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2 The Hydrodynamic Theory for Dielectrics 2.1 Lorentz Transformation and Thermodynamics The starting point of every hydrodynamic theory is the thermodynamic theory. The energy density ɛ tot is a function of all the other conserved quantities, the entropy density s and the field variables B and D. For a two-component liquid we have, in the rest frame, dɛ tot =dε + c 2 dρ = T ds +(µ + c 2 )dρ +µc dρc + H · dB + E · dD , (1) where the constraints ∇·B =0, ∇·D = ρel (2) are satisfied in equilibrium. (ρel is the charge density.) Note that ɛ tot is the total energy including the rest mass. The mass densities, ρ and ρc, are connected to the particle numbers n1 and n2 by ρ = m1 n1 + m2 n2 , ρc = m2 n2, (3) where m1 and m2 denote the respective masses. The choice of µ + c 2 as the conjugate variable to ρ renders the expansion in the small parameter ε/ρ c 2 simple. (At most densities, the rest mass is certainly the by far dominating contribution.) The equilibrium fluxes of energy and momentum in the rest frame are [5], Q = c E × H , (4) Πij =(Ts+ µρ+ µc ρc + E · D + H · B − ε) δij − 1 2 [Hi Bj + Ei Dj +(i ↔ j))] . (5) These variables and their fluxes constitute the energy-momentum 4-tensor Π µν , Π 00 = ɛ tot , Π 0k =Π k0 = Qk/c = cg tot k , Π ik =Πik =Πki . (6) The local conservation laws ensure that Π µν satisfies 4
∂νΠ µν = 0 (7) and is symmetric. (The greek indices go from 0 to 3, the latin ones from 1 to 3.) The Lorentz transformation will now yield these thermodynamic expressions for an arbitrary inertial frame. Denoting the rest frame quantities with the superscript 0 , the Lorentz transformation Π µν =Λ µ α Π αβ 0 Λ ν β , (8) for v = v ex employs with the matrix where Λ ν ⎛ ⎞ ⎜ γ γβ 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ γβ γ 00⎟ µ = ⎜ ⎟ ⎜ ⎟ , ⎜ 0 0 1 0 ⎟ ⎝ ⎠ (9) 0 0 0 1 β = v/c , γ =1/ 1 − β2 . (10) Up to order v2 c 2 ,wehave ɛ tot =(1+v 2 /c 2 )ɛ tot,0 +2 v [1 + v 2 /c 2 ] · g tot,0 +vi Π 0 ij vj/c 2 , (11) cg tot i = Qi/c = vi/c (1 + v 2 /c 2 ) ɛ tot,0 +[1 + v 2 /(2 c 2 )] cg tot,0 i +3 vi cg tot,0 l vl/(2 c 2 ) +[1 + v 2 /(2 c 2 )] vl Π 0 li/c +vi vl (Π 0 lk/c) vk/(2 c 2 ), (12) Πik = vi vk ɛ tot,0 /c 2 +[1+v 2 /(2 c 2 )] vi g tot,0 k +[1 + v 2 /(2 c 2 )] vk g tot,0 i + vi vk g tot,0 l vl/(vc 2 ) +Π 0 ik +Π 0 il vl vk/(2 c 2 ) +Π 0 lk vl vi/(2 c 2 ) . (13) As ε ≪ ρc2 , we shall only include terms of linear order in v, except for terms c connected to the rest mass, ρc2 , where quadratic order will be included. (In the 5
Page 1 and 2: Boundary Conditions of the Hydrodyn
Page 3: fields are involved. This is the re
Page 7 and 8: Putting all this together, the Π 0
Page 9 and 10: (E, B) µν ⎛ ⎞ ⎜ 0 −Ex −
Page 11 and 12: 0= ˙ D + j D el + ρel v − c ∇
Page 13 and 14: The hydrodynamic equations are with
Page 15 and 16: vt = v − (v · n) n , (82) vn =(v
Page 17 and 18: ⎛ ⎜ ⎝ −ΠD,eff t,i HD × n
Page 19 and 20: ∆(D D n + Dn) =−σsf , (118)
Page 21 and 22: available, such as for a turbulent
Page 23 and 24: To obtain a finite βs, we add a th
Page 25 and 26: with H D z = αc A e−x/λ λ A =

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