  This paper is the second in a sequence of three devoted to the formulation of
a theory of self-gravitating anisotropic fluids in both Newtonian gravity and
general relativity. In this second paper we develop the Newtonian theory,
inspired by a real-life example of an anisotropic fluid, the (nematic) liquid
crystal. We apply the theory to the construction of static and spherical
stellar models. In addition to the usual fluid variables (mass density,
velocity field), the Newtonian theory features a director vector field, whose
length provides a local measure of the size of the anisotropy, and whose
direction gives the local direction of anisotropy. The theory is defined in
terms of a Lagrangian which implicates all the relevant forms of energy:
kinetic energy (with contributions from the velocity field and the time
derivative of the director vector), internal energy (with isotropic and
anisotropic contributions), gravitational interaction energy, and
gravitational-field energy. This Lagrangian is easy to motivate, and it
provides an excellent starting point for a relativistic generalization in the
third paper. The equations of motion for the fluid, and Poisson's equation for
the gravitational potential, follow from a variation of the action functional,
given by the time integral of the Lagrangian. Because our stellar models
feature a transition from an anisotropic phase at high density to an isotropic
phase at low density, a substantial part of the paper is devoted to the
development of a mechanics for the interface fluid, which mediates the phase
transition.
