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arXiv:cond-mat/0405555v3 [cond-mat.stat-mech] 2 Feb 2005

Thermodynamic theory of incompressible hydrodynamics∗

Santosh Ansumali

ETH-Z¨ urich, Institute of Energy Technology, CH-8092 Z¨ urich, Switzerland†

Iliya V. Karlin

ETH-Z¨ urich, Institute of Energy Technology, CH-8092 Z¨ urich, Switzerland

‡

Hans Christian¨Ottinger

ETH-Z¨ urich, Department of Materials, Institute of Polymers, CH-8093 Z¨ urich, Switzerland§

(Dated: February 2, 2008)

The grand potential for open systems describes thermodynamics of fluid flows at low Mach num-

bers. A new system of reduced equations for the grand potential and the fluid momentum is derived

from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are

the quasi-stationary solution to the new system. It is argued that the grand canonical ensemble is

the unifying concept for the derivation of models and numerical methods for incompressible fluids,

illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.

PACS numbers: 05.20.Dd, 47.11.+j

The classical incompressible Navier-Stokes equation

(INS) is mechanical description of fluid flows at low Mach

numbers (Mach number, Ma = U0/cs, is the ratio of the

characteristic flow velocity U0 to the isentropic sound

speed cs defined at some reference temperature T and

density ρ). The INS equation can be written in the Eu-

lerian coordinate system as,

∂tuα+ uβ∂βuα+ ∂αP =

1

Re∂β∂βuα,∂αuα= 0, (1)

where u is the fluid velocity, P is the pressure and Re

the Reynolds number, which characterizes the relative

strength of the viscous and the inertial forces [1]. The

pressure in (1) is not an independent thermodynamic

variable but is rather determined by the condition of the

incompressibility,

∂β∂βP = −(∂βuα)(∂αuβ).(2)

Thus, in order to obtain the pressure at a point, one has

to solve the Laplace equation (2) in a domain, and the re-

lationship between the pressure and the velocity becomes

highly nonlocal. The physical meaning of (2) is that in a

system with infinitely fast sound propagation, any pres-

sure (and thus density) disturbance induced by the flow

is instantaneously propagated into the whole domain.

A textbook justification for the thermodynamics of the

INS description is usually based on the isentropic flow as-

sumption. If this assumption is valid for all times (the

entropy density is simply convected by the flow), then the

thermodynamic pressure depends only on the acoustic

(density) variations. For low Mach number flows, these

variations adjust to the flow on every spatial scale in

∗Physical Review Letters, accepted for publication (2005).

the long time dynamics [1, 2]. The way this adjustment

takes place is nontrivial, and it was given considerable

attention recently [3, 4, 5]. In particular, it was proved

that weak solutions of the isentropic compressible Navier-

Stokes equations converge to that of the INS equation for

some special boundary conditions (adiabatic absorbing

walls) [5]. However, it remains a challenge to give a ther-

modynamic derivation of the incompressibility without

the isentropic flow assumption. Such a thermodynamic

derivation goes far beyond academic interest. Indeed, as

is well known, the INS equations are extremely hard to

study, both analytically and numerically. Therefore, an

extended system where the flow is coupled to a dynamic

equation for a scalar thermodynamic variable provides a

better starting point for numerical and theoretical stud-

ies of the incompressible hydrodynamics. Indeed, in the

computational fluid dynamics, at least two undeniably

successful routes to avoid the “elliptic solver problem”,

that is, avoiding the nonlocality of the pressure (2) are

well known. The first is the so-called artificial compress-

ibility method introduced by Chorin [6] and Temam [7],

where an evolution equation for the pressure is postu-

lated instead of the constraint (2) (see e. g. [8] for a

recent review). The second route is kinetic-theory mod-

els such as the lattice Boltzmann method [9], where a

Boltzmann-like equation is obtained for the compressible

fluid flow in the low Mach number limit. The thermo-

dynamics of the lattice Boltzmann method was clarified

[10], and the method enjoys a thermodynamically sound

derivation from the Boltzmann equation [11]. Can the

compressibility methods be modified in a way to make

them physical models? Furthermore, for emerging fields

of fluid dynamics such as flows at a micrometer scale,

corrections to the incompressibility assumption become

crucial [12].

