  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, key being the distinction between each
velocity and its canonically conjugate momentum. Dissipation is incorporated to
second order in the ``thermodynamic forces'' via the approach pioneered by
Onsager. 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.
