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By Lee Y., Wang A. N., Wu D.

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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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