Advances in Mathematical Physics
Volume 2011 (2011), Article ID 606757, 26 pages
Research Article

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

Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Received 11 March 2011; Revised 12 May 2011; Accepted 14 June 2011

Academic Editor: Luigi Berselli

Copyright © 2011 Lorenzo Fusi and Angiolo Farina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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