thermodynamic change is adiabatic, i.e., that the change does not involve

simultaneous heat exchange with a thermal bath. In real physical systems

contact with thermal baths can only be realized at boundaries and is thus

incorporated in boundary conditions. There is one exception: in an environ-

ment with given temperature the system is in interaction with a radiation

¬eld with a black-body distribution typical for that temperature and ab-

sorbs and emits radiation, ¬nally leading to thermal equilibration with the

radiation ¬eld. We may, however, for most practical purposes safely as-

sume that the rate of equilibration with the radiation ¬eld is negligibly slow

compared to thermal conduction within the system and over its boundaries.

The adiabaticity assumption is therefore valid in most practical cases. In

simulations, where unphysical heat baths may be invoked, the adiabaticity

assumption may be arti¬cially violated.

For small changes, the adiabatic relation between pressure and density

change is given by the adiabatic compressibility κs :

1 dρ

κs = , (9.19)

ρ dp S

or

dp 1

= . (9.20)

dρ κs ρ

S

A special case is an ideal gas for which pV cp /cV remains constant under an

adiabatic change. This implies that

cp p

dp

= . (9.21)

dρ cV ρ

S

For dense liquids the compressibility is so small that for many applications

the ¬‚uid can be considered as incompressible, and ρ taken as constant in a

9.2 The macroscopic equations of ¬‚uid dynamics 287

coordinate system that moves with the ¬‚uid. This means that the divergence

of the ¬‚uid velocity vanishes (see (9.5)) and the Navier“Stokes equation

(9.16) simpli¬es to (9.17).

9.2.6 Heat conduction and the conservation of energy

Note on notation In this section we need thermodynamic quantities per unit

mass of material. We use overlined symbols as notation for quantities per unit

mass in order not to cause confusion with the same thermodynamic quantities

used elsewhere (without overline) per mole of component. These are intensive

thermodynamic properties; the corresponding extensive properties are denoted by

capitals (temperature is an exception, being intensive and denoted by T ). For

internal energy we use u, not to be confused with ¬‚uid velocity u. We de¬ne the

following quantities:

(i) Internal energy per unit mass u. This is the sum of the kinetic energy due

to random (thermal) velocities and the potential energy due to interactions

within the system.3 It does not include the kinetic energy per unit mass 1 u2

2

as a result of the ¬‚uid velocity u. It has SI units J/kg or m2 s’2 .

(ii) Enthalpy per unit mass h = u + p/ρ [J/kg].

(iii) Entropy per unit mass s [J kg’1 K’1 ],

(iv) Thermodynamic potential per unit mass μ = h ’ T s [J/kg].

Due to adiabatic changes, and to dissipation caused by frictional forces,

heat will be locally produced or absorbed, and the

temperature will not be homogeneous throughout the system. Tempera-

ture gradients will cause heat ¬‚ow by conduction, and this heat ¬‚ow must

be incorporated into the total energy conservation. If it is assumed that

the heat ¬‚ux J q (energy per unit of time ¬‚owing through a unit area) is

proportional to minus the temperature gradient, then

J q = ’»∇T, (9.22)

where » is the heat conduction coe¬cient.

The energy per unit mass is given by u + 1 u2 + ¦ext , and hence the energy

2

1 2 + ρ¦ext . Here ¦ext is the potential energy

per unit volume is ρu + 2 ρu

per unit mass in an external ¬eld (such as gz for a constant gravitational

¬eld in the ’z-direction), which causes the external force per unit volume

f ext = ’ρ∇¦(r). Note that the external force per unit volume is not equal

to minus the gradient of ρ¦. The energy of a volume element (per unit

volume) changes with time for several reasons:

(i) Reversible work is done on the volume element (by the force due to

pressure) when the density changes: (p/ρ)(‚ρ/‚t).

3 For a discussion on the locality of energy, see Section 17.7.

288 Coarse graining from particles to ¬‚uid dynamics

(ii) Reversible work is done by external forces; however, this work goes at

the expense of the potential energy that is included in the de¬nition

of the energy per unit volume, so that the energy per unit volume

does not change.

(iii) Energy is transported with the ¬‚uid (kinetic energy due to ¬‚uid velo-

city plus internal energy), when material ¬‚ows into the volume elem-

ent: ’(u + 1 u2 + ¦ext )∇ · (ρu).

2

(iv) Heat is produced by irreversible transformation of kinetic energy into

heat due to friction: ’u · [·∇2 u + (ζ + 1 ·)∇(∇ · u)].

3

(v) Heat ¬‚ows into the volume element due to conduction: ∇ · (»∇T ).

Summing up, this leads to the energy balance equation

‚ p ‚ρ 1

’ u + u2 + ¦ext ∇ · (ρu)

ρu + 1 ρu2 + ρ¦ext =

2

‚t ρ ‚t 2

1

’u · ·∇2 u + ζ + · ∇(∇ · u) + ∇ · (»∇T ). (9.23)

3

This concludes the derivation of the ¬‚uid dynamics equations based on

the assumptions that local density and local ¬‚uid velocity can be de¬ned,

and local thermodynamical equilibrium is de¬ned and attained. In the next

secion we return to a more realistic molecular basis.

9.3 Coarse graining in space

In this section we consider a classical Hamiltonian system of N particles with

masses mi , positions r i , and velocities v i , i = 1, . . . , N . The particles move

under the in¬‚uence of a conservative interaction potential V (r 1 , . . . , r N ) and

may be subject to an external force F ext , which is minus the gradient of a

i

potential ¦(r) at the position r i .

Instead of considering the individual particle trajectories, we wish to de-

rive equations for quantities that are de¬ned as “local” averages over space

of particle attributes. We seek to de¬ne the local averages in such a way

that the averaged quantities ful¬ll equations that approximate as closely

as possible the equations of continuum ¬‚uid dynamics, as described in the

previous section. Exact correspondence can only be expected when the av-

eraging concerns an in¬nite number of particles. For ¬nite-size averaging we

hope to obtain modi¬cations of the ¬‚uid dynamics equations that contain

meaningful corrections and give insight into the e¬ects of ¬nite particle size.

The spatial averaging can be carried out in various ways, but the simplest

is a linear convolution in space. As stated in the introduction of this chapter,

9.3 Coarse graining in space 289

we consider for simplicity an isotropic ¬‚uid consisting of particles of one

type only. Consider the number density of particles n(r). If the particles

are point masses at positions r i , the number density consists of a number

of δ-functions in space:

N

δ(r ’ r i ).

0

n (r) = (9.24)

i=1

The coarse-grained number density is now de¬ned as

N

w(r ’ r i ),

n(r) = (9.25)

i=1

where w(r) is a weight function, with dimension of one over volume. We

shall take the weight function to be isotropic: w(r), with the property that

it decays fast enough with r for the integral over 3D space to exist. The