  A physical model of a three-dimensional flow of a viscous bubbly fluid in an
intermediate regime between bubble formation and breakage is presented. The
model is based on mechanics and thermodynamics of a single bubble coupled to
the dynamics of a viscous fluid as a whole, and takes into account multiple
physical effects, including gravity, viscosity, and surface tension.
Dimensionless versions of the resulting nonlinear model are obtained, and
values of dimensionless parameters are estimated for typical magma flows in
horizontal subaerial lava fields and vertical volcanic conduits.
  Exact solutions of the resulting system of nonlinear equations corresponding
to equilibrium flows and traveling waves are analyzed in the one-dimensional
setting. Generalized Su-Gardner-type perturbation analysis is employed to study
approximate solutions of the model in the long-wave ansatz. Simplified
nonlinear partial differential equations (PDE) satisfied by the leading terms
of the perturbation solutions are systematically derived. It is shown that for
specific classes of perturbations, approximate solutions of the bubbly fluid
model arise from solutions of the classical diffusion, Burgers,
variable-coefficient Burgers, and Korteweg-de Vries equations.
