By Lee Y., Wang A. N., Wu D.
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Additional info for A Bridge Principle for Harmonic Diffeomorphisms between Surfaces
52) is an example of a heat equation. The second term in the RHS of Eq. 52) represents the viscous heating, and can be neglected for small and moderate gradients. 4 Hydrodynamic equations 27 is second order in the dissipative ﬂuxes, while ∇ · Q is ﬁrst order. For practical reasons it is convenient to switch variables in Eq. 52), from entropy density s(r, t) to temperature T (r, t). 53) ρ where cV is the speciﬁc heat capacity at constant volume. 49) into the RHS of Eq. 5) one obtains: T αp λ ∂T ∇2 T − ∇ · v.
1) into the balance laws; and ﬁnally each function Cαβ (r, r ) is determined by imposing that the entropy must be a maximum, so that the equal-time (static) correlation functions among the ﬂuctuating ﬁelds have the entropy as probability generating functional‡ (Landau and Lifshitz, 1958, 1959; Fox and Uhlenbeck, 1970a). In this book we adopt a less formal approach and in Eq. 3) we have already anticipated the result to be obtained. In Sect. 3 we evaluate the statistical properties of the ﬂuctuating thermodynamic ﬁelds using Eq.
2). Consequently, to specify the equations of ﬂuctuating hydrodynamics for a binary ﬂuid mixture, we have to consider three random terms: a random energy ﬂow δQ , a random deviatoric stress tensor δΠ and a random diﬀusion ﬂow δJ. As was the case for a one-component ﬂuid, δΠ does not couple with the other two dissipative ﬂuxes, so the FDT for δΠ is the same as Eq. 7c) for a one-component ﬂuid, depending on whether or not the ﬂuid is assumed to be incompressible. The other two dissipative ﬂuxes do couple, and the relevant phenomenological laws are given by Eqs.
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