In this work, we present the thermodynamic descrip-

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tion of incompressible fluid flows. In step one, we will

argue that the grand potential is the proper thermo-

dynamic potential to study the onset of incompressibil-

ity. Use of the grand potential instead of the entropy

enormously simplifies the equations of compressible hy-

drodynamics in the low Mach number limit. We show

that after a short time dynamics during which acous-

tic waves are damped by viscosity, the fast dynamics of

the grand potential becomes singularly coupled to the

slow dynamics of momentum, and reduced compressible

Navier-Stokes equations are derived (RCNS). The incom-

pressible Navier-Stokes equation is the quasi-stationary

solution of the RCNS, when Mach number tends to

zero. RCNS equations are a thermodynamically consis-

tent generalization of the compressibility schemes. Fi-

nally, by writing the grand potential for the Boltzmann

equation, we show that, upon an appropriate discretiza-

tion of velocities, the present constructions leads to the

entropic lattice Boltzmann method. Correctness of the

present almost-incompressible description is illustrated

with a simulation of a microflow setup.

A low Mach number flow is a setup where only small

spatial deviations of the entropy and the density from

the equilibrium value exist. The grand potential is the

natural thermodynamic variable to describe such a setup

(and the corresponding to it grand canonical ensemble is

the natural statistical thermodynamics framework). This

happens because the balance laws (compressible Navier-

Stokes equations or the Boltzmann equation) are always

written for a sufficiently small volume element in the

Eulerian frame of reference (i.e. the volume element is

fixed in space). From a thermodynamic standpoint, this

volume element is an open system.

thermodynamics we know that in an open system, the

thermodynamic equilibrium is conveniently described in

terms of the grand potential Ω(ψ,T), where ψ is chemical

potential, and T is the temperature.

From elementary

Now, we shall find the expression for the grand po-

tential in the Eulerian coordinate system for a vol-

ume element δV , in thermodynamic equilibrium.

the co-moving system, the grand potential is written as

ΩL(ψ,T) = −P(ψ,T)δV , where P is the pressure. The

transition to the Eulerian (fixed) system is done by fixing

the momentum, ΩE(ψ,T,m) = −PδV +λαmαδV , where

λα are Lagrange multipliers, and m is the momentum

density. For small values of momentum, the Lagrange

multipliers can be specified by noting that the energy in

the Eulerian coordinate system is [ǫ+(m2/2ρ)]δV , where

ǫ is the internal energy density. Using the relationship be-

tween the energy and the grand potential for the thermo-

dynamic equilibrium, we find that λα= mα/2ρ+O(m3).

Thus, in the Eulerian coordinate system, the grand po-

tential, up to the higher-order terms in momentum, is

written as,

In

ΩE=

?

−P +m2

2ρ

?

δV.(3)

The difference of the pressure and the kinetic energy is

the (negative of) density of the grand potential, and it

will be used below as the natural thermodynamic poten-

tial for the low Mach number flows:

G = P −m2

2ρ.

(4)

Dynamic equations for the set of variables ρ, m and

G are written using the standard compressible Navier-

Stokes equations (CNS) for Newtonian fluids [1, 2]. Note

that CNS are usually written in terms of a different set

of variables (for example, in terms of the entropy density

S instead of G). The recomputation from either form of

the CNS equations to the present set of variables poses

no difficulties, and we here write the final result:

∂tρ + ∂αmα= 0,

∂tmα+ ∂β

??

G +m2

?

2ρ

?

?

δαβ+mαmβ

ρ

+ Παβ

?

?

?

= 0,

∂tG + ρ∂P

∂ρ

????

S

∂α

?mα

ρ

− ∂α

mα

?m2

?????

2ρ2

?

+mβ

ρΠαβ

+

1 +

1

ρCV

∂P

∂T

ρ

Παβ∂α

?mβ

ρ

=

1

ρCV

∂P

∂T

?????

