fluid-equations.rst

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

∇²Φ = 4πGρ.

Here, ρ, P, T, S and $\vv$ are the fluid density, pressure, temperature, specific entropy and velocity; Φ 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 κ is the opacity and a the radiation constant.

The fluid equations are augmented by the thermodynamic relationships between the four state variables (P, T, ρ 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 variables¹, and the other two are presumed to be derivable from them:

ρ = ρ(P, S),  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

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1. This may seem like a strange choice, but it simplifies the switch between adiabatic and non-adiabatic calculations↩