What Makes Gas Micro Flows So Complicated: Non-classical ...

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What Makes Gas Micro Flows So Complicated: Non-classical Physical Laws and their Morphing into Gas-Surface Interaction June 6 Wed, 2012 (8:45-9:30AM) R. S. Myong D t fA dS t E i i Dept. of Aerospace and System Engineering Gyeongsang National University South Korea [email protected] Presented at 1 st European Conference on Gas Micro Flows Skiathos Island, Greece

Transcript of What Makes Gas Micro Flows So Complicated: Non-classical ...

Page 1: What Makes Gas Micro Flows So Complicated: Non-classical ...

What Makes Gas Micro Flows So Complicated: Non-classical Physical Laws and

their Morphing into Gas-Surface Interaction

June 6 Wed, 2012 (8:45-9:30AM)

R. S. MyongD t f A d S t E i iDept. of Aerospace and System Engineering

Gyeongsang National UniversitySouth Korea

[email protected]

Presented at 1st European Conference on Gas Micro Flows pSkiathos Island, Greece

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Many thanks toMany thanks to organizers of this conference

(Profs Colin Frijns Valougeorgis)(Profs. Colin, Frijns, Valougeorgis)for making this opportunity possible.

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Introduction to gas micro flowsg

Micro and nanoscale cylindrical flow

Hard disk drive: Kn=0.6, M=0.7

Gas flow in micro/nano devicesMolecular interaction between gas particles and solid atomsMolecular interaction between gas particles and solid atomsVelocity shear dominated flows (high Kn, but relatively low M)

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Theory and modeling of gas micro flowsy g g

Continuum approachChapman Enskog: BurnettChapman-Enskog: BurnettMoment method: Grad (1949), Eu (1992),

R-13 (2005)1/

D pρ

ρ⎡ ⎤−⎡ ⎤⎢ ⎥⎢ ⎥ + ∇ +⎢ ⎥⎢ ⎥

uu I Π 0

Top-down

(constitutive equations taking into accountthe microscopic nature of gas molecules)

( )t

pDt

E pρ +∇ ⋅ + =⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥ + ⋅ +⎣ ⎦ ⎣ ⎦

u I Π 0I Π u Q

molecules)

Molecular approachDSMC

Bottom-up

DSMCLinearized Boltzmann equationLattice-Boltzmann method

Hybrid approachDSMC ti li (b d d i d iti )

Lattice Boltzmann method

DSMC-continuum coupling (based on domain decomposition)Multi-scale method (non-conserved variables calculated by MD)

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Research goal of present studyResearch goal of present study

Develop a unified computational model for rarefied and micro- & nano-scale gases g

upon which others can build efficient (3-D) CFD d (lik t t f th t FLUENT k )codes (like state-of-the-art FLUENT package)

http://acml gnu ac kr <Open knowledge>http://acml.gnu.ac.kr <Open knowledge>

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Challenges and emphasisg p

Challenges in gas micro flows: unknown problem (deviation from classical physics)- unknown problem (deviation from classical physics)

- not easily testable (limited information)- theoretical barrier (G.E. and B.C.)

Emphasis in this talk:

theoretical barrier (G.E. and B.C.)- cross-disciplines

Emphasis in this talk: - clear demonstration of knowns (fully exact analytic approach, focusing on simple problems)- qualitative over quantitative (putting many abnormal behaviors in proper context, rather than one-time snap-shot

t)agreement)- sharing failure and struggle (rather than making conclusive)- verification & validation of computational simulation of gasverification & validation of computational simulation of gas micro flows

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ScopeFocusing on (1-D velocity shear dominated) benchmark flows:

C tt ( t d b i ll)- Couette (generated by moving wall)- force-driven Poiseuille (pure 1-D)- pressure-driven Poiseuille (testable)- pressure-driven Poiseuille (testable)- verification & validation (V & V)

Based on published journal papersBased on published journal papers- Physics of Fluids (2011)- Computers & Fluids (2011)Co pute s & u ds ( 0 )- Int. J. Heat Mass Transfer (2006; with Lockerby & Reese)- Physics of Fluids (2004)- Physics of Fluids (1999)

and on going work on Couette and pressure drivenand on-going work on Couette and pressure-driven Poiseuille flows

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Part I.

