On the Solution of a Hyperbolic One-Dimensional Free Boundary Problem for a Maxwell Fluid

We study a hyperbolic (telegrapher's equation) free boundary problem describingthe pressure-driven channel flow of a Bingham-type fluid whose constitutive modelwas derived in the work of Fusi and Farina (2011). The free boundary is the surface that separates the inner core (where thevelocity is uniform) from the external layer where the fluid behaves as an upper convectedMaxwell fluid. We present a procedure to obtain an explicit representation formula for the solution. Wethen exploit such a representation to write the free boundary equation in terms of the initialand boundary data only. We also perform an asymptotic expansion in terms of a parametertied to the rheological properties of the Maxwell fluid. Explicit formulas of the solutions forthe various order of approximation are provided