The Equations of Fluid Dynamics and Their Connection with the Boltzmann

Astrofysikaliska dynamiska processer, VT2008

The equations of ﬂuid dynamics and their connection with the Boltzmann equation

Lecture Notes

Susanne H¨ofner

Department of Physics and Astronomy Uppsala University

1 1 Relation of Kinetic Theory to Fluid Mechanics

1.1 Distribution Function and Boltzmann equation Distribution Function To describe the state of the gas statistically, we define the distribution function f(x, v, t) such that f d3x d3v is the average number of particles contained in a volume element d3x about x and a velocity-space element d3v about v at time t. We demand that f ≥ 0 and that f → 0 as the components of the particle velocity → ∞, sufficiently rapidly to guarantee that a finite number of particles has a finite energy. Macroscopic properties of the gas can be computed from f(x, v, t), e.g.:

∞ n(x, t) = f(x, v, t) d3v (1) −∞ Z ρ(x, t) = m n(x, t) (2) ∞ u(x, t) = n−1 f(x, v, t) v d3v = hvi (3) −∞ Z i.e. the number density of the particles n, the mass density ρ (where m is the mass of a single particle) and the average velocity of an element of gas u (= macroscopic ﬂow velocity). To study macroscopic properties of the gas it is useful to decompose the particle velocity v into the average velocity u and a random velocity relative to the mean ﬂow w:

v = u + w (4)

Boltzmann Transport Equation We now search for an equation describing the changes of f(x, v, t) with time. Consider a group of particles located in a phase-space volume element (dx0, dv0) around (x0, v0). We assume that an external force F(x, t) is acting on the particles so that they experience an acceleration a(x, t) = F(x, t)/m. For the moment, we ignore collisions (and interactions) between the particles. The phase-space element will evolve into (dx, dv) around (x, v) where

x = x0 + v0 dt (5) v = v0 + a dt (6)

and the phase volume of the new element is related to the original one by

3 3 3 3 d x d v = J(x, v/x0, v0) d x0 d v0 (7)

with J denoting the Jacobian of the transformation (note that the components of x and v are independent sets of variables). It can be easily shown that J = 1 + O(dt2). Consequently, to ﬁrst order in dt the volume of the phase-space element remains constant, i.e. 3 3 3 3 d x d v = d x0 d v0. (8)

2 The number of particles inside the original phase-space element is

3 3 δN0 = f(x0, v0, t0) d x0 d v0. (9)

In the absence of collisions all these particles must end up in the new element and therefore δN0 = δN where

3 3 3 3 δN = f(x, v, t) d x d v = f(x0 + v0 dt, v0 + a dt, t0 + dt) d x d v. (10)

Together with Eq. 8 this implies that

f(x0 + v0 dt, v0 + a dt, t0 + dt) = f(x0, v0, t0) (11) i.e. in the absence of collisions the phase-space density of a group of particles is invariant. By expanding to first order in dt we find ∂f ∂f ∂f + vi + ai = 0 (12) ∂t ∂xi ∂vi which is known as the collisionless Boltzmann equation, or Vlasov’s equation. When particles suffer a collision during dt their velocities will be changed and they will not, in general, end up in the velocity element (dv) centered around v but in a different velocity element (dv)0 centered around v0 with v0 =6 v. On the other hand, particles not contained in the original element (dv0) may end up in (dv) due to collisions. Therefore, f is no longer invariant. To take these processes into account we have to add a term on the right-hand side of Eq. 12 which gives the net rate at which particles are entering into Df the phase-space element under consideration. We write this term symbolically as Dt coll and obtain the so-called Boltzmann transport equation

∂f i ∂f i ∂f Df + v i + a i = . (13) ∂t ∂x ∂v Dt !coll

Note that f is a one-particle distribution function, i.e. we assume that the probability of ﬁnding a particle at a particular point in phase space is independent of the coordinates of all other particles in phase space. Such a description is only valid for a dilute gas. The most general treatment of a system of N particles involves using a N-particle distribution function which gives the joint probability of ﬁnding the N particles in a given 6-N- dimensional phase-space element. The time evolution of this function is described by Liouville’s equation. Eq. 13 can be derived as a special case from this general approach.