Governing equations andand

boundary conditions

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Governing equationsg q

Assumption: Non-classical phenomena are related to what happens in b lk fl i d h t h th lid ll d t th ibulk flow region and what happens near the solid wall and to their combination.

Boltzmann equation (1844-1906) collision integral

[ ]2,);,( ffCtft v =⎟

⎠⎞

⎜⎝⎛ ∇⋅+∇⋅+∂∂ vrav );,( vrtf

Boltzmann equation (1844-1906) g

t ⎠⎝ ∂ force:a

( , ; )mf tρ = r v the statistical definition

( , ; ) Differentiating

m f tρ ≡u v r v

x y zdv dv dv= ∫∫∫L Land with the Boltzmann equation

with timethen combining

( ) aΠIu ρρ =+⋅∇+ pDtD ( ) ),(,,,,, rQΠu tT Lρ

w t t e o t a equat o

Moment equation

shear stress tensor

heat flux vector

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Derivation of continuum model

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

Conservation laws (exact) in vector form

1/ 0 0

t

D pDt

E p

ρρ ρ

ρ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+∇ ⋅ +∇ ⋅ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ + ⋅⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

uu I Π a

u Π u Q a u[ ]( 2)( )Π

[ ]( 2) 2 / 2, m f mc f≡ ≡Π cc Q c

t p ρ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦u u Q a u[ ]

[ ]

( 2)( )

( 2)( )2,[ ] ( , , )

/

m f

m f fC qpη

Π

Π

≡ = −

ψ cc c

ccΠΛ Π QLConstitutive equation (exact) in vector form

[ ] [ ](2) (2)( ) ( )2 2 pt

Π Π∂+ ⋅∇ = −∇ ⋅ − ⋅∇ − ∇ +

∂Π u Π ψ Π u u Λ

KinematicHigher-order Thermo. driving DissipationConvectionUnsteady Kinematic Higher order Thermo. driving Dissipation Convection Unsteady

Key observations:

- NSF limit dictated by Euler if[ ](2) 22~ ~ ~ ;MKn M

⋅∇⋅

Π u Π or p ρ∇ ⋅ < ∇ ⋅ ∇ ⋅Π I uuNSF limit dictated by Euler if

- NSF limit recovered from the last two terms and a relaxation (BGK) approximation

[ ](2) ;Re2

Kn Mpp ∇u

( , , ) 1q ≈Π QL

p ρ

( )

[ ] [ ](2) (2)0 small 2 1 2/

pp

ηη

= − ∇ − ⇒ = − ∇Πu Π u

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Derivation of algebraic constitutive relations∂Π

Original Grad closure and 2 2 22 1 1 15 2 2 2jk ik ijm f mc f mc f mc fδ δ δ⎛ ⎞

= + +⎜ ⎟⎜ ⎟⎝ ⎠

ccc c c c ( , , ) 1q ≈Π QL

[ ] [ ](2) (2)( ) ( )2 2 pt

Π Π∂+ ⋅∇ = −∇ ⋅ − ⋅∇ − ∇ +

∂Π u Π ψ Π u u Λ

no convective

5 2 2 2i j k⎜ ⎟⎝ ⎠

[ ] [ ] [ ](2) (2) (2)2 2 2 15 /

pt pη

∂+ ⋅∇ = − ∇ − ⋅∇ − ∇ −

∂Π Πu Π Q Π u u

no convectivein velocity-shear steady yielding partial differential type!u v

x y∂ ∂+

∂ ∂

5 /t pη∂

y

New closure and 2/1

2:~ where),,( ⎟⎟

⎞⎜⎜⎝

⎛ ⋅+=

kTq QQΠΠsinhQΠ

ηκ

κκ

L[ ](2)( ) 2Π∇ ⋅ < ⋅∇ψ Π u

[ ] [ ](2) (2)small 2 2 ( , , )/

p qt pη

∂+ ⋅∇ = − ⋅∇ − ∇ −

∂Π Πu Π Π u u Π QL

yielding algebraic type!yielding algebraic type!

I t ti l i l h i l lInteresting non-classical physical laws

can be identified!

