Conformal Invariants by Thomas J. M. PDF

By Thomas J. M.

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14) and −ρ0D ηθ˙ + 1 ˙ 2 DT · C + 1 Q · ∇ θ − Kˆ ≥ 0. 2). 16) where is the dissipation potential and U η , UT , and UQ are the constitutive residuals, subject to ˙ + UQ · ∇ θ = 0. U η θ˙ + UT · C Thus, they do not contribute to the dissipation of energy. ,2 Q , θ ˙ Y(X, t) = {θ˙ (X, t), C(X, t), ∇ θ(X, t)}, ω (X, t) = {C(X, t), θ(X, t)}, U = {U η , UT , UQ }. 15) may be expressed in the form of a scalar product J · Y − Kˆ ≥ 0. 19) The collections Y(X, t) and J(X, t) are, respectively, called the thermodynamic force and the thermodynamic flux.

6a) σ Stokes’ Theorem. 6b) where ϒ is any tensor and q and A are vectors, all being continuously differentiable in the star-shaped domains. 2). 5]. 2) may be transformed to V −σ ∇ · v) + (tkl,k + ρfl − ρ v˙l )vl −( + 21 v · v)(ρ˙ + ρ∇ −ρ ˙ + tkl vl,k + qk,k + ρh dv + σ [ − ρ( + 21 v · v)(vk − uk ) + tkl vl + qk ]nk da = 0. 4) of the conservation of mass becomes V −σ ∇ · v) dv + (ρ˙ + ρ∇ [ρ(vk − uk )]nk da = 0. , for the entire body. 1. 6a). V − σ and ∂V − σ excludes points of σ . V ρˆ dv + V −σ Rˆ da = 0.

2 Second Law of Thermodynamics 25 where η is the entropy density and (θ > 0, inf θ = 0) is the absolute temperature. 2b) where the entropy residuals γˆ and ˆ are subject to V −σ ˆ da = 0. 3) implies that V −σ ˆ da = 0. 40a), to cast the energy equation into the form −ρ(ψ˙ + ηθ ˙ + ηθ˙ ) + tkl vl,k + qk,k + ρh = 0. 2a) we obtain the (C–D) inequality V −σ 1 1 ˙ + tkl dkl + qk θ,k dv ≥ 0. 10). , η = R η + D η, tkl = R tkl + D tkl . 5), ψ is considered to be a fully static variable and, of course, the heat q has no static part.

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Conformal Invariants by Thomas J. M.

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