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![Page 1: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/1.jpg)
Polymer Physics
Hans Christian Öttinger
Department of Mat..., ETH Zürich, Switzerland
Hydrodynamics, Thermodynamics, andMathematics
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Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations
2. example: hydrodynamics
3. solving vs. coarse graining
4. numerical solutions
5. boundary conditions
![Page 3: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/3.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics
3. solving vs. coarse graining
4. numerical solutions
5. boundary conditions
![Page 4: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/4.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining
4. numerical solutions
5. boundary conditions
![Page 5: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/5.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions
5. boundary conditions
![Page 6: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/6.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions
![Page 7: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/7.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 8: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/8.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 9: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/9.jpg)
![Page 10: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/10.jpg)
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Polymer Physics
GENERIC Structure
General equation for the nonequilibriumreversible-irreversible coupling
metriplectic structure (P. J. Morrison, 1986)
dxdt------ L x( ) δE x( )
δx--------------⋅ M x( ) δS x( )
δx--------------⋅+=
L x( ) δS x( )δx
--------------⋅ 0= M x( ) δE x( )δx
--------------⋅ 0=
L antisymmetric, M Onsager/Casimir symm., positive-semidefinite Jacobi identity
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Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 17: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/17.jpg)
Polymer Physics
Nonequilibrium Thermodynamics
ρ r( ) M r( ) ε r( ) structural variables, , ,( )mass density
momentum density
internal energy density
balance equations structured time-evolutionequations (GENERIC)
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Polymer Physics
Hydrodynamics
E 12--- M r( )2
ρ r( )--------------- ε r( )+ r3d∫= S s ρ r( ) ε r( ),( ) r3d∫=
L is associated with the Lie group of space transformations: unique construc-tion by following arguments:
• are scalar densities
• is a (contravariant) vector density
ρ s,M
• separation of mechanical (C) and ther-modynamic (D) effects
• C: well-known for relaxational and transport processes and given E
• D: viscosity, thermal conductivity, and diffusion coefficient
M C D CT⋅ ⋅=
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Polymer Physics
Diffusion in Hydrodynamics
v Mρ-----=
v reference velocity (constant)
CT ∂∂r----- v ∂
∂r-----⎝ ⎠
⎛ ⎞ T ∂∂r-----1
2---v2 v ∂
∂r-----v⎝ ⎠⎛ ⎞ T⋅ 1
2---v2 α+⎝ ⎠⎛ ⎞ ∂
∂r-----+–=
constantα
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Polymer Physics
Diffusion in Hydrodynamics
δEδx------
12---v2–
v1
=v M
ρ-----=
v reference velocity (constant)
CT ∂∂r----- v ∂
∂r-----⎝ ⎠
⎛ ⎞ T ∂∂r-----1
2---v2 v ∂
∂r-----v⎝ ⎠⎛ ⎞ T⋅ 1
2---v2 α+⎝ ⎠⎛ ⎞ ∂
∂r-----+–=
constantα
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Polymer Physics
Hydrodynamic Equations
∂ρ∂t------ ∂
∂r----- M jρ–( )⋅–=
∂M∂t
-------- ∂∂r----- 1
ρ---MM jρv– p1 τ+ +⎝ ⎠⎛ ⎞⋅–=
∂s∂t----- ∂
∂r----- 1
ρ---Ms js+⎝ ⎠⎛ ⎞⋅– dissipation+=
j D'ρ----- ∂
∂r----- µ
T--- α ∂
∂r----- 1
T---–⎝ ⎠
⎛ ⎞=
∂s∂ρ------ µ
T---–= ∂s
∂ε----- 1
T---=
µ µ 12--- v v–( )2–=
js 1T---j
qµ α–( )ρ
T---j+=
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Polymer Physics
Properties of Hydrodynamic Equations
• Momentum density and mass flux are two different quantities.
• Both momentum density and mass flux are conserved quantities.
• The conservation law for the mass flux is equivalent to the conserva-tion law for the booster (expressing center-of-mass motion).
• Angular momentum is a conserved quantity.
• A rigidly rotating fluid is a solution of the hydrodynamic equations.
• Galilean invariance is satisfied.
• A new type of inertial forces occurs in the driving force for diffusion.
