5.6 Characteristic speeds in a viscous, heat-conducting gas

We recall from Section 2.4, in particular (14View Equation), that the jumps δu across acceleration waves and their speeds of propagation are to be calculated from the homogeneous system
∂F A ϕ,A ----δu = 0. (109 ) ∂u
In the present context, where the field equations are given by (79View Equation, 80View Equation) this homogeneous algebraic system spreads out into three equations, viz.
ϕ,AδAA = 0, ϕ,AδAAB = 0, ϕ,AδAABC = 0. (110 )
By (89View Equation) and (91View Equation) this is a fully explicit system, if the thermal equation of state p = p(α, T) is known. The vanishing of its determinant determines the characteristic speeds. Seccia & Strumia [44] have calculated these speeds – one transversal and two longitudinal ones – for non-degenerate gases and obtained the following results in the non-relativistic and ultra-relativistic cases

2 ∘ ----- mkcT- ≫ 1 : Vtrans = 75kmT, ∘ ----- Vl1ong = 43mkT, ∘ -------- Vl2ong = 5.18mkT,
∘ -- mc2 ≪ 1 : Vtrans = 1c kT 5, ∘ -- Vl1ong = 1c 3, ∘ -- Vl2ong = 3c 5.
(111)

All speeds are finite and smaller than c. Inspection shows that in the non-relativistic limit the order of magnitude of these speeds is that of the ordinary speed of sound, while in the ultra-relativistic case the speeds come close to c.


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