1.2 The Collision Integral Eq. 13 is of little use until we specify the right-hand side. In this section we will derive an explicit expression.

3 Dynamics of Binary Collisions We consider elastic collisions between two classic point particles (no internal degrees of freedom, non-relativistic regime). Such a collision is subject to momentum conservation

0 0 m1v1 + m2v2 = m1v1 + m2v2 (14) and energy conservation

1 2 2 1 02 02 (m1v1 + m2v2) = (m1v1 + m2v2 ) (15) 2 2 where m1 and m2 are the masses of the two particles (labeled ‘1’ and ‘2’), v1 and v2 0 0 are their velocities before the collision, and v1 and v2 their velocities after the collision. 0 0 These two equations together with mass conservation (m1 + m2 = m1 + m2) are known as the summational invariants of the collision. For the description of the collision it is useful to deﬁne the relative velocity of the two particles as q21 = v2 − v1. (16) Using momentum and energy conservation it can be shown that

0 q ≡ |q21| = |q21|, (17)

i.e. the relative velocity of the two particles is changed only in direction but not in mag- nitude by the collision.

Basic Form of the Collision Integral The right-hand side of Eq. 13 is deﬁned as the net rate at which particles are entering into the phase-space element under consideration. Thus, we can write

Df = Rin − Rout (18) Dt !coll

where Rin and Rout are the rates at which particles are scattered in and out of a given phase-space element due to collisions. 3 3 First, we calculate the rate at which particles of type 1 are scattered out of d x1 d v1. For each particle of type 1 the number of particles of type 2 moving with velocities in the range (v2, v2 + dv2), incident within a range of impact parameters (b, b + db) and 3 within azimuth range dΦ in a unit time is (f2 d v2) q b db dΦ where q = |v2 − v1| and f2 ≡ f(x, v2, t). The total number of collisions is given by integrating over all impact parameters, azimuths and incident velocities, and then multiplying by the number of 3 3 particles of type 1, i.e. f1 d x1 d v1. We obtain

3 3 3 3 3 Rout d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1. (19) Z Z Z

4 Rin can be evaluated by considering the inverse enounters, i.e. collisions where initial 0 0 velocities v1 and v2 are changed to the ﬁnal values v1 and v2 within the considered phase-space element. We ﬁnd

3 3 0 0 0 3 0 3 3 0 Rin d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1 (20) Z Z Z 0 0 0 0 0 where f1 ≡ f(x, v1, t) and f2 ≡ f(x, v2, t). It can be shown that q = q and that 3 3 3 0 3 0 d v1 d v2 = d v1 d v2. Therefore we can re-write Rin as

3 3 0 0 3 3 3 Rin d x1d v1 = f1f2 q b db dΦd v2 d x1 d v1. (21) Z Z Z Thus we obtain Df1 0 0 3 = (f1f2 − f1f2) q b db dΦ d v2. (22) Dt !coll Z Z Z Introducing the cross section σ(Ω) dΩ = b db dΦ instead of the impact parameter and azimuth we can write the Boltzmann equation, accounting for binary collisions, as

∂f1 i ∂f1 i ∂f1 0 0 3 1 2 Ω 2 + v1 i + a i = (f1f2 − f f ) q σ( ) dΩ d v . (23) ∂t ∂x ∂v1 Z Z Properties of the Collision Integral

Let Q(v1) be any function of the particle velocity v1 and deﬁne

0 0 3 3 Df1 3 I ≡ Q(v1) (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2 = Q(v1) d v1. (24) Z Z Z Z Dt !coll If we just interchange the labeling 1 and 2 of the particles then obviously the value of I doesn’t change. Adding these two expressions we get

