0217.1

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.More specifically, in an external gravitational field (see Box 12.2 for the more general caseof fluids with significant self gravity), the energy density retains its standard perfect-fluidform1U = � v2 + u + � (17.2)2[Eq.(12.45)], and the energy flux (12.65) gets modified by the addition of the diffusiveheat-flow term (17.1):1F = �v v2 + h + � - ��v - 2�� v - �"T.(17.3)2Assuming there are no sources or sinks of energy beyond those already included in these Uand F (no nuclear or chemical reactions, radiation emission or absorption,.), then the lawof energy conservation takes the standard form"U+ " � F = 0.(17.4)"tAs in our discussion of the influence of viscosity on energy conservation (Sec.12.4), soalso here, we can derive a law for the evolution of entropy by combining energy conservation(17.4) with mass conservation and the first law of thermodynamics; see Ex.17.1 for details.The result can be written in either of two equivalent forms.The first, a conservation law forentropy, says:"(�s) 2�� : � + ��2+ " � [�sv - �" ln T ] = �(" ln T )2 + , (17.5)"t Twhere the colon signifies a double contraction of the second rank rate of shear tensor withitself.The quantity �s is obviously entropy density (since s is entropy per unit mass), and �sv1This is only true non-relativistically.In relativistic fluid dynamics, it remains true in the fluid s restframe; but in frames where the fluid moves at high speed, the diffusive energy flux gets Lorentz transformedinto heat-flow contributions to energy density, momentum density, and momentum flux.4is obviously a contribution to entropy flux produced by the motion of the entropy-endowedfluid.The term -�" ln T = Fheat/T is a flux of heat divided by temperature and thus, sinceheat divided by temperature is entropy, it must be a flux of entropy carried by the flowingheat (i.e.carried, microscopically, by the anisotropy N1 in the momentum distributions ofthe molecules and other particles).The left side of Eq.(17.5) would vanish if entropy wereconserved, so the right side must be the rate of production of entropy in a unit volume.Itcontains the viscous heating terms that we discussed in Sec.12.4, and also a new term:dS 1= �(" ln T )2 = Fheat � ".(17.6)dtdV TThis entropy increase per unit volume is the continuum version of the thermodynamic lawthat, when an amount of heat dQ is transfered from a reservoir with high temperature T1 toa reservoir with lower temperature T2, there is a net entropy increase given by1 1dS = dQ - ; (17.7)T1 T2cf.Ex.17.2.Note that the second law of thermodynamics (entropy never decreases), applied toEq.(17.5), implies that the thermal conductivity �, like the viscosity coefficients � and�, must always be positive.Our second version of the law of entropy evolution, derivable from Eq.(17.5) by combiningwith mass conservation, says that the entropy per unit mass in a fluid element evolves at arate given byds 2�� : � + ��2�T = �"2T + (17.8)dt Tcf.Ex.17.2.For a viscous, heat-conducting fluid moving in an external gravitational field, the gov-erning equations are the standard law of mass conservation (12.23) or (12.24), the standardNavier-Stokes equation (12.59), the first law of thermodynamics [Eq.(2) or (3) of Box 12.1],and either the law of energy conservation (17.3) or the law of entropy evolution (17.5) or(17.8).This set of equations is far too complicated to solve, except via massive numerical simu-lations, unless some strong simplification is imposed.We must therefore introduce approx-imations.Our first approximation is that the thermal conductivity � is constant; for mostreal applications this is close to true, and no significant physical effects are missed by assum-ing it.Our second approximation, which does limit somewhat the type of problem we canaddress, is that the fluid motions are very slow slow enough that the squares of the shearand expansion (which are quadratic in the fluid speed) are neglibibly small, and we thus canignore viscous dissipation.This permits us to rewrite the entropy evolution equation (17.8)asds�T = �"2T.(17.9)dtWe can convert this entropy evolution equation into an evolution equation for temper-ature by expressing the changes ds/dt of entropy per baryon in terms of changes dT/dt of5temperature.The usual way to do this is to note that T ds (the amount of heat depositedin a unit mass of fluid) is given by CdT , where C is the fluid s specific heat.However,the specific heat depends on what one holds fixed during the energy deposition: the fluidelement s volume or its pressure.As we have assumed that the fluid motions are very slow,the fractional pressure fluctuations will be correspondingly small.(This does not precludesignificant temperature fluctuations, provided that they are compensated by density fluctu-ations of opposite sign.However, if there are temperature fluctuations, then these will tendto equalize through thermal conduction in such a way that the pressure does not changesignificantly.) Therefore, the relevant specific heat for a slowly moving fluid is the one atconstant pressure, CP , and we must write T ds = CP dT.2 Eq.(17.9) then becomes a linearpartial differential equation for the temperaturedT "Ta" + v � "T = �"2T , (17.10)dt "twhere� = �/�Cp (17.11)is known as the thermal diffusivity and we have again taken the easiest route in treatingCP and � as constant.When the fluid moves so slowly that the advective term v � "T isnegligible, then Eq.(17.10) says that the heat simply diffuses through the fluid, with thethermal diffusivity being the diffusion coefficient for the temperature.The diffusive transport of heat by thermal conduction is similar to the diffusive transportof vorticity by viscous stress [Eq.(13.3)] and the thermal diffusivity � is the direct analogof the kinematic viscosity �.This motivates us to introduce a new dimensionless numberknown as the Prandtl number, which measures the relative importance of viscosity and heatconduction (in the sense of their relative abilities to produce a diffusion of vorticity and ofheat):�Pr = (17.12)�For gases, both � and � are given to order of magnitude by the product of the mean molecularspeed and the mean free path and so Prandtl numbers are typically of order unity.(Forair, Pr [ Pobierz całość w formacie PDF ]

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