A flux-conservative formalism for convective and dissipative multi-fluid systems, with application to Newtonian superfluid neutron stars

Abstract

We develop a flux-conservative formalism for a Newtonian multi-fluid system, including dissipation and entrainment (i.e. allowing the momentum of one fluid to be a linear combination of the velocities of all fluids). Maximum use is made of mass, energy and linear and angular momentum conservation to specify the equations of motion. Also used extensively are insights gleaned from a convective variational action principle, the key being the distinction between each velocity and its canonically conjugate momentum (which is modified because of entrainment). Dissipation is incorporated to second order in the ‘thermodynamic forces’ via the approach pioneered by Onsager, which makes it transparent how to guarantee the law of increase of entropy. An immediate goal of the investigation is to understand better the number, and form, of independent dissipation terms required for a consistent set of equations of motion in the multi-fluid context. A significant, but seemingly innocuous detail is that one must be careful to isolate ‘forces’ that can be written as total gradients, otherwise errors can be made in relating the net internal force to the net externally applied force. Our long-range aim is to provide a formalism that can be used to model dynamical multi-fluid systems both perturbatively and via fully nonlinear 3D numerical evolutions. To elucidate the formalism we consider the standard model for a heat-conducting, superfluid neutron star, which is believed to be dominated by superfluid neutrons, superconducting protons and a highly degenerate, ultra-relativistic gas of normal fluid electrons. We determine that in this case there are, in principle, 19 dissipation coefficients in the final set of equations. A final reduction of the system is made by neglecting heat conduction. This leads to an extension of the standard two-fluid model for neutron star cores, which has been used in a number of previous applications, and illustrates how mutual friction is represented in our formalism.