Archive for Rational Mechanics and Analysis (Arch. Ration. Mech. Anal.) (20120101), 205, no.~1, 1-26. ISSN: 0003-9527 (print).eISSN: 1432-0673.
Subject
35 Partial differential equations -- 35Q Equations of mathematical physics and other areas of application 35Q35 PDEs in connection with fluid mechanics
A number of recent works have been dedicated to the modeling and mathematical analysis of {\it incompressible} viscoelastic fluids. The assumption of incompressibility allows one to avoid complicated thermodynamical considerations because the gradient of the pressure may be seen as the Lagrangian multiplier corresponding to the divergence-free constraint. In the present work the authors are concerned with comparing four models of a {\it compressible} viscoelastic fluid: \roster \item B.~J. Edwards and A.~N. Beris' model initially treated by considering weakly compressible fluids [J. Non-Newton. Fluid Mech. {\bf 36} (1990), 411--417, \doi{10.1016/0377-0257(90)85021-P}]; \item a compressible Oldroyd B model proposed by F. Belblidia, I.~J. Keshtiban and M.~F. Webster in [J. Non-Newton. Fluid Mech. {\bf 134} (2006), no.~1-3, 56--76, \doi{10.1016/j.jnnfm.2005.12.003}]; \item a model for a Maxwell fluid formally derived by K. Wilmański [{\it Continuum thermodynamics. Part I}, Ser. Adv. Math. Appl. Sci., 77, World Sci. Publ., Hackensack, NJ, 2008; MR2482665 (2010f:74002)] using extended thermodynamics; \item a model for non-isothermal polymeric fluids proposed by M. Dressler, Edwards and H.~C. Ottinger [Rheol. Acta {\bf 38} (1999), no.~2, 117--136, \doi{10.1007/s003970050162}]. \endroster \par In the second half of the paper, the authors propose other approaches to find a generalization of viscoelasticity to compressible fluids. The first one is based on basic elasticity principles and the principle of objectivity (Section 4) while Section 5 is based on a generalization of the integral form of the constitutive equation. The last one (Section 6) uses a differential geometric route.