ρ

∂α(κ∂αT),

(5)

where Παβis the stress tensor of a Newtonian fluid,

Πα β= −µ

?

∂α

?mβ

ρ

?

+ ∂β

?mα

ρ

?

− δαβ

?2

D− λ

?

∂γ

?mγ

ρ

??

,

with D the spatial dimension, µ the shear viscosity, and

λ the ratio of bulk to shear viscosity. In (5), κ is the

thermal conductivity, CVis the specific heat at constant

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volume, and the temperature T is known from the equa-

tion of state. Variations of material parameters such as

viscosity µ will be ignored in the further discussion. Note

that equations (5) are “exact” in the sense that they are

just the standard CNS equations written for ρ, m, and

the function G (4). However, the physical meaning of the

function G (4) as the density of the grand potential is

valid only up to the lowest order in momentum, and thus

the form of the CNS (5) is an important intermediate step

in the study of the low Mach number hydrodynamics.

Since the speed of sound, cs=?∂P/∂ρ|S, contributes

instructive to rewrite (5) as:

to the dynamic equation for the grand potential, it is

∂tG + ∂α(c2

smα) −mα

ρ

∂α(ρc2

s) − ... (6)

We expect the following scenario of the onset of incom-

pressibility, as it can be inferred from (6): If the speed of

sound is “large”, then, after a “short-time” dynamics of

the density leading to ρ ≈ const, the third term in (6) can

be neglected, whereas the second term becomes the dom-

inant contribution to the time derivative of G. The dy-

namic equation for the grand potential becomes then sin-

gularly perturbed, and represents the “fast” mode cou-

pled to the “slow” dynamics of momentum. The contri-

butions to the time derivative of the grand potential not

displayed in (6) are responsible for corrections to the in-

compressibility. We now proceed with quantifying these

statements.

For small deviations from the no-flow situation,

(|mαmβ| ≪ P ρ, which implies Ma ≪ 1), we are in-

terested in the long time solutions of the CNS equations

(times of the order of the momentum diffusivity time

tmd∼ ρL2/µ, where L is a characteristic length associ-

ated with the flow). We define a dimensionless number,

Kn = µ/(ρcsL), the ratio of the sound propagation time

L/csand the momentum diffusivity time, as the Knud-

sen number for a general fluid, and we are considering

Kn ≪ Ma ≪ 1. For the sake of simplicity, we assume

that the Prandtl number Pr ∼ 1 in the subsequent anal-

ysis. The short time dynamics is isentropic and linear,

and it is well known that, away from boundaries, any

density perturbation at a distance r away from the dis-

turbance source decays as (see [1], p. 300):

δρ(r,t) ∝ (LarL)−1/2exp

?

−(r − cst)2

2LarL

?

. (7)

Here a new dimensionless number La is defined as:

La = Kn

?

2 −2

D+ λ

?

+Kn(γ − 1)

Pr

, (8)

where γ the ratio of specific heat at constant pressure and

volume. La generalizes the notion of Knudsen number

for an arbitrary fluid (we call it the Landau number in

the honor of Landau, who explained the relevance of this

number in the context of acoustic damping [1], p. 300).

Thus, the short term hydrodynamics reveals the following

length scale La and the time scale ta (since we assume

Pr ∼ 1, we need not distinguish between La and Kn for

the present purpose):

ta∼

√Kn

?L

cs

?

, La∼

√KnL,(9)

At the time scale larger than ta, and on the spatial scale

larger than La, the density of the fluid can be safely

treated as a constant (in the usual isentropic theory, the

characteristic time for the onset of incompressibility is of

the order L/cs≫ ta). Note that the length scale Lawas

also found in the derivation of the sub-grid model from

kinetic theory [13]. On the time-space scale larger than

(9), we can neglect the density variation, and the tem-

perature variation δT (from the globally uniform value

T) becomes a function of the grand canonical potential,

δT ≈∂T

∂P

????

ρ

?

G +m2

2ρ

?

.