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Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 24: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/24.jpg)
Polymer Physics
GENERIC Structure
General equation for the nonequilibriumreversible-irreversible coupling
metriplectic structure (P. J. Morrison, 1986)
dxdt------ L x( ) δE x( )
δx--------------⋅ M x( ) δS x( )
δx--------------⋅+=
L x( ) δS x( )δx
--------------⋅ 0= M x( ) δE x( )δx
--------------⋅ 0=
L antisymmetric, M Onsager/Casimir symm., positive-semidefinite Jacobi identity
![Page 25: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/25.jpg)
Polymer Physics
Statistical Mechanics: Entropy
relative entropy
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Polymer Physics
Statistical Mechanics: Friction Matrix
D 12∆t--------- ∆x( )2⟨ ⟩=cf.
Green-Kubo
EinsteinMjk1
2kBτ------------ ∆τΠj ∆τΠk⟨ ⟩x=
Mjk1kB----- Πk
f· eiL0Qt
Πjf·
⟨ ⟩x td
0
τ
∫= Πj⟨ ⟩x xj=
τ1 τ2
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Polymer Physics
HilbertGradChapman-Enskog
Newton’s (Hamilton’s) equations of motionor Liouville equation
Boltzmann’s equation
Hydrodynamics
birth of irreversibility!
more irreversibility?
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Polymer Physics
From Grad’s 10-Moment Equations to Navier-Stokes
∂ρ∂t------ ∂
∂r----- vρ( )⋅–=
∂v∂t----- v ∂
∂r-----v⋅– 1
ρ--- ∂∂r----- π⋅–=
∂T∂t------ v ∂
∂r-----T⋅– 2m
3ρk---------π:κ–=
κ ∂∂r-----v⎝ ⎠⎛ ⎞ T
=
π p1 σ+=
∂σ∂t------ ∂
∂r----- vσ( )⋅– κ π⋅ π κT⋅–– 2
3---1 π:κ( ) 1
τ---σ–+=
σ η κ κT 23--- 1:κ( )1–+–= η pτ=
Singular perturbation theory:
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Polymer Physics
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Polymer Physics
Functional Integrals
For a systematic calculation of thecoarse grained hydrodynamic entropy:
Ω ρ T,( ) ρ2m------- det 1 σ
p---+⎝ ⎠
⎛ ⎞ d3rln∫⎩ ⎭⎨ ⎬⎧ ⎫
σexp∫=
Functional integrals occur most naturallyin systematic coarse graining!
![Page 31: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/31.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 32: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/32.jpg)
Polymer Physics
Symplectic Integrators
dxdt------ L
δE∆tδx
-----------⋅=
reproduces values of integration scheme at times n∆t
J. Moser (1968):
Alternatives:• canonical transformations in each time step
• Lagrangian formulation (variational principle instead of symplectic structure)
![Page 33: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/33.jpg)
Polymer Physics
GENERIC Integrators
dxdt------ L
δE∆tδx
-----------⋅ M∆tδSδx------⋅+=
L δSδx------⋅ 0= M∆t
δE∆tδx
-----------⋅ 0=
L antisymmetric, M∆t Onsager/Casimir symm., positive-semidefinite Jacobi identity
![Page 34: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/34.jpg)
Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
Menu
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
![Page 35: Hydrodynamics, Thermodynamics, and Mathematics · Polymer Physics Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications](https://reader030.fdocuments.us/reader030/viewer/2022040206/5d51029088c9934a328bc046/html5/thumbnails/35.jpg)
Polymer Physics
Boundary Thermodynamics
3d variables
2d variables
time evolution
time evolution
sourcesboundaryconditions
distance from interface
conc
entra
tion
excess at interface
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Polymer Physics
GENERIC Structure
General equation for the nonequilibriumreversible-irreversible coupling
dxdt------ L x( ) δE x( )
δx--------------⋅ M x( ) δS x( )
δx--------------⋅+=
dAdt------- δA
δx------ dx
dt------⋅ δA
δx------ L δE
δx------⋅ ⋅ δA
δx------ M δS
δx------⋅ ⋅+= =
A E, A S,[ ]+=Poisson & dissipative brackets
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Polymer Physics
Example: Diffusion Cell
System: Solute particle number densities• P in the bulk• p at the wall
δSδP------ kB
PP0------ ,ln–= δS
δp------ kB
pp0-----ln–=
