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The Fundamental Equations of Gas Dynamics

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The Fundamental Equations of Gas Dynamics The Fundamental Equations of Gas Dynamics 1 References for astrophysical gas (fluid) dynamics 1. L. D. Landau and E. M. Lifshitz Fluid mechanics 2. F.H. Shu The Physics of Astrophysics Volume II Gas Dynam- ics 3. L. Mestel Stellar Magnetism Fundamental equations © Geoffrey V. Bicknell 2 The equation of continuity 2.1 Derivation of the fundamental equation n Define v = density (1) vi = velocity components S Mass within volume V is V M = dV (2) V Fundamental equations 2/78 Rate of flow of mass out of V is vinidS S Therefore, the mass balance within V is given by: d ----- dV = – v n dS (3) dt i i V S Use the divergence theorem: v n dS = v dV (4) i i xi i S V Fundamental equations 3/78 and d ----- dV = dV (5) dt t V V to give + v dV = 0 (6) t xi i V Since the volume V is arbitrary, then +0vi = (7) t xi This is the equation of continuity. Fundamental equations 4/78 Aside: Differentiation following the motion Suppose we have a function fxi t which is a function of both space and time. How does this function vary along the trajectory of a fluid element described by xi = xit ? We simply calculate d f f dxi f f fxit t ==+ ------- + vi (8) dt t xi dt t xi Fundamental equations 5/78 Useful result from the equation of continuity Write the above equation in the form: vi ++vi = 0 (9) t xi xi Going back to the equation of continuity, then d vi vi 1d ------ = – = –--------- (10) dt xi xi dt Fundamental equations 6/78 Putting it another way: 1d vi --------- = – (11) dt xi This tells us that in a diverging velocity field vi xi 0, the density decreases and in a converging velocity field vi xi 0 the density increases. Fundamental equations 7/78 2.2 Examples of mass flux Wind from a massive star Massive stars such as O and B stars produce winds with veloci- ties of around 1 000 km s–1 and mass fluxes of around 10–6 solar masses per yr. We can use these facts to estimate the density in the wind as follows. Mass flux = vinidS (12) r S = 4r2r vr Fundamental equations 8/78 where the integral is over a sphere of radius r. We assume that the flow is steady so that the mass flux integrated over any 2 sur- faces surrounding the star is the same. Hence, · M ==4r2r vr constant (13) We shall show in later lectures that far from the star, the velocity is constant, i.e. vr==v constant (14) Therefore the density is given by: M· r = ------------------- (15) 2 4r v Fundamental equations 9/78 For typical parameters: M· = 10–6 solar masses per yr 15 r ==0.1 pc 3.0910 m (16) –1 6 –1 v ==1 000 km s 10 m s Units 30 16 1 solar mass= 210 kg 1 pc= 3.0910 m (17) 7 1year= 3.1610 seconds The density of the wind at 0.1 pc from the star is: –22 = 5.210 kg m–3 (18) Fundamental equations 10/78 This is not particularly informative by itself. We usually express the density in terms of particles cm–3. Assuming the gas is to- tally ionised, then = nHmp ++nHemHe ...... (19) Now for solar cosmic abundances, nHe = 0.085nH (20) and mHe 4m (21) where m is an atomic mass unit –27 m = 1.6610 kg (22) Fundamental equations 11/78 so that nHm + 4 0.085nHm = 1.34nHm (23) Therefore, for the O-star wind: –22 –3 1.34nHm 5.210 kg m –22 5.210 n = -------------------------m–3 (24) H 1.34m 5 = 2.310 Hydrogen atoms m–3 Fundamental equations 12/78 Often, instead of the Hydrogen density we use the total number density of ions plus electrons. In this case we put = nm (25) ==mean molecular weight 0.6156 Hence, –22 5.210 5 n ==------------------------- 5.110 particles m–3 (26) 0.6156m Fundamental equations 13/78 3 Conservation of momentum n Consider first the rate of change of momentum within a volume as a re- vivj sult of the flux of momentum. Let i = total momentum, then S vi · d i = ----- v dV (27) V dt i V is the rate of change of momentum in –pni the volume, V. Fundamental equations 14/78 The flux of momentum out of the surface S is vivjnjdS (28) S 3.1 Surface force There are two complementary ways that we can look at the other aspects of the conservation of momentum. In a perfect fluid there is a force per unit area perpendicular to the surface that exerts a force on the gas inside. This is the pes- sure p Force on volume = – pn dS (29) within S i S Fundamental equations 15/78 Therefore the total momentum balance for the volume is: d ----- v dV = – v v n dSpn– dS (30) dt i i j j i V S S The surface integrals Using the divergence we can write vivjnjdS = vivj dV (31) xj S V The surface integral of the pressure can be written pnidS ==pijnjdS pij dV (32) xj S S V