Once the time and space scales (9) are identified, we

complete the reduction of the CNS equations (5) by

merely rescaling the variables. The momentum is scaled

by the characteristic momentum ρU0 (known from the

initial or boundary condition), j = m/(ρU0), and we

introduce the reduced grand canonical density, Θ =

G/(ρU2

La, (t → t/ta, x → x/La), neglecting variations of the

density, and taking into account the thermodynamic re-

lation for the temperature mentioned above, the two last

equations in (5) reduce to the following scale-independent

closed set of equations for the dimensionless grand poten-

tial and momentum:

0). Making time dimensionless with ta, space with

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∂tjα= −Ma∂β

?

jαjβ+ δαβ

?

Θ +j2

2

??

+

√Kn

?

1 + λ −2

D

?

∂α∂βjβ+

√Kn∂β∂βjα,

∂tΘ = −

1

Ma∂α

?

jα

?

1 −Ma2j2

2

??

+

√

Kn

?

γ

Pr∂α∂αΘ +

?γ

Pr− 1

?

∂α∂αj2

2+ (∂αjβ)(∂αjβ)

+∂P

∂T

?????

ρ

1

2CV(∂αjβ+ ∂βjα)(∂αjβ+ ∂βjα) −

?

1 + λ −2

D

?

∂β(jβ∂αjα)

?

.

(10)

Note that all “material parameters” appearing in (10)

(CV, γ, λ, κ) are evaluated at equilibrium at ρ and T.

The reduced set of compressible Navier-Stokes equa-

tions (10) is valid for Kn ≪ Ma ≪ 1, and, as we ex-

plained above, on the scales larger than acoustic scales

(9). Roughly speaking, (10) is what survives from the

compressible Navier-Stokes equations just before the in-

compressibility sets on. Indeed, the time derivative of Θ

becomes singularly perturbed as the Mach number tends

to zero, and we recover the incompressibility condition,

∂αjα= 0, as the quasi-stationary solution of the system

(10). This solution, when substituted in the momentum

equation, recovers the INS equation (1) with the usual

accuracy of the order O(Ma2). Note that the velocity in

the INS equations recovered from (10) is u = j/ρ. Cor-

rections to the quasi-stationary solution can be found in a

systematic way (see, e. g. [14, 15]), and we do not address

this here. The following point needs to be stressed: The

dissipation terms (proportional to

glected in the equation for the grand potential (10) and

simultaneously kept in the momentum equation. This

is at variance with the artificial compressibility method

[6, 7]. In other words, the RCNS is the minimal thermo-

dynamic system for incompressible hydrodynamics.

In this Letter, we reported a new basic physical fact of

fluid dynamics: Grand potential (3) for low Mach number

flows gives the thermodynamic description of the incom-

pressible phenomena. The resulting system (10) includes

a local (non-advected) equation for the scalar thermody-

namic field. What follows from this fact? Let us list

some of the consequences:

√Kn) cannot be ne-

• With the corrections mentioned above, the numer-

ical schemes of the artificial compressibility kind

become a firm status of physical models.

• The structure of the coupling between the flow and

the thermodynamic variable hints at that the true

incompressible flows are attractors of system (10),

with the INS as the leading-order approximation.

Because of the singular perturbation nature, it may

be easier to study attractors of (10), rather than of

the INS equations.

• System (10) can be a starting point for a systematic

derivation of nonlinear models for heat transport

in the nearly-incompressible fluids such as multi-

phase fluids, polymeric liquids and melts etc (this

is relevant even in the linear case, see, e. g. [16]).

• In the celebrated Kolmogorov theory, equation for

the kinetic energy is used to make predictions about

the structure of the fully developed turbulence.

The present thermodynamic approach unambigu-

ously delivers the density of the grand potential as

the scalar field associated with the incompressible

fluid flow, and thus can be relevant to develop the-

ories of turbulence through studying the resulting

balance equation.

