Extended thermodynamics of fluids and virial equations of state
Starting out with slight modified supply-free balance equations of extended thermodynamics for thirteen field-densities (mass, momentum, momentum flux and energy flux) the constitutive equations for the fluxes and productions in these balance equations are presupposed as functions of these thirteen fields. Using material axioms which the constitutive equations have to satisfy (entropy principle, hyperbolicity, material frame-indifference) restrictions of the possible constitutive functions are obtained: Objective constitutive quantities (tensors under Euclidean transformation) are isotropic functions of the objective parts of the thirteen density-fields representing ten thermodynamic variables. The objective parts of the fluxes, of the productions, of the entropy density, and of the entropy flux are approximated up to second - or third - order terms in the thermodynamic variables. In first order a usual Gibbs equation, and the entropy flux density as heat flux density over temperature is achieved by the axiom that both flux densities are continuous on ideal partitions. Exploiting hyperbolicity inequalities are derivable which includes the usual stability conditions. The linearized balance equations are the usual ones (mass, momentum, internal energy) and additional six further balances for the non-convective parts of the antisymmetric momentum flux and the energy flux. By using an iterative procedure the momentum- and the energy-flux can be eleminated so obtaining 5-field balance equations and constitutive equations with vanishing bulk viscosity. The thermal equation of state for an ideal gas is derived as \(p=T^{1+(n/2)}f(\rho T^{-(n/2)})\), if its caloric equation of state is presupposed as \(\epsilon =(n/2)(p/\rho)+c\). (p=pressure, T=absolute temperature, n=degree of freedom of a gas molecule, f=arbitrary function, \(\rho=\)mass density, \(\epsilon=\)internal energy density, c=constant). For moderately dense gases the virial coefficients of the virial expansion in the mass density are restricted by integrability conditions derived above. Virial coefficients for monoatomic gases and for the van der Waals gas are derived explicitly.
scientific article; zbMATH DE number 3936740
scientific article