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Linear uncoupled Navier-Fourier equations: l i l h i l lclassical physical laws

Navier (1822) (2)η= − ∇⎡ ⎤⎣ ⎦Π uFourier (1822) k T= − ∇Q

( ) / 3 ( ) / 2u u v u vΠ Π − + +⎡ ⎤ ⎡ ⎤( ) / 3 ( ) / 22 ( )

( ) / 2 ( ) / 3xx xy x x y y x

yx yy x y y x y

u u v u vT

v u v u vη

Π Π − + +⎡ ⎤ ⎡ ⎤← −⎢ ⎥ ⎢ ⎥

Π Π + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

case (compression and expansion)

2 /3 02 ( )

x

x

u

uTη

⎡ ⎤− ⎢ ⎥

case (velocity shear only)

0 / 22 ( )

y

y

u

uTη

⎡ ⎤− ⎢ ⎥

Newtonian orli Uncoupled

2

2 ( )0 / 3

Not like ( )

x

x

Tu

u

η ⎢ ⎥−⎣ ⎦2 ( )

/ 2 0

Always vanishing normal stress or y

xx yy

Tu

η ⎢ ⎥⎢ ⎥⎣ ⎦

Π Π

linear Uncoupled

2( ) . Not like ( )x xx

Q Tk T T

Q T⎡ ⎤ ⎡ ⎤

← −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦y yQ T⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

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Non-classical laws embodied in nonlinear coupled constitutive relations (NCCR)coupled constitutive relations (NCCR)

Thermodynamic Shear stressf

Algebraic Π(2)2η− ∇⎡ ⎤⎣ ⎦up÷ p×

driving force heat fluxgNCCR Qk T

⎣ ⎦− ∇

2PrpC T

p÷2Pr

pC Tp×

pxyxx ,Π1

1.5

Shear stress (Navier−Stokes)

yyyyxy Π⎟⎟⎞

⎜⎜⎛ Π+⎟⎟

⎞⎜⎜⎛Π

132

0.5

Shear stress (monatomic NCCR)

Nonlinear (shear thinning) pppyyyyy

⎟⎟⎠

⎜⎜⎝+−=⎟⎟

⎠⎜⎜⎝

12

kinematic stress xyΠ

−0.5

0

Normal stress (Navier−Stokes)

constraint!

−1

Normal stress (monatomic NCCR)Coupled(non-zero normal stress)

xxΠ

0 1 2 3 4 5 6−1.5

/ /du dy pη−velocity shear case

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Non-Fourier law by the coupling of force and shear stressand shear stress

( ) ( )0 0

0

2

3 ; is force3 2y y xy

xy

Q Q a a= + Π+ Π

) ) )))

3

ˆ /xy xy pΠ ≡ Π

Non-Fourier behavior!

( )0xy

( ), , / /(2 Pr)x y x y pQ Q p C T≡)

lux

0

1

2

Qy

behavior!Fourier law

y yQ Q=) )H

eat f

l

−2

−1

Q

0y yQ Q

1

2

0

5−3

−2

−1

0

−5Q

y0Π

xy0

Temp. gradient Vel. gradient

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Slip/jump models (boundary conditions)

General comments:B C should describe the molecular interaction of the- B.C. should describe the molecular interaction of the

gas particles with the solid surface atoms, involving the kinetic theory of gases and solid state physics.y g p y- Boltzmann equation based on collision of gas particles only may not be valid.y y- Concept based on gaseous adsorption may give a hint for parameters governing the gas-surface atom molecular interaction. Also, surface diffusion can be built from adsorption.

W h t li ith i l B C h M ll- We may have to live with simple B.C. such as Maxwell model in case of continuum-based computational modelsmodels.

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Langmuir slip model based on gaseous adsorption isotherm (PoF 2004)adsorption isotherm (PoF 2004)g c

g + s cs

c

Condense on the surface, being held by the field of force of the surface atoms and subsequently evaporate from the surface

)(moleculesgaswithginteractin)(sitesofnumber: msN

atoms, and subsequently evaporate from the surface⇒ time lag ⇒ adsorption ⇒ slip and jump

becomes constant mequilibriu Then thecovered are which sites ofnumber :

)(molecules gaswith ginteractin)(sites ofnumber :

KN

msNα

[ ] . where1

is, that ,)1(/

becomes constant mequilibriu Then the

wBwBsm

c

TkK

βpβpα

NTkpN

CCCK

K

=+

=−

== βα

α

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Heat of adsorption vs accommodation coeff.