Let us dwell on the use of (10) for numerical simu-

lation of incompressible flows. Recall that the spectral

methods [17] for the INS (1) based on the Fast Fourier

Transform (FFT) for solving the Laplace equation (2)

are very efficient for high Reynolds number flows in sim-

ple geometries. On the other hand, thanks to a rela-

tively simple structure of the system (10) (the equation

for the scalar variable contains no convected derivative),

it can be addressed by a host of discretization methods

(see e. g. [8]). The system (10) can be useful for simula-

tion of flows in complex geometries and especially non-

stationary problems, that is, in the situations where one

seeks to avoid solving the Laplace equation by restoring

to some relaxation schemes (artificial compressibility, lat-

tice Boltzmann etc). Special attention should be payed to

the fact that the system (10) contains terms of different

order of magnitude. This situation is typical for all relax-

ational schemes such as, for example, the lattice Boltz-

mann method, and we do not discuss here how to deal

with this issue numerically (see, e. g. [9, 10, 11]). Note,

however, the important smoothing effect of the diffusion

term (inversely proportional to the Prandtl number) in

the equation for Θ which makes the physical system (10)

more amenable to numerics, that is, less numerically stiff

than artificial compressibility methods. These numerical

issues remain out of the scope of the present Letter, and

will be addressed in a separate publication.

While the new system of reduced compressible Navier-

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−0.50

Y

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

ux

Simulation

Analytical

FIG. 1: Velocity profile of the 2D body force driven Poiseuille

flow at Kn = 0.035 and Ma = 0.01.

Stokes equations (10) is our central result, we con-

clude this Letter with a more general statement that

the grand potential (and the relevant grand canoni-

cal ensemble) can be implemented for eventually any

more microscopic setup (and not only for compressible

Navier-Stokes equations as above). This viewpoint pro-

vides a unified setting for derivations of a variety of

mesoscopic or molecular dynamics models for numerical

simulation of incompressible and nearly-incompressible

flows. As an illustration here, let us assume Boltzmann’s

description with the one-particle distribution function

f(v,x,t). Starting from the general form of grand po-

tential, G(f,α,β,λ) =?f?lnf + α + λαvα+ βv2?dv, it

fined from δG = 0), we have G(feq) = Geq(α,β) +

(λ2/2)?feq(v,α,β,0)(v2/2)dv for small λ. When the

Hermit quadrature with the weight exp(βv2) at fixed β,

one obtains the entropy function of the lattice Boltzmann

method [10, 11]. The present alternative derivation based

on the grand potential is new. Thus, the entropic lattice

Boltzmann method is a valid physical model for nearly-

incompressible flows, and can be used for finite but small

Knudsen low Mach number flow problems often encoun-

tered in the microflows [12]. In Fig. 1 we present ex-

cellent comparison between the analytical solution to the

Bhatnagar-Gross-Krook(BGK) kinetic equation [18] and

the entropic lattice Boltzmann simulation for the body-

force driven 2D Poiseuille microflow. A detailed study of

microflows within the entropic lattice Boltzmann setting

is presented elsewhere [19].

While this paper was in the revision process, we

learned about a very recent paper [20] which states the

usefulness of the grand potential in the context of plasma

turbulence simulations. Although [20] does not deal with

is easy to show that at equilibrium feq(v,α,β,λ) (de-

velocity integral G(f,α,β,λ) is evaluated with the Gauss-

the incompressibility per se, it indicates that grand po-

tential may be a relevant thermodynamic variable in flow-

ing systems way beyond “ordinary” fluids.

Acknowledgments. Discussions with Alexander Gor-

ban were important starting point of this work. Use-

ful comments of C. Frouzakis, M. Grmela, V. Kumaran,

and A. Tomboulides are kindly acknowledged. IK and

SA were supported by the Swiss Federal Department of

Energy (BFE) under the project Nr. 100862 “Lattice

Boltzmann simulations for chemically reactive systems

in a micrometer domain”.

†Electronic address: ansumali@lav.mavt.ethz.ch

‡Electronic address: karlin@lav.mavt.ethz.ch; Also at In-

stitute of Computational Modeling RAS, Krasnoyarsk

660036, Russia

§Electronic address: hco@mat.ethz.ch

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