Post on 26-Aug-2020
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
GIAN Lecture 7-2 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
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
GIAN Lecture 7-3 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
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!
GIAN Lecture 7-4 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
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!
GIAN Lecture 7-5 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
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).
GIAN Lecture 7-6 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
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
GIAN Lecture 7-7 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
I. Method of moment
GIAN Lecture 7-8 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
I. Method of moment
GIAN Lecture 7-9 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
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
GIAN Lecture 7-10 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
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
GIAN Lecture 7-11 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
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?
GIAN Lecture 7-12 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
Memo (a preparation before tackling 2nd law)
Recall lecture 3! Calortropy?
GIAN Lecture 7-13 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
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
GIAN Lecture 7-14 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
II. 2nd law of thermodynamics
GIAN Lecture 7-15 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
II. 2nd law of thermodynamics
GIAN Lecture 7-16 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
II. 2nd law of thermodynamics
GIAN Lecture 7-17 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
II. 2nd law of thermodynamics
GIAN Lecture 7-18 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
II. 2nd law of thermodynamics (???)
Just a little extra push is all it takes!
GIAN Lecture 7-19 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
II. 2nd law of thermodynamics
GIAN Lecture 7-20 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
III. Cumulant expansion
GIAN Lecture 7-21 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
III. Cumulant expansion
GIAN Lecture 7-22 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
III. Cumulant expansion
GIAN Lecture 7-23 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
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
GIAN Lecture 7-24 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
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)
GIAN Lecture 7-25 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
Q & A