Langmuir model (Dirichlet type)4)1( Knp ω

)(exp)(

41,)1(

)1/(21 −+

=⎟⎟⎞

⎜⎜⎛−⎟⎟

⎞⎜⎜⎛

=

+=−+=

DTfnDT

Knppuuu

ew

rw

ννωω

ωααα

ν

kcal/mol] )10~10([ adsorption ofHeat :

),,(exp)(

1

0

=⎟⎟⎠

⎜⎜⎝

⎟⎟⎠

⎜⎜⎝

=

OD

DTfnTkT

e

ewwBr

ννωω

Maxwell model (Neumann type)

coeff.tion)(accommodaslip;2 , vMMw

uuu σωω −≡⎟

⎠⎞

⎜⎝⎛ ∂+= l )(p;,

vM

wMw n σ

⎟⎠

⎜⎝ ∂

Assignment of physical meaning to accommodation coeff.A i l l ti b d b l i d i

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

−+

ewv DT exp)(-2)1/(21 ν

νωωσ

An equivalence relation can be proved by solving pressure-driven gas flow in a microchannel with two slip models

⎟⎟⎠

⎜⎜⎝−⎟⎟

⎠⎜⎜⎝

=rwBrv TkT

exp)(~ 0 νωωσ

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Morphing of NCCR into B.C.

Troubles found during the implementation- how to determine reference velocity in the Langmuir modelhow to determine reference velocity in the Langmuir model

rw uuu )1( αα −+=

- not successful in showing the velocity gradient singularity (identified by Lilley PRE 2007) in Couette flow by NCCR and B C (M ll & L i )B.C. (Maxwell & Langmuir)

Morphing of nonlinearity and coupling (of NCCR) intoMorphing of nonlinearity and coupling (of NCCR) into B.C.- back to the original concept of Maxwell model (degree of slip proportional to the degree of non-equilibrium near the wall)

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Morphing of NCCR into B.C.: Nonlinear coupled Maxwell modelcoupled Maxwell model

Conventional linear velocity slip and temperature jump models

/ 2/ 2 / 2

3( / 2) ( ) , ( / 2) ( )4V w T

hh h

u T Tu h V l T h T ly T x y

ησ σρ

⎤ ⎤∂ ∂ ∂⎤= + ⋅ − − = + ⋅ −⎥ ⎥⎥∂ ∂ ∂⎦⎦ ⎦

y p p j p

/ 2/ 2 / 24 hh hy T x yρ∂ ∂ ∂⎦⎦ ⎦

Nonlinear coupled velocity slip and temperature jump models

/ 2 / 2 / 2

3 ( 1)( / 2) , ( / 2)4 / Pr

yxV w T

h h h

QQu h V l T h T lp k

γσ ση γ

⎤⎤⎤Π −= + ⋅ + = + ⋅ ⎥⎥⎥

⎦ ⎦ ⎦

- Non-Navier-Stokes shear thinning behavior morphs into shear stress- Non-Fourier law morphs into heat fluxp- Morphing of NCCR (tangential heat flux generated by velocity shear) into velocity slip- Capable of describing the velocity gradient singularityCapable of describing the velocity gradient singularity

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Summary of G.E. and B.C. (I)Kn=M/Re

Rare

Nan

Increase of thermal

Hard disk drive

efied

o/Micro

Kn = 0.1

Increase of thermal

non-equilibrium

Re-entry vehicle trajectory

0.

Solid-gas

interaction

/2 = O(1)M ReKn = 0.03

Rarefied

Nonlinear

MM= 30

Rarefied

Hypersonic

C li

2 -1= O(10 )M /ReLocal thermal

equilibriumMM 30

High temperature effect boundary

Coupling

Non-isothermal

[ ]2( , ) ,f C f f⋅∇ =v r v pρ ⋅∇ +∇ ⋅ + ∇ ⋅ =u u I Π 0

Two terms: Kn Three terms: M, KnMp ⋅Π Kn~/Main parameter

(not Kn alone!)

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Summary of G.E. and B.C. (II)

Boltzmann (or similar) in

f

Nonlinear

Coupled

Kn*M

bulk flow

Gas-surface atom

Coupled

Nonlinear Kn⎞⎛

Morphing

(Seamless transition)

atom interaction Coupled

⎟⎟⎠

⎞⎜⎜⎝

⎛−

wB

e

v

vTk

Dexp~-2σσ

Combination of two nonlinearities and two couplings—nonlinear coupled constitutive relation for bulk flow and nonlinear coupled boundary conditions at the wall—may be critical within the y ycontinuum framework.