1 0 0 3 3 I = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2. (25) 2 Z Z Z If we replace the collision by its inverse the integral still has the same value because for every collision exist an inverse collision with the same cross section. Therefore we get

1 0 0 0 0 0 0 0 3 0 3 0 I = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω ) dΩ d v1 d v2 (26) 2 Z Z Z 1 0 0 0 0 3 3 = [Q(v1) + Q(v2)] (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2 (27) 2 Z Z Z 0 0 0 3 3 3 0 3 0 where we have used that σ(Ω ) = σ(Ω), dΩ = dΩ, q = q and d v1 d v2 = d v1 d v2. Adding the expressions (25) and (27) for I we get

1 0 0 3 3 I = − [δQ(v1) + δQ(v2)] (f1f2 − f1f2) q σ(Ω) dΩ d v1 d v2. (28) 4 Z Z Z

5 where we have deﬁned δQ(v) ≡ Q(v0) − Q(v). (29) By deﬁnition

0 0 δQ(v1) + δQ(v2) = [Q(v1) + Q(v2)] − [Q(v1) + Q(v2)] (30)

Thus, if Q is a quantity which is conserved during the collision (mass, momentum, energy) δQ(v1) + δQ(v2) = 0, i.e. the integrand in Eq. 28 vanishes and consequently I ≡ 0. We will use this property when deriving the conservation laws of a ﬂuid from the Boltzmann equation.

1.3 Moments of the Boltzmann Equation The equations of ﬂuid dynamics can be derived by calculating moments of the Boltzmann equation for quantities that are conserved in collisions of the particles.

The Conservation Theorem

We form moments of Eq. 23 by multiplying by a quantity Q(v1) and integrating over the velocity:

∂f1 i ∂f1 i ∂f1 3 Df1 3 1 1 Q + v1 i + a i d v = Q d v = I(Q) (31) Z ∂t ∂x ∂v1 ! Z Dt !coll 0 0 where I(Q) is deﬁned by Eq. 24. If Q is conserved during the collision (v1, v2) → (v1, v2) in the sense that 0 0 Q(v1) + Q(v2) = Q(v1) + Q(v2) (32) then it follows immediately from Eq. 28 that I(Q) ≡ 0. In this case Eq. 31 reduces to a conservation law. In the following discussion we assume that Q is conserved. Eq. 31 can be transformed into the conservation theorem ∂ ∂ ∂Q (nhQi) + (nhQvii) − n aih i = 0 (33) ∂t ∂xi ∂vi where we deﬁned the average value hAi of any quantity A as

3 A f d v −1 3 hAi = 3 = n A f d v. (34) R f d v Z To obtain Eq. 33 from Eq. 31 we haRve used partial integration of all three terms, made use of the fact that t, xi and vi are independent variables (change the order of integration and diﬀerentiation with respect to diﬀerent variables) and used the fact that as vi → ∞, f → 0 so strongly that (Qaif) vanishes. For a gas consisting of particles which have no inner structure we have 5 conserved quantities Q, i.e. the mass m, the three components of the momentum mv and the energy 1 2 2 mv .

6 The Equation of Continuity If we choose Q = m and insert this into Eq. 33 we get ∂ ∂ (n m) + (n mhvii) = 0. (35) ∂t ∂xi Using n m = ρ and ui = hvii we obtain the continuity equation

∂ρ + ∇· (ρ u) = 0. (36) ∂t

The Momentum Equations If Q = mvi (the ith component of the particle momentum) Eq. 33 becomes

∂ ∂ (n mhvii) + (n mhvivji) − n m ajδi = 0. (37) ∂t ∂xj j Using the decomposition vi = ui + wi (particle velocity = mean ﬂow velocity + random component) and keeping in mind that hwii = 0 and hvii = ui (by def.) we obtain the momentum equation ∂ ∂ (ρui) + (ρuiuj + P δij − πij) = ρ ai. (38) ∂t ∂xj Here we have used ρhwiwji = P δij − πij (39) where 1 P ≡ ρhw2i (40) 3 is the gas pressure, and 1 πij ≡ ρ( hw2iδij − hwiwji) (41) 3 is the viscous stress tensor.

