Fluid Equations

The starting point is the fluid equations, comprising the conservation laws for mass

\[\pderiv{\rho}{t} + \cdot \nabla \left( \rho \vv \right) = 0\]

and momentum

\[\rho \left( \pderiv{}{t} + \vv \cdot \nabla \right) \vv = -\nabla P - \rho \nabla \Phi;\]

the heat equation

\[\rho T \left( \pderiv{}{t} + \vv \cdot \nabla \right) S = \rho \epsnuc - \nabla \cdot (\vFrad + \vFcon);\]

and Poisson’s equation

\[\nabla^{2} \Phi = 4 \pi G \rho.\]

Here, \(\rho\), \(P\), \(T\), \(S\) and \(\vv\) are the fluid density, pressure, temperature, specific entropy and velocity; \(\Phi\) is the self-gravitational potential; \(\epsnuc\) is the specific nuclear energy generation rate; and \(\vFrad\) and \(\vFcon\) are the radiative and convective energy fluxes. An explicit expression for the radiative flux is provided by the radiative diffusion equation,

\[\vFrad = - \frac{c}{3\kappa\rho} \nabla (a T^{4}),\]

where \(\kappa\) is the opacity and \(a\) the radiation constant.

The fluid equations are augmented by the thermodynamic relationships between the four state variables (\(P\), \(T\), \(\rho\) and \(S\)). Only two of these are required to uniquely specify the state (we assume that the composition remains fixed over an oscillation cycle). In GYRE, \(P\) and \(S\) are adopted as these primary variables[1], and the other two are presumed to be derivable from them:

\[\rho = \rho(P, S), \qquad T = T(P, S).\]

The nuclear energy generation rate and opacity are likewise presumed to be functions of the pressure and entropy:

\[\epsnuc = \epsnuc(P, S), \qquad \kappa = \kappa(P, S).\]