rarefied & Microscale Gases and Viscoelastic Fluids: a Unified Framework R. S. Myong, Gyeongsang National University, South Korea Feb. 23 – March 2, 2017 – IIT Kanpur, India

Content

I. Method of moment

II. 2nd law of thermodynamics

III. Cumulant expansion

IV. Balanced closure


I. Method of moment

2( , , ) ,f t C f ft

v r v

0d

pdt

u

I Π

( , , )f t r vBoltzmann transport equation (BTE): 1023

( , , )

Enormous reduction of information

x y z

mf t

dv dv dv

r v

),(,,,,, rQΠu tT

the statistical definition

( , , )

and

with BTE

Differentiating

m f t

with time

then combining

u v r v

Conservation laws & constitutive equations: 13

Nonlinear collision

integral


I. Method of moment

2

2

( , , ) ,

the statistical definition ( , , ) and

with BTE ( , , are independent and )

[ , ]

Here

f t C f ft

Differentiating m f t with time

then combining t

fm f m m f m C f f

t t

m f

v r v

u v r v

r v v u c

v v v v v

v v

(2)

2

where Tr( ) / 3 , [ ] ,

After the decomposition of the stress

and using the collisional invariance of the momentum, [ , ] 0,

we have

( )0.

m f m f

m f p p m f m f

m C f f

pt

vv uu cc

P cc I Π cc Π cc

v

uuu I Π

Exact consequence of the original BTE!


I. Method of moment

2The same for the heat flux, the derivation of

the energy cons

/ 2 ,

ervation law

w h

.

itmc fQ c

In fact, this procedure can be generalized for arbitrary molecular expressions

of general moment h(n)

( ) ( ) ( ) ( ) ( ) ( )2

(2) (2)(1)

(2) (2)( ) (1)2

( )

[ , ]

For , the constitutive equation of the viscous shear stress

( / )2 2 [ , ] ,

Tr( )

n n n n n ndh f h f h f f h f h h C f f

t dt

h m m f

dp h C f f

dt

m f m f

u c c

cc Π cc

ΠΠ u u

ccc ccc

/ 3.I

Again exact consequence of BTE

No approximation introduced so far!


I. Method of moment

( ) ( ) ( ) ( ) ( ) ( )2

(3) 2 2

( )

[ , ]

For / 2 , the constitutive equation of the heat flux / 2

(assuming in monatomi0

( / )

c gas)

n n n n n n

p

Q

dh f h f h f f h f h h C f f

t dt

h mc mC T mc f

m f

d dm f

dt dt

u c c

c Q c

J c

Q uccc u Π Q u

(3) ( ) 22[ , ] , / 2 ( ).

p p

Qp

C T pC T

h C f f mc f C T p

Π

cc I Π

Again no approximation!

This critical fact can not be found on codified textbooks!

This constitutive equation was not presented in Grad’s 1949 work, since his 13-

moment closure was already in place. In fact, to the best of my knowledge, it

was never derived explicitly until B. C. Eu (1992).


I. Method of moment

In the derivation of constitutive equation of heat flux, the following relations are

used

(3)

(3) 2 2

(3) 2

,

/ 2 / 2 ,

/ 2

= ,

p

p p

p p

p p p

h f C T

h f mc mC T f mc f C T

f h f mc f C T m mfC T Et t t t t t

C T C T C Tt t t t

Q J

c c c cc P

c u uc c P I

u uJ Π J

(3) 2 2 2

(3) 2 2 2

/ 2 / 2 ( ) / 2

( ) ,

/ 2 / 2 ( ) / 2

p

p p p p

p

p p

f h f mc mf C T fmc fm c

mf C T mfC T E C T C T

f h f mc mf C T fmc fm c

mf C T mfC T

u u c u c u c u c

u c u c u u u u P J u u u

c c c c c c c c c

c c c c

( )

( ) 2

,

, / 2 , / , .

