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Time-Dependent Density-Functional TheoryConcepts and Applications$

Carsten A. Ullrich

Print publication date: 2011

Print ISBN-13: 9780199563029

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199563029.001.0001

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(p.450) Appendix I A brief review of classical fluid dynamics

(p.450) Appendix I A brief review of classical fluid dynamics

Source:
Time-Dependent Density-Functional Theory
Publisher:
Oxford University Press

In this appendix, we summarize some of the basic notions of classical fluid dynamics, with particular emphasis on those concepts that are important for TDCDFT. This review is by no means exhaustive; more information and practical examples can be found in the numerous excellent books on this subject, such as the volume by Landau and Lifshitz (1987).

I.1 Basics and ideal fluids

Fluid dynamics is concerned with the forces on and within a continuous fluid body and the resulting motion. The state of the system is conventionally described by three variables: the mass density ϱ(r, t), the velocity distribution u(r, t), and a third ther-modynamic variable such as the pressure distribution p(r, t). The relation between density and pressure is established via an appropriate thermodynamical equation of state, in which other quantities of interest such as the temperature also appear.1

The most basic relation in fluid dynamics is the continuity equation,

(I.1)
t ϱ + ( ϱ u ) = 0 ,
which expresses the fact that any change in the mass density ϱ in a given spatial region can result only from a net mass current density ϱ u entering or leaving that region.

Next, we wish to establish an equation of motion for the fluid under the influence of internal and external forces. According to Newton's second law, this equation should have the form (mass density) × (acceleration) = (force density).

Let us talk first about the proper treatment of acceleration. Figure I.1 shows the path of a fluid element moving along a streamline. As it travels from position 1 to position 2, the velocity vector of this fluid element changes in accordance with

(I.2)
u ( r + Δ r , t + Δ t ) = u ( r , t ) + u x u x Δ t + u y u y Δ t + u z u z Δ t + u t Δ t .

The last term on the right-hand side is simply the change in the velocity vector at a particular point in space. But we want the rate of change of the velocity vector riding (p.451)

Appendix I A brief review of classical fluid dynamics

Fig. I.1 Change in the velocity of a fluid volume element traveling along a streamline.

along with a particular fluid element, and this is why we need to include the three additional terms on the right-hand side of eqn (I.2). The total acceleration thus comes out as (u·∇)u+∂u/∂t. The general equation of motion in fluid dynamics can therefore be written in the following way:
(I.3)
ϱ [ ( u ) u + u t ] = f total ,
where f total is the total force density, which describes the sum of all internal and external forces acting on a fluid element.

We distinguish between ideal and viscous fluids. There is no energy dissipation in an ideal fluid, which means that there are no processes of internal friction or heat exchange between different fluid elements. Under such circumstances, the equation of motion (I.3) becomes

(I.4)
( u ) u + u t = p ϱ + 1 ϱ f ext ,
where –∇p is the pressure force on the fluid element and f ext is caused by external force fields, usually a gravitational field. Equation (I.4) is known as the Euler equation of fluid dynamics.

If one combines the Euler equation (I.4) and the continuity equation (I.1), it only takes a few elementary steps to show that the rate of change of the momentum density is given by

(I.5)
t ( ϱ u μ ) = v v μ v + f ext , μ ,
where μ denotes one of the Cartesian components x, y, or z, and
(I.6)
μ v = ϱ u μ u v + p δ μ v
is the momentum flux density tensor. The element Πμν of this tensor gives the amount of the μth component of the momentum flowing through a unit area element perpendicular to the r ν-axis per unit time.

Equation (I.5) can be written in integral form, which gives the rate of change of the total momentum in a volume V bounded by a surface S:

(I.7)
t V d 3 r ϱ u μ = v S μ v d s v V d 3 r f ext, μ ,

(p.452) where we have used Green's theorem to transform the volume integral over the diver-gence of the momentum flux density tensor into a surface integral.

Now, assume that the system is finite and let the boundary surface go to infinity: the surface integral in eqn (I.7) then vanishes (since the density and pressure are zero outside the system), and only the term involving f ext,μ survives. Thus, as expected, only external forces can cause a change in the total momentum of a finite system.

I.2 Viscous fluids and dissipation

At the end of the previous section, we learned an important lesson: if we describe all internal forces or interactions within a system as the divergence of a momentum flux density tensor, it is automatically guaranteed that Newton's third law is satisfied. This will provide us with a useful guideline for how to describe viscous effects in a fluid.

Viscous fluids are subject to energy dissipation caused by internal frictional forces between fluid elements in motion. We would like to extend the Euler equation (I.4) or, equivalently, eqn (I.5) to include these internal forces and thus arrive at a more general description of the dynamics of real fluids. To do this, let us add a new term to the momentum flux tensor:

(I.8)
μ v = ϱ u μ u v + p δ μ v σ μ v visc ,
where σ μ v VISC is the viscous stress tensor (the minus sign in front of it is a convention). We have to derive an explicit expression for it, which we will do in a minute, making use of some general physical principles; but whatever the outcome may be, we can rest assured that the resulting theory will satisfy Newton's third law and conserve total momentum.

