In an Astrophysics & Cosmology summer programme I attended last summer at UCL we were taught that stars seem to transport matter from the regions near the core to the surface through convective currents, and, if I remember correctly, these currents also take place near the surface. Nonetheless, we were also told there is no general theory for convective currents inside stars yet, and this brings me to my question. Considering that matter inside stars being transported by convection currents moves at non-relativistic speeds with respect to any other point in the star, which seems a natural assumption, then why could we not run simulations of the interior of stars using a continuity equation? A valid candidate would be, of course, the equation regarding fluid dynamics:
$$\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot (\rho\vec{u})=0$$
where $\rho$ is the fluid density and $\vec{u}$ is the flow velocity vector field. Or, if the matter next to the star's core is ionised (which is, again, very logical to assume due to the high temperatures in said region) then local charge conservation should also apply:
$$\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot \vec{J}=0$$
This last equation being written, of course, under the assumption that there is no generation of new charged particles, which is generally false since the nucleus is, I understand, constantly spitting out ionized particles. Therefore, we would need to add a term to our equation to account for this generation of new ions.
Again, are these continuity equations of any theoretical value when studying convection currents and energy transport mechanisms inside stars?
