279

280 Coarse graining from particles to ¬‚uid dynamics

degrees of freedom” as was discussed in Chapter 8. The “important” degrees

of freedom are now the density ρ(r) as a function of space, which is described

with a limited precision depending on the way the spatial averaging is carried

out. All other degrees of freedom, i.e., the particle coordinates within the

restriction of a given density distribution, form the “unimportant” degrees

of freedom, over which proper ensemble-averaging must be done. The forces

that determine the evolution of density with time consist of three types:

(i) systematic forces, depending on the coarse-grained density distribu-

tion (and temperature) itself;

(ii) frictional forces, depending on the coarse-grained velocities;

(iii) random forces that make up the unpredictable di¬erence between the

exact forces and the systematic plus frictional forces.

In analogy with the behavior of a system with a reduced number of degrees of

freedom (Chapter 8), we expect the random force to become of relatively less

importance when the spatial averaging concerns a larger number of particles,

and, in fact, a decrease in standard deviation with the square root of that

number. If the spatial averaging is characterized by a smoothing distance

a, then the relative standard deviation of the noise in mechanical properties

is expected to be proportional to a’3/2 . As an example of a speci¬c type

of coarse graining, we can consider to simplify the description of particle

positions by a density on a cubic spatial grid with spacing a. Instead of

na3 particles (where n is the number density of the particles) we now have

one density value per grid cell. So we must sum mechanical properties over

roughly na3 particles: correlated quantities will become proportional to a3

and the noise will be proportional to the square root of that value. In Section

9.3 more precise de¬nitions will be given.

There are three reasons for obtaining the macroscopic equations for the

behavior of ¬‚uids by a process of coarse graining:

(i) The assumptions on which the macroscopic equations rest (as validity

of local density, bulk ¬‚uid velocity, and pressure) are made explicit.

(ii) The limits of application of the macroscopic equations become clear

and correction terms can be derived.

(iii) The macroscopic equations valid as approximation for a system of

real particles are also an approximation for a system of di¬erent and

larger particles if their interactions are appropriately chosen. Thus

the macroscopic problem can be solved by dynamic simulation of a

many-particle system with a much smaller number of particles, be

9.2 The macroscopic equations of ¬‚uid dynamics 281

it at the expense of increased noise. This is the basis of dissipative

particle dynamics described in Chapter 11.

In Section 9.2 an overview is given of the macroscopic equations of ¬‚uid

dynamics. This is done both as a reminder and to set the stage and notation

for the systematic derivation of the macroscopic equations from microscopic

equations of motion of the constituent particles, given in Section 9.3. Note

that in Section 9.3 the macroscopic quantities are properly de¬ned on the

basis of particle properties; in the macroscopic theory these quantities (den-

sity, ¬‚uid velocity, pressure, etc.) are not really de¬ned, and their existence

and validity as spatially-dependent thermodynamic quantities is in most

textbooks assumed without further discussion.

9.2 The macroscopic equations of ¬‚uid dynamics

Note on notation We shall use vector notation as usual, but in some cases (like

the derivatives of tensors) confusion may arise on the exact meaning of compound

quantities, and a notation using vector or tensor components gives more clarity.

Where appropriate, we shall give either or both notations and indicate cartesian

components by greek indexes ±, β, . . . , with the understanding that summation is

assumed over repeated indexes. Thus ‚vβ /‚x± is the ±β component of the tensor

∇v, but ‚v± /‚x± is the divergence of v: ∇ · v.

The principles of single-component ¬‚uid dynamics are really simple. The

macroscopic equations that describe ¬‚uid behavior express the conservation

of mass, momentum and energy. The force acting on a ¬‚uid element is “ in

addition to an external force, if present “ given by a thermodynamic force

and a frictional force. The thermodynamic force is minus the gradient of

the pressure, which is related to density and temperature by a locally valid

equation of state, and the frictional force depends on velocity gradients. In

addition there is heat conduction if temperature gradients exist. Since we

assume perfect homogeneity, there is no noise.

Our starting point is the assumption that at every position in space the

bulk velocity u(r) of the ¬‚uid is de¬ned. Time derivatives of local ¬‚uid

properties can be de¬ned in two ways:

(i) as the partial derivative in a space-¬xed coordinate frame, written as

‚/‚t and often referred to as the Eulerian derivative;

(ii) as the partial derivative in a coordinate frame that moves with the

bulk ¬‚uid velocity u, written as D/Dt and often referred to as the

Lagrangian derivative or the material or substantive derivative.

282 Coarse graining from particles to ¬‚uid dynamics

The latter is related to the former by

D ‚ D ‚ ‚

+ u · ∇,

= = + u± . (9.1)

Dt ‚t Dt ‚t ‚x±

Some equations (as Newton™s equation of motion) are simpler when material

derivatives are used.

The next most basic local quantity is the mass density ρ(r) indicating the

mass per unit volume. It is only a precise quantity for locally homogeneous

¬‚uids, i.e., ¬‚uids with small gradients on the molecular scale, on which no

real ¬‚uid can be homogeneous). We now de¬ne the mass ¬‚ux density J (r)

as the mass transported per unit time and per unit area (perpendicular to

the ¬‚ow direction):

J = ρu. (9.2)

9.2.1 Conservation of mass

The continuity equation expresses the conservation of mass: when there is

a net ¬‚ow of mass out of a volume element, expressed (per unit of volume)

as the divergence of the mass ¬‚ux density, the total amount of mass in the

volume element decreases with the same amount:

‚ρ ‚ρ ‚J±

+ ∇ · J = 0, + = 0. (9.3)

‚t ‚t ‚x±

The continuity equation can also be expressed in terms of the material

derivative (using the de¬nition of J ):

Dρ

+ ρ∇ · u = 0. (9.4)

Dt

¿From this formulation we see immediately that for an incompressible ¬‚uid,

for which ρ must be constant if we follow the ¬‚ow of the liquid, Dρ/Dt = 0

and hence the divergence of the ¬‚uid velocity must vanish:

∇·u=0 (incompressible ¬‚uid). (9.5)

9.2.2 The equation of motion

Next we apply Newton™s law to the acceleration of a ¬‚uid element:

Du

= f (r) = f int + f ext ,

ρ (9.6)

Dt

where f (r) is the total force acting per unit volume on the ¬‚uid at position r.

The total force is composed of internal forces arising from interactions within

9.2 The macroscopic equations of ¬‚uid dynamics 283

the system and external forces, arising from sources outside the system.

Internal forces are the result of a pressure gradient, but can also represent

friction forces due to the presence of gradients in the ¬‚uid velocity (or shear

rate). Both kinds of forces can be expressed as the divergence of a stress

tensor σ:1

‚σ±β

f int = ∇ · σ, int

f± = . (9.7)

‚xβ

Thus Newton™s law reads

Du ‚u

+ ρ (u · ∇)u = ∇ · σ + f ext ,

ρ = ρ

Dt ‚t

‚σ±β

Du± ‚u± ‚u± ext

ρ = ρ + ρuβ = + f± . (9.8)