Fundamental equations 16/78 Entire momentum equation Again, we take the time derivative inside and obtain: vi + vivj + pij dV = 0 (33) t xj xj V Since the volume is arbitrary, then vi +0vivj + pij = (34) t xj We often write this equation in the form: p vi + vivj = – (35) t xj xi Fundamental equations 17/78 3.2 2nd way - momentum flux Across any surface in a gas there is a flux of momentum due to the random motions of atoms in the gas. n n We write the flux per unit area of ij j momentum due to molecular mo- v v i j tions as ijnj The flux of momentum out of the S vi volume due to molecular motions is V then: n dS (36) –pni ij j V Fundamental equations 18/78 For a perfect fluid, the relation between ij and p is ij = pij (37) In viscous fluids (to be treated later) the tensor ij is not diago- nal. Fundamental equations 19/78 Momentum balance When we adopt this approach, the momentum balance of the vol- ume of gas is: d ----- v dV = – v v n dS– n dS dt i i j j ij j V S S (38) vi + vivj + ij dV = 0 t xj V The corresponding partial differential equation is: vi +0vivj + ij = (39) t xj Fundamental equations 20/78 The point to remember from this analysis is that pressure in a flu- id is the result of a flux of momentum resulting from the micro- scopic motions of the particles. Particles of mass m crossing a surface within the fluid with ran- dom atomic velocity ui (relative to the bulk velocity) contribute an amount muiuj to the flux of momentum. A particle with an equally opposite velocity contributes exactly the same amount to the momentum flux. Hence, atomic motions in both directions across the surface contribute equal amounts to the pressure. Fundamental equations 21/78 3.3 Alternate forms of the momentum equations Consider, the isotropic case where ij = pij. Then, p vi + vivj = – (40) t xj xi Expand the terms on the left: vi vi p vi ++ vi vj + vj= – (41) t t xj xj xi Using the continuity equation to eliminate the blue terms: vi vi p + vj = – (42) t xj xi Fundamental equations 22/78 In terms of the derivative following the motion: dvi p ------- = – (43) dt xi 3.4 Additional forces We can in general have additional forces acting on the fluid. In particular, we can have a gravitational force derived from a grav- itational potential. This is not a surface force like the pressure but a body force which is proportional to the volume of the re- gion. We write: Gravitational force per unit mass = – (44) xi Fundamental equations 23/78 where is the gravitational potential. Therefore the gravitational force acting on volume V is Fi = – dV (45) xi V We add this term to the momentum equations to obtain: vi + vivj + ij dV = – dV (46) t xj xi V V implying that: vi + vivj + ij = – (47) t xj xi Fundamental equations 24/78 or the alternative form: vi vi ij + vj = – – (48) t xj xj xi When the momentum flux is diagonal ij = pij (49) p vi + vivj = – – t xj xi xi (50) vi vi p + vj = – – t xj xi xi Fundamental equations 25/78 3.5 Hydrostatic application of the momentum equations X-ray emitting atmosphere in an elliptical galaxy Elliptical galaxies have extended hot atmospheres extending for 10– 100 kpc from the centre of the galaxy. Our first approach to understanding the distribution of gas in such an atmosphere is to consider an hydrostatic model. In this case vi = 0 and the momentum equations reduce to 1p --- = – (51) xi xi Fundamental equations 26/78 Let us restrict ourselves to the case of spherical symmetry: 1p GM r --- ==– –----------------- (52) r r r2 G = Newtons constant of gravitation –11 = 6.6710 SI units (53) Mr= Mass within r This can be used to estimate the mass of the galaxy. Put kT pnkT==---------- (54) m Fundamental equations 27/78 then 1 d kT GM r -------- ---------- = –----------------- dr m r2 (55) 1 kT d k dT GM r --- -------------- + -------------- = –----------------- m dr m dr r2 Rearrangement of terms gives: r2 kT 1d 1dT Mr= –------------- --------- + --------- (56) G m dr T dr Fundamental equations 28/78 Usually, at large radii, the temperature of the atmosphere is iso- thermal and the density of the X-ray emitting gas may be approx- imated by a power-law: T constant r– 0.7 r d (57) --------- = – dr Therefore, r2 kT 1d 1dT kT Mr==–---------------- --------- + --------- --------------- r G mp dr T dr mpG (58) 11 T r = 3.110 --------- ---------------- solar masses 107 10 kpc Fundamental equations 29/78 The amount of matter implied by such observations is larger than can be accounted for by the stellar light from elliptical galaxies and implies that elliptical galaxies, like spiral galaxies, have large amounts of dark matter.
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