Pp p

Pp

C T C T

m f E mc f C T p E p

Q u u P J u

ccc P I Π


I. Method of moment


I. Method of moment

Finally, the conservation laws and constitutive equations, all of which are an

exact consequence of the Boltzmann equation

1/ 0 0

0 ,

0t

dp

dtE p

u

u I Π

u Π u Q

(2)

(2)( ) ( )

( ) ( ) ( )

2/ 2

./ :Q P Q

pp

pdddt C p TC Tdt

Π uΠ Λu

uQ u ΛΠ Q u Π

( ) ( )

Key observations: 1) Even though we used various , disappeared except for

(kinematic high order) and (dissipation)

2) Thus the

f f

Λ

Monitor exactly w

number of places

here they go from

to close is two, rather one.

Lesson micro (kinetic) to macro (ave: rage)

(2)

2( , , ) , 2f t C f ft

v r v Π u


I. Method of moment

Let us tackle the dissipation term first, since we cannot get away from it.

( ) (2)2[ ] [ , ]m C f f Λ cc

But, before we move on , let us think of what we are trying to do here first:

the seat of energy dissipation coming from the collision integral. Therefore, we

might need to recall the 2nd law of thermodynamics (lecture 3)

1

0 where is calortropy

( ) : extended Gibbs relation

d

d T dE dW dN

nonequilibrium

Let us start from the balance equation for the calortropy (nonequilibrium entropy)

the t

ˆ

hermodynamic branch of the Boltzmann equati

( , ) ln ( , , ) 1 ( , , )

wher one

cB

c

t k f t f t

f

r v r v r


Memo (a preparation before tackling 2nd law)

Arnold Sommerfeld (1868-1951):

“Thermodynamics is a funny subject. The first time you go through it, you don't

understand it at all. The second time you go through it, you think you understand it,

except for one or two small points. The third time you go through it, you know you don't

understand it, but by that time you are so used to it, it doesn't bother you anymore.”

What?

2nd-law?

Why is it anything to do with here?

Is it dead animal?


Memo (a preparation before tackling 2nd law)

Recall lecture 3! Calortropy?


2

differentiating the local calortropy density with time and combining it with BTE

the positive calortropy producti

By

ˆ(ln 1) ln ln [ , ] .

After assuming ,

c c cB B c B

c

d dk f f k f f k f C f f

dt dt

f f

c c

12 2

2* * * *

2 12 2 2 2 20 0

( )

on

Then, if the calortropy production is worked out first, instead of the dissipation term

in the

becomes ( )

1 ln(

conventi

/ )( ) 0

o

.4

c

c c c c c c c cc B

nc

g

k d d d db bg f f f f f f f f

v v

v v

Λ

( )

nal approach, and if there is a direct relation between them,

a thermodynamically consistent form of be obtained.

From the logarithmic form of the calortropy production, it will be conven

ca

t

n

ien

nΛ

2 ( ) ( )

1

2 ( ) ( )

1

to write

the distribution function in the exponential form, instead of Grad's polynomi

1exp ,

2

1 1 1exp( ) exp ,

al form;

2

c n n

n

n n

d Bn

f mc X h N

N mc X hn k

.T



II. 2nd law of thermodynamics (???)

Just a little extra push is all it takes!


IV. Balanced closure (Myong PoF 2014)

(2) (2)( ) (2 ) ( ) ( ) ( )

12 2 1 2

1

A thermodynamically-consistent constitutive equation, still exact to BTE,

( / ) 12 2 ( , , ).l l

l

dp R X q

dt g

Π

Π u u

The simplest closure in 2nd-order level is

while the 2nd-order closure of RH term is Then

( ) 0,

(21) (1) ( )

12 2 1( ).R X q

This is the 2nd-order constitutive equation beyond the two-century old

Navier-Stokes equation and recovers NS,

(2)

2 / .NSp p u Π

(2) (2)

1/21/4 1/41

1 11

1

( / )2 2 ,

sinh ( ) :( ) , .

2

( )NS

B

NS NS

d pp

dt

mk T Tq

p kd

q

ΠΠ u u

Π Q /

Π

Π Q


The lessen: Type of closure may not be important. Rather, balanced

treatment of two open terms! Remember ‘balanced’ does not mean

anything in case of only one term.

Though looking trivial at first glance, we will appreciate the beauty and real

power of the balanced closure, when we tackle the high Mach number

shock singularity in Lecture 11.

IV. Balanced closure (Myong PoF 2014)


Q & A

lecture 7 method of moment, 2nd law of thermodynamics,...

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