# 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:

$\epsnuc = \epsnuc(P, S), \qquad \kappa = \kappa(P, S).$

Footnotes