  This paper presents an analytical investigation of the solutions to a control
volume model for liquid films flowing down a vertical fibre. The evolution of
the free surface is governed by a coupled system of degenerate nonlinear
partial differential equations, which describe the fluid film's radius and
axial velocity. We demonstrate the existence of weak solutions to this coupled
system by applying a priori estimates derived from energy-entropy functionals.
Additionally, we establish the existence of traveling wave solutions for the
system. To illustrate our analytical findings, we present numerical studies
that showcase the dynamic solutions of the partial differential equations as
well as the traveling wave solutions.
