An Application of Classical Thermodynamics for a Newtonian Fluid

We first consider the following mechano-thermal coupling problem in a fluid, showing how classical thermodynamics may be applied within the framework of the principle of conservation of energy. 

Let r and p be the Cauchy stress and pressure (compression positive), respectively. We first introduce the Kirchhoff energy stress o- (at, t) and the corresponding pressure p (x,t) by 

Recall that the fluid is a material in which the residual stress, reduced by the reversible part of the stress (i.e., the pressure), is dissipative and irreversible therefore, for an isotropic Newtonian fluid, we have 

We assume that the internal energy u(v, i) and Helmholtz free energy /(v, T) exist. Here v is the volume, s is the entropy and T is the absolute temperature (details of these variables will be discussed later). The increments are written as 

Since —p dv/p = —p dv is the elastically stored internal energy due to stress, it can be written as 

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