The Energy Equation 1 2 If we choose Q = 2 mv Eq. 33 assumes the form

∂ 1 ∂ ( n mhv2i) + (n mhv2vji) − n m ajhv i = 0. (42) ∂t 2 ∂xj j Using the same decomposition of vi and the same deﬁnitions as before we obtain the energy equation

∂ 1 2 ∂ 1 2 j ij ij j j ρu + ρe + j ( ρu + ρe)u + (P δ − π )ui + C = ρ a uj (43) ∂t 2 ∂x 2

7 where 1 e ≡ hw2i (44) 2 is the specific internal energy (per gram), and 1 C ≡ ρ hw2wi (45) 2 is the conduction heat flux. These conservation equations are exact for the adopted model of the gas but have no practical value until we can evaluate πij and Cj. Using kinetic theory this can be done from first principles.

1.4 Conservation Equations for Equilibrium Flow The gas can be considered to be in local equilibrium if particle mean free paths are very small compared to characteristic length scales of the ﬂow and if gradients are suﬃ- ciently small. Then we can assume that the distribution function f(x, v, t) is given by a Maxwellian velocity distribution

m 3/2 f(w) = n exp(−m w2/2kT ) (46) 2πkT with a local temperature T (x) and particle density n(x) (w is the random part of the particle velocity as before). Using this distribution function it can be shown that πij and Cj as deﬁned in the previous section are zero. Therefore, the conservation equations become ∂ρ ∂ + (ρuj) = 0, (47) ∂t ∂xj ∂ ∂ ∂P (ρui) + (ρuiuj) = − + ρ ai (48) ∂t ∂xj ∂xi and ∂ 1 2 ∂ 1 2 j ∂ j ρu + ρe + j ( ρu + ρe)u = − j (P uj) + ρ a uj. (49) ∂t 2 ∂x 2 ∂x References

A.R. Choudhuri, The Physics of Fluids and Plasmas, An Introduction for Astrophysicists, Cambridge University Press, 1998, chapters 1-3 F.H. Shu, The Physics of Astrophysics, Vol.II: Gas Dynamics, University Science Books, 1992, p. 14-28

8 2 The Equations of Fluid Dynamics

In this section we will discuss the basic equations which describe the flow of a gas from a macroscopic point of view. We will re-derive and interpret the equations obtained in the previous section by looking at the changes of macroscopic quantities within a fixed volume. Considering ideal fluids only, we neglect the effects of viscosity and heat conduction which are of little importance in many astrophysical flows. Furthermore, we will explicitly discuss the relevant external forces and energy exchange processes which were taken into acount only in a formal way or not at all in the previous section.

2.1 The Conservation Laws Consider an arbitray but ﬁxed volume V with a surface A (with local unit normal nˆ): the time rate of change of a quantity inside this volume will be given by the sum of explicit changes of this quantity (volumetric contributions) and surface eﬀects (net transport across the surface).

Mass Conservation The time rate of change of the mass contained in the volume V is equal to the (negative) value of the mass ﬂux ρ u across the surface: d ρ dV = − ρ u · nˆ dA = − ∇·(ρ u) dV (50) dt ZV IA ZV where the last expression is obtained by using the divergence theorem. V is ﬁxed, so we can write ∂ρ + ∇·(ρ u) dV = 0 (51) ZV " ∂t # and, since V is completely arbitrary, the integrand must vanish. Thus we obtain the equation of continuity: ∂ρ + ∇·(ρ u) = 0. (52) ∂t