There may exist many cases in how to combine them makingThere may exist many cases in how to combine them, making qualitative & many-facets agreement more difficult.

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Part II.Part II.

Couette flowCouette flow

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Gaseous Couette flow (Comp. Fluids 2011 and on going)on-going)

VwallT

( )u y

0xyd⎡ ⎤Π ⎡ ⎤⎢ ⎥ ⎢ ⎥

yh

00

with

yy

xy y

d pdy

u Q

⎢ ⎥ ⎢ ⎥+Π =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Π + ⎣ ⎦⎣ ⎦

wall

wall wallK

VM

RT

RT

γ

ηπ

wallT

h

, ,

with

( , ) fn ,

li & l d

xy xx x ydu dTQ kdy dy

η⎛ ⎞

Π = − −⎜ ⎟⎝ ⎠

wall wallKn2 (0)p h

≡V

Non-classical behaviors (well-understood parts)

nonlinear & coupled

- nonlinear velocity profile (non-isothermal)- non-zero tangential heat flux and normal shear stress (coupling)- small shear stress (shear thinning)( g)However, unsolved problems persist (within continuum framework)- velocity gradient singularity (power law identified from DSMC)velocity gradient singularity (power law identified from DSMC)- smaller slip (or larger velocity slope)- exact role of non-isothermal physics

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NCCR + nonlinear Maxwell

With NCCR + linear Maxwellcapable of describing non zero- capable of describing non-zero

tangential heat flux and normal shear stress

b t t bl t d ib th- but not able to describe the velocity gradient singularity in the Knudsen layer

With NCCR + nonlinear (coupled) Maxwell( p )-capable of explaining the origin of the velocity gradient singularity- at the same time being able toat the same time being able to describe all other abnormal behaviors in qualitative agreement with DSMC

3 ( 1)( / 2) xV

Qu h V l γσ⎤⎤Π −

= + ⋅ + ⎥⎥agreement with DSMC/ 2 / 2

( / )4 / PrV

h h

u h V lp

ση γ ⎥⎥⎦ ⎦

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Summary of Part II: Well-posedness of Couette flowflow

0 M≤ < ∞

Initial data Physical constraints imposed

/ /du dy T Tη

0 Kn0 wallT< < ∞

<

,Kn, wallM TWall boundary conditionsu (slip) P i l tiwall ave/ , /du dy T Tη

(0) (0)T ρ

u (slip) T (jump)

p RTρ=

Primary solutions

(0), (0)T ρ

( ), ( ), ( )T y u y yρ , , 0p T ρ >

p RTρ=

, ,,xy xx yy zzΠ Π Π Π 0xx yy zzΠ Π Π+ + =

( ), ( )x yy yQ Q Conservation lawssatisfied exactly

Note: T(0) itself is the part of solution. Without it, the analysis remains incomplete. (For example, Marques et al. 2001 in Contin. Mech. Thermo.)

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Part III.Part III.

Force-driven andForce driven and pressure-driven

PoiseuillePoiseuille (monatomic)

gas flowsgas flows

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1-D force-driven compressible Poiseuille gas flowy

wallTwallT

h 0xy a

d pρ⎡ ⎤Π ⎡ ⎤

⎢ ⎥ ⎢ ⎥+Π =⎢ ⎥ ⎢ ⎥

x( )reservoir,p T ( )reservoir

,p TwallT

T

0

with

yy

xy y

pdy

auu Q

d dT

ρ+Π⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥Π + ⎣ ⎦⎣ ⎦

⎛ ⎞wallTuniform driving

force a

RTah ηπ

, ,( , ) fn ,

nonlinear & coupled

xy xx x ydu dTQ kdy dy

η⎛ ⎞

Π = − −⎜ ⎟⎝ ⎠

wall wall

wall reservoirwall

, Kn2

: Richardson no. : Knudsen no.h

RTah

RT p h

ηπε ≡ ≡

Non-classical behaviors (easy parts)3- central temperature minimum

- non-zero tangential heat flux and normal shear stress( ) ( )02

33 2 NSF

NSF

y y xyxy

Q Q a= + Π+ Π

) ) )))

However, difficult parts (within continuum framework)- concave cross-stream pressure distribution with central temp. minimum- correct cross-stream density distributioncorrect cross stream density distribution- exact role of non-isothermal physics (for example, Knudsen minimum in mass flow rate)

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Exact analytic solutions (with only one assumption : PoF 2011 )[ ](2)( ) 2Π∇ < ∇ψ Π uassumption : PoF 2011 )[ ]2∇⋅ < ⋅∇ψ Π u

Rigorous treatment of thermal property through/ 2

/ 20 1

0

2 / 2Average quantities , h

r r hhu udy T

h T dy−= =∫

∫0* * *

1/2 1/2* * * * * * *

A new (temperature scaled) variable and auxiliary relations 1 1 1 1, ,

T ds dy

s y u T ds T ds

=

⎛ ⎞= = = =⎜ ⎟⎝ ⎠

∫ ∫0 0, ,

2 2 2 2s y u ds ds⎜ ⎟⎝ ⎠ ∫ ∫

Analytic form of concave pressure distribution

( ) ( )2 30, , and 2

2 1 2

xyyy xy yy yy

dd p a p RT pdy dy

ρ ρΠ

+Π = = = Π = − +Π Π

⎛ ⎞ ⎛ ⎞* 2 * * * 2 * *2 1 2 1 tan , tan , 3 3

1 3 2

w wh w yy h wp T s T sNδ

ε ε⎛ ⎞ ⎛ ⎞

⇒ = + Π = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ Very surprising tangent profile!* * *1 3 2 tan

2 3 wxy h wT sNδ

ε⎛ ⎞

Π = ⎜ ⎟⎜ ⎟⎝ ⎠

y p g g p

Pressure more fundamental (than velocity, temperature)!

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Exact analytic solutions: stressesQualitative agreement with DSMC

0.4

NCCR

wall( , Kn=0.1)0.6hε =

1.06

1.08

1.1

1.12

pressure

0.1

0.2

0.3• : NCCR

O: DSMC

Solid line: NSF stresses

1

1.02

1.04

−0.2

−0.1

0normal stress yy−component

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.94

0.96

0.98

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.4

−0.3

shear stress xy−component

0.20.02

y y

0.1

0.15

normal stress xx−component

−0.01

0

0.01 pxxΠstresses

0

0.05

p

−0.05

−0.04

−0.03

−0.02

Existence of the i !

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.1

−0.05 normal stress zz−component

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.08

−0.07

−0.06

y

stress constraint!

pxyΠ

Page 30: What Makes Gas Micro Flows So Complicated: Non-classical ...

Exact analytic solutions: temp., density, heat fluxes

0.021.02

After applying the Langmuir slip/jump model

temperature

0.005

0.01

0.015

0.98

0.99

1

1.01temperature

normal heat flux

−0.01

−0.005

0

0.95

0.96

0.97

• : NCCR

O: DSMC

Solid line: NSF

14x 10

−3−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.02

−0.015

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.92

0.93

0.94

yy

8

10

12

1.06

1.08

1.1

1.12

density

tangential heat flux

2

4

6

1

1.02

1.04

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−2

0

y y−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0.94

0.96

0.98

y

Page 31: What Makes Gas Micro Flows So Complicated: Non-classical ...

Exact analytic solutions: Knudsen minimumCapturing all the qualitative features predicted by the DSMC

and providing insights DSMC can not !

0.8

1

1.2

1.3

1.4

1.5

u* (0)

central velocity

0.4

0.6

0.8

0.9

1

1.1

Tw*

velocity

Responsible for Knudsen average temp

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

−2 −1.5 −1 −0.5 0 0.5 10.5

0.6

0.7

Solid line: NSF y

0 5 / 2h

Kn

minimum !average temp.

0.35

0.4

0.45

0.5 / 2

0/ 2

0

2

2

h

hr w w

udym hRT RT dy

ρ

ρ γ γ ρ≡

⎛ ⎞

∫∫

&

Mass flow rateaverage

temperature

0.15

0.2

0.25

0.3

2 2*1/ 2

* *1/ 2

1 4 Kn13(1 1 4 Kn) tan1

1 4 KnKn (0) 216 21 1 4 K

wh w

ave

S Tu S T

ωε ωπ

ωγ

⎛ ⎞−⎜ ⎟ ⎛ ⎞ ⎛ ⎞+⎝ ⎠= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

p

−2 −1.5 −1 −0.5 0 0.5 10

0.05

0.11 1 4 Knω+

Knslip central

velocity

Stress

constraint

Page 32: What Makes Gas Micro Flows So Complicated: Non-classical ...

What happened here: Knudsen minimum

1.01

What happened here: Knudsen minimum

2

1.005 Kn=5.0

temperature

22 ~ w

aver w

Tm hTRTρ γ

⎛ ⎞⎜ ⎟⎝ ⎠

&

1 Kn=0.25

average

temperature

0.995

Kn=0.1

0.985

0.99 Responsible for Knudsen minimum !

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.98

yy

Page 33: What Makes Gas Micro Flows So Complicated: Non-classical ...

Pressure-driven compressible Poiseuille gas flowy

wallTwallT

h Conservation laws

xreservoirHigh pwallT

T, ,

Conservation lawswith

( , ) fn ,xy xx x ydu dTQ kdy dy

η⎛ ⎞

Π = − −⎜ ⎟⎝ ⎠

reservoirLow p

wallT

RTp hηπ

nonlinear & coupledy y⎝ ⎠

**

* ( )dp xdx

wall wall

reservoir

, 2

: , pressure ratio Kn : length-to-height ratioin

out

RTp

p p h

hL

ηπ ε≡ =

Easy (or already-done) parts- the benchmark problem most studied in the past- experimental data available (but, mass flow rate and stream-wise pressure distribution only)p y)Difficult parts- overall flowfields physics remain illusive (for example, concave or convex cross-stream pressure distribution)convex cross stream pressure distribution)- how subtle the effect of channel length-to-height ratio is- exact role of non-isothermal physics (in particular, high Knudsen flows)

Page 34: What Makes Gas Micro Flows So Complicated: Non-classical ...

Exact analytic solutions (on-going work)y ( g g )

Analytic form of sectional pressure distribution andAnalytic form of sectional pressure distribution and effect of length-to-height ratio on the mass flow rate

** *2 *

* *

( ln )( ) 2 1 tan3

m

m

d pp y h yp L dx

⎛ ⎞−= + ⎜ ⎟⎜ ⎟

⎝ ⎠concave (tangent-type)

( )2

3** *1/ 2 *

1/ 23 2 * * *1/ 2

tan( ln ) ( ln )24 = 1L

L

m mSd p d pLRTm S

H p dx S dxη

⎝ ⎠

⎛ ⎞− −= + +⎜ ⎟⎜ ⎟

⎝ ⎠& L

1/ 2

**1/ 2 *

( ln )2 1where 3 2

L L

L

m

m

H p dx S dx

d phSL dx

⎝ ⎠

−≡

N l i l t3 2L dx Non-classical term related to the stress constraint

Page 35: What Makes Gas Micro Flows So Complicated: Non-classical ...

Summary of Part IIIIn contrast with Couette flow, the boundary condition seems to remain secondary in case of the force (orseems to remain secondary in case of the force (or pressure) driven Poiseuille flows.

In the force-driven flow, non-isothermal effect has a definite role in high Knudsen flows. For example, the change of temperature profile (measured by the averagechange of temperature profile (measured by the average temperature) is responsible for the Knudsen minimum.

The concave cross-stream pressure distribution is due to the non-classical stress constraint.

Further experimental study on cross-stream pressure distribution and temperature flowfield is stronglydistribution and temperature flowfield is strongly recommended.

Page 36: What Makes Gas Micro Flows So Complicated: Non-classical ...

Part IV.

Diatomic gas gand

verification and validation

(V & V) ( )

Page 37: What Makes Gas Micro Flows So Complicated: Non-classical ...

Diatomic gas caseSurprisingly, there is even no consensus what the proper master

kinetic equations would be for describing diatomic gases like nitrogen in thermal non-equilibriumnitrogen in thermal non equilibrium.A proposed master equation: Boltzmann-Curtiss equation with a

moment of inertia I and an angular momentum j: excess normal stressΔ

[ ]( , , , , )j f t C ft I

ψψ

⎡ ⎤∂ ∂+ ⋅∇ + ⋅∇ + =⎢ ⎥∂ ∂⎣ ⎦

vv a v r j

[ ] [ ](2) (2)ll 2 2( ) ( )∂∇ ∇ Δ ∇ Δ

Π ΠΠ Π Π Q

: excess normal stressΔ

1.1

1.15

NCCR (monatomic; fb=0)

[ ] [ ]( ) ( )small 2 2( ) ( , , , )/

p qt pη+ ⋅∇ = − ⋅∇ − + Δ ∇ − Δ

∂u Π Π u u Π Q L

monatomic

0.95

1

1.05

* 2 * *7 1 tanh2 wh wp T sε

⎛ ⎞= − ⎜ ⎟⎜ ⎟

⎝ ⎠

0.85

0.9

0.95

NCCR (diatomic; fb=1)

NSF & NCCR (diatomic; fb=2/9)

⎝ ⎠

Convex hyperbolic tangent profile! (PoF 2011)

Pressure profile across channel−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0.75

0.8 diatomic

Page 38: What Makes Gas Micro Flows So Complicated: Non-classical ...

Why V & V in gas micro flows are difficult?difficult?

Too many computational models (governing equation, boundarycondition)Cf. Only one in near-equilibrium: Linear uncoupled NSF + no-slip

DSMC is not immune since it is also highly subject to theboundary condition and post-processing employed.

Lack of experimental data (how to measure exotic propertiessuch as temperature jump?)

Lack of theories:- no consensus what the proper master kinetic equations would

be for describing diatomic gases like simple nitrogen inbe for describing diatomic gases like simple nitrogen inthermal non-equilibrium

- no rigorous gas-surface molecular interaction theory

Page 39: What Makes Gas Micro Flows So Complicated: Non-classical ...

Microscopic sampling vs macroscopicp p g pDSMC solutions of the two-dimensional lid-driven cavity gas

flow are considered (Kn=0.1).The exact values of slip velocity and temperature jump in the

DSMC method are prone to how these properties are obtained from the simulation resultsobtained from the simulation results.

Page 40: What Makes Gas Micro Flows So Complicated: Non-classical ...

Microscopic sampling vs macroscopic(courtesy of Dr Roohi)(courtesy of Dr. Roohi)

Direct microscopic sampling of the molecular properties of particles that strike the wall surfacevsmacroscopic approach that ppaccounts for all molecules in the adjacent cellj

Page 41: What Makes Gas Micro Flows So Complicated: Non-classical ...

Assessment of internal errors in DSMCKey observation: The conservation laws must be satisfied

irrespective of computational models. The relative internal error of numerical solutions for one-

dimensional gas flow can be checked (Comp. Fluids 2011).

0-momentum

( ) ( )error

(0)

ρ

ρ

Π −≡ ∫

y

xyx

y

y y ady

ah

0energy

( ) ( ) ( ) ( ) ( )error

(0) (0)

ρ

ρ

Π + −≡ ∫

y

xy yy u y Q y y u y ady

ahu

The relative error of the DSMC increases from the center to the solid wall and reaches non-solid wall and reaches nonnegligible value near the solid wall (Kn=0.1).Force-driven compressible Poiseuille gas flow

wallwall

Page 42: What Makes Gas Micro Flows So Complicated: Non-classical ...

Assessment of internal errors: Another lexamples

The relative errors of the DSMC is not

DSMCDSMC is not negligible.

In case of a PDE-type higher-order model, large errors are found i th t llin the center as well as near the wall.DSMC: traceless error

PDE-type constitutive

llll

PDE type constitutive equation model

wallwall

Page 43: What Makes Gas Micro Flows So Complicated: Non-classical ...

Concluding remarks: What makes gas micro flowsgas micro flows

so complicated—and difficult?

Complicated physics coming from non-classical physics(nonlinearity and couplings in G E and B C )(nonlinearity and couplings in G. E. and B. C.).

Lack of theories in some problems; in particular, rotational non-equilibrium and gas surface molecular interactionequilibrium and gas-surface molecular interaction.

Presence of exotic properties such as tangential heat flux( t l diffi lt t i t ll )(extremely difficult to measure experimentally).

Need of theoretical and experimental investigation on the wholeflowfields, including diatomic gases, beyond a reduced quantitysuch as the mass flow rate. Showing a snap-shot agreement onething, but describing many features simultaneously quite another.g g y y q