A unified hyperbolic model for viscoelastic liquids Amr Guaily * , Marcelo Epstein University of Calgary, Calgary, Alberta, Canada T2N 1N4 article info Article history: Received 11 October 2009 Received in revised form 7 December 2009 Available online 14 December 2009 Keywords: Viscoelastic liquids Hyperbolic equations Least-squares Finite element abstract We present a unified purely hyperbolic model for compressible and incompressible viscoelastic liquids. The proposed model is then solved numerically using the least-squares finite element method (LSFEM). Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction A viscoelastic liquid is a fluid that exhibits a physical behaviour intermediate between that of a viscous liquid and an elastic solid. For this reason, both the mathematical formulation and the experimental techniques used to describe the response of visco- elastic liquids are substantially different from their viscous liquid counterparts. In particular, the numerical implementation of the governing system of equations contains important qualitative differences, such as the character of the equations, the choice of independent variables and the enforcing of boundary conditions. To the best of our knowledge, this is the first attempt to formulate the governing equations of viscoelastic liquids as a purely hyper- bolic model describing the compressible and the incompressible flow regimes in a unified way. Being a totally hyperbolic system is of great importance for many reasons, the most important of which are: (i) the boundary conditions can be determined without ambiguity, which may not be the situation for other types of sys- tems. The issue of the boundary conditions for hyperbolic systems is well presented in Godlewski and Raviart (1996). (ii) A major source of difficulty in numerical simulation for viscoelastic liquids is the change of type of the system of equations, as described in Joseph (1990). Phelan et al. (1989) implemented a hyperbolic mod- el for the incompressible flow regime only. Their results appear to be very sensitive to the discretization grid and their solutions are limited to very coarse spatial discretizations. In the current work the Upper Convected Maxwell Model (UCMM) is used to account for the fluid viscoelasticity. There are many reasons for utilizing this particular constitutive equation. Elastic effects in a UCM fluid are modeled, albeit in a very simple way. The model depends on only two parameters: a relaxation time k and a viscosity coefficient or elastic viscosity g. Other models, such as the Giesekus model or the White–Metzner model, can be obtained as straightforward generalizations of the UCMM (for details, see Renardy (2000)). There is also a link to molecular theories, e.g. the linear dumbbell model. Also the UCMM is known to be the most challenging consti- tutive equation for numerical analysis as it overestimates stresses at higher shear rates, as explained by Renardy (2000). We should also mention that researchers have found that crude oil (Tsiklauri and Beresnev, 2001), flour dough (Schofield and Scott-Blair, 1932), and toluene (Reiner, 1960) also obey, at least approxi- mately, the Maxwell model. In fact, many well-known fluids have been found to exhibit viscoelastic behaviour (see the tables in Jo- seph et al. (1986)). In particular, the Maxwell model serves as a simplified description of dilute polymeric solutions or melts (Raik- her and Rusakov, 1996; Speziale, 2000). Tanner (1962) gave the ex