Momentum Conservation The time rate of change of the fluid momentum in volume V equals the surface integral of the momentum flux due to fluid flow across A, plus the effect of the pressure P acting on the fluid across the surface A, plus the contribution of forces (e.g. gravity) acting on every point of V (causing an acceleration a): d ρ ui dV = − (ρ ui) ujnˆj dA − P δijnˆj dA + ρ ai dV (53) dt ZV IA IA ZV Applying the divergence theorem to transform the surface integrals into volume integrals we obtain the momentum equation ∂ ∂ ∂P (ρ ui) + (ρ uiuj) = − + ρ ai (54) ∂t ∂xj ∂xi

9 Energy Conservation The time rate of change of the total fluid energy (kinetic energy of fluid motion plus internal energy) equals the surface integral of the energy flux (kinetic + internal), plus the surface integral of the work done by the pressure plus the volume integral of the work done by external forces (e.g. gravitation):

d 1 2 1 2 ρu + ρe dV = − ρu + ρe u·nˆ dA− uiP δijnˆj dA+ ρ u·a dV (55) dt ZV 2 IA 2 IA ZV Using the divergence theorem we obtain the total energy equation: ∂ 1 1 ρu2 + ρe + ∇· ρu2 + ρe u = −∇·(P u) + ρ u · a (56) ∂t 2 2 We see that the fluid equations have the general form ∂ (density of quantity) + ∇·(flux of quantity) = sources − sinks. ∂t 2.2 External Forces and Radiation So far, we have taken external forces formally into account by considering the acceleration a acting on the gas. In most astrophysical situations gravity is the force which determines the large-scale structure of an object. In this case we have a = g where g is the the gravitational acceleration, given by Poisson’s equation ∇·g = −4πGρ. (57) In some circumstances we can not ignore the interaction of the gas with a radiation field. The force per unit volume exerted by the radiation on the gas is ρ ∞ frad = κν Fν dν, (58) 0 c Z were ν denotes the frequency, κν is the opacity of the gas (absorption plus scattering) and Fν is the monochromatic radiative energy flux. In this case we have to add

frad to the right-hand side of Eq. 54, and

u · frad to the right-hand side of Eq. 56. In addition to the work done by the radiation ﬁeld on the matter we also have to take radiative sources and sinks of heat into account by adding Γ − Λ to the right-hand side of Eq. 56. Here Γ and Λ denote the volumetric gains and losses of energy due to local sources and sinks (absorption and emission of radiation). Note in this context that this simple treatment of the interaction of radiation and matter can only be used if the ﬂow velocity is much smaller than the speed of light.

10 2.3 Limit Cases of the Energy Transport Depending on which processes are relevant for the transport of energy in the system under consideration, the energy equation can be rather complicated. However, in many situations this equation can be replaced by a barotropic equation of state, i.e. a relation

P = P (ρ). (59)

One example for such a case is a ﬂuid consisting of quantally degenerate matter. Two other examples are the limiting cases of negligible and extremely eﬃcient heat transport.

Adiabatic Flow Consider an element of gas which neither gains nor loses heat by contact with its surroundings (no thermal conduction, radiative energy exchange, etc.) and the internal energy only changes due to work performed by the element on its surroundings, and vice versa. Then the entropy of this element remains constant and from thermodynamics we know that the pressure and temperature of the gas are related by the adiabatic relationship

P = K ργ.

Here γ is the usual ratio of the speciﬁc heats (e.g. γ = 5/3 for a monatomic gas) and K is a constant related to the entropy of the element.

Isothermal Flow On the other hand, in the limit of extremely eﬃcient radiative heating and cooling the energy equation reduces to Γ − Λ ≡ 0. For a gas of given composition under optically thin conditions Γ−Λ can often be expressed as a function of only density and temperature. Then Γ − Λ ≡ 0 associates a unique temperature for a given density, which implies, together with the equation of state for an ideal gas, that P = P (ρ).

References

A.R. Choudhuri, The Physics of Fluids and Plasmas, An Introduction for Astrophysicists, Cambridge University Press, 1998, p. 53-58, 61-62 F.H. Shu, The Physics of Astrophysics, Vol.II: Gas Dynamics, University Science Books, 1992, p. 44-53, 64-65