Frictional forces arise between neighboring fluid elements that have different veloc-ities, causing an irreversible tendency for neighboring fluid elements to reach the same final velocity, at which point the frictional forces disappear. Therefore, σ μ v VISC can only depend on the derivatives of the velocity, μ υ ν, and not on the fluid velocity itself.

Two generic situations are illustrated in Fig. I.2. The left-hand side shows a fluid flowing in the x-direction, with a velocity that is increasing along the y-direction, which causes shear stresses between neighboring fluid elements. For a flat cell of cross-sectional area ∆S, we find the shear stress—measured as a force per unit area—as

(I.9)
Δ F x Δ S = η Δ u x Δ y ,
where η is the coefficient of shear viscosity. This leads to the following shear contribution to the viscous stress tensor:
(I.10)
σ μ v visc | shear = η ( δ u μ δ r v + δ u v δ r μ ) .

The same coefficient η is used for all μ, ν, since the fluid is isotropic. Furthermore, eqn (I.10) must contain the symmetric combination ν υ μ + μ υ ν due to the requirement that there can be no friction if the system is in a state of uniform rotation.2

(p.453)

Appendix I A brief review of classical fluid dynamics

Fig. I.2 Left: a velocity gradient perpendicular to the direction of fluid flow causes a viscous shear stress. Right: a velocity gradient parallel to the direction of fluid flow causes a viscous volume stress (or tensile/compressional stress). The latter is absent in an incompressible fluid.

The right-hand side of Fig. I.2 shows a velocity field where the change in velocity occurs in the same direction as the flow of the fluid. This gives a volume contribution to the viscous stress tensor, associated with tensile or compressional strain:
(I.11)
σ μ v visc | volume = ζ u δ μ v ,
where ζ′ is often referred to as the second viscosity coefficient. Taking the contributions together, we have
(I.12)
σ μ v visc = η ( v u μ + μ u v ) + ζ u δ μ v .

This, however, is not the standard way in which the viscous stress tensor is written. Instead, we redefine the second viscosity coefficient as ζ ' = ζ 2 3 η so that

(I.13)
σ μ v visc = η ( v u μ + μ u v 2 3 u δ μ v ) + ζ u δ μ v .

This expression has the property that the trace of the first term (associated with the shear viscosity) vanishes.

We thus obtain the following generalization of the Euler equation (I.4):

(I.14)
ϱ ( u μ t + v u v v u μ ) = μ p + f ext , μ + v v σ μ v visc .

In general, the viscosity coefficients η and ζ can be complicated functions of space owing to their dependence on pressure and temperature. However, it is often a good approximation to assume that they are constants, and eqn (I.14) then simplifies to

(I.15)
ϱ ( u t + ( u ) u ) = p + f ext + η 2 u + ( ζ + 1 3 η ) ( u ) .

Equation (I.15) is known as the Navier-Stokes equation and represents a cornerstone of classical fluid dynamics.

(p.454) Lastly, let us discuss the energy dissipation caused by viscoelastic stresses. The total kinetic energy in the fluid is given by

(I.16)
E kin = d 3 r 1 2 ϱ u 2 ,
where the integral is over all space and we assume the system to be either finite (i.e., confined between hard boundaries) or such that the fluid is at rest at infinity. The rate of change of the total kinetic energy is given by
(I.17)
t E kin = d 3 r ( 1 2 ϱ t u 2 + ϱ μ u μ u μ t ) .

This expression can be analyzed further by making use of the continuity equation [eqn (I.1)] and the equation of motion (I.14). We will not carry out a full discussion here (see Landau and Lifshitz, 1987), but merely state the obvious fact that in the absence of external forces, the sum of the total kinetic energy and the total internal energy is conserved. In general, exchange between kinetic and internal energy can take place, even without any viscous forces, owing to the compressibility of the fluid. The total kinetic energy remains separately conserved only for incompressible ideal fluids without external forces.

What is of interest to us here is the irreversible change in the total kinetic energy caused by the viscosity of the fluid. We obtain

(I.18)
t E kin | visc = μ v d 3 r u μ v σ μ v visc .

The change in kinetic energy per unit time due to the fluid viscosity corresponds to a dissipative power loss, and we define

(I.19)
t E kin | visc = P diss .

Equation (I.18) can then be recast into

(I.20)
P diss = 1 2 μ v d 3 r σ μ v visc ( v u μ + μ u v ) .

Inserting the explicit form of the viscoelastic stress tensor gives

(I.21)
P diss = 1 2 μ v d 3 r n ( v u μ + μ u v ) 2 d 3 r ( ζ 2 3 η ) ( u ) 2 .

This expression for the dissipated power must always be negative to indicate loss of mechanical energy due to internal friction, which places constraints on the allowable values of the viscosity coefficients. In the special case of an incompressible liquid (characterized by ∇ · u = 0), this implies that the shear viscosity coefficient η must always be a positive number.

Notes:

(1) If the medium is conducting, quantities such as the electric current and the magnetic field will be of interest; this falls within the realm of magnetohydrodynamics.

(2) If the system is rotating with a uniform angular velocity Ω, then the velocity is given by u = Ω×r, and it can be easily verified that the combination ∇ν υ μ + ∇μ υ ν vanishes.