kB A B,[ ] Db∂∂r-----δAδP------ ∂
∂r-----δBδP------⋅ P 3rd
V∫ Ds∂∂r-----δA
δp------ ∂
∂r-----δB
δp------⋅ p 2rd
W∫+=
νsδAδp------ ΩδA
δP------– δB
δp------ ΩδB
δP------– p 2rd
W∫+ Ω 1=νs: ad/desorption rate
Brenner & Ganesan, Phys. Rev. E 61 (2000) 6879hco, Phys. Rev. E 73 (2006) 036126
wall
wall
reservoir(P1)
reservoir(P2)V
W
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Polymer Physics
Diffusion Cell: Results
A S,[ ] δAδP------ dP
dt-------
irr
3rd⋅V∫
δAδp------ dp
dt------
irr
2rd⋅W∫ Jirr
A 2rd∂V∫+ +=
dPdt------- ∂
∂r----- Db
∂P∂r------⋅=
dpdt------ ∂
∂r----- Ds
∂p∂r------⋅ n Db
∂P∂r------⋅–=
n– Db∂P∂r------⋅ νsp HP
p--------ln=
boundary condition on wall:evolution equations:
H p0 P0⁄=characteristic length scale
open boundaries: wall:JirrA δA
δP------n Db
∂P∂r------⋅–= Jirr
A 0=
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Polymer Physics
Example: Wall Slip
tethered subchains(semiloops and tails)
System
bulk variables
boundary variables
ρ M s, , Ψ Q( )
tangential on wall segment cdf
c s, ψ R( )
entropy density subchain cdf
temporary network model
R
Ω
∂Ω1
∂Ω3
∂Ω2n
n
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Polymer Physics
Wall Slip: Building Blocks
Energy: internal energiesEntropy: trivially availabledepend on
s ρ snet Ψ( ) 3Qd∫–
s steth c ψ,( ) 3Rd∫–
Poisson bracket: from a naturalaction of space transformationsthat leave the wall invariant
Dissipative bracket:A B,[ ] heat conduction (bulk, wall) polymer relaxation (bulk, wall)+=
TTRK------- 1
T---δAδs------ 1
T---δAδs------– 1
T---δBδs------ 1
T---δBδs------– 2rd
∂Ω1
∫+ Kapitza resistance
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Polymer Physics
Wall Slip: Results
n τ p1+( )⋅ T ψ ∂∂R-------
∂steth∂ψ
------------- 3Rd∫– s ∂T∂r ||
----------+=
∂c∂t-----
ceq
τsav,eq
------------ cτs
av-------– r
ceq c–
τsav
----------------+=
strength of binding
requires ψ (or ansatz!) 1τs
av------- 1
τs R( )-------------ψ R( )
3Rd∫=
τshearwall vslipckBT τs
∂ ψln∂R ||
-------------⎝ ⎠⎛ ⎞ 2
ψ 3Rd∫=
τshearwall ceqkBT ψeq
∂∂R ||
------------ ψψeq--------⎝ ⎠⎛ ⎞ 3Rdln∫=
Giesekus-type
τshearwall vslip
ceqkBTτsav
Req2
-------------------------=
ψψeq--------⎝ ⎠⎛ ⎞ln R R⟨ ⟩⋅
Req2
------------------=R⟨ ⟩ vslipτs
av=
Wall-slip law:
⇒
Yarin & Graham, JoR (1998)
τshearwall
vslip
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Polymer Physics
Next Steps
• Free boundaries
• Moving interfaces
• Viscoelastic interfaces
• More general relations between bulk and boundary variables
• Variables characterizing the geometry of interfaces
• Functional calculus
• Statistical mechanics
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Polymer Physics
Thermodynamic admissibility and mathematical well-posedness
1. structure of equations & geometry, implications for solutions
2. example: hydrodynamics & diffusive mass flux
3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom
4. numerical solutions & discrete version of structure
5. boundary conditions & boundary thermodynamics
Thanks to Miroslav Grmela, Howard Brenner, Henning Struchtrup, Mario Liu
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Polymer Physics
GENERIC with Fluctuations
dxdt------ L x( ) δE x( )
δx--------------⋅ M x( ) δS x( )
δx--------------⋅ B x( )
dWtdt
---------⋅ kBδδx----- M x( )⋅+ + +=
B x( ) B x( )T⋅ 2kBM x( )=
fluctuation-dissipation theorem
dWtdt
---------⟨ ⟩ 0=dWtdt
---------dWt'dt'
----------⟨ ⟩ δ t t'–( )1= Wienerprocess
Itô
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Polymer Physics
GENERIC with Fluctuations
dxdt------ L x( ) δE x( )
δx--------------⋅ M x( ) δS x( )
δx--------------⋅ B x( )
dWtdt
---------⋅ kBδδx----- M x( )⋅+ + +=
B x( ) B x( )T⋅ 2kBM x( )=
fluctuation-dissipation theorem
dWtdt
---------⟨ ⟩ 0=dWtdt
---------dWt'dt'
----------⟨ ⟩ δ t t'–( )1= Wienerprocess
Itô
Some important issues:• Partial stochastic differential equations• Fluctuation renormalization (long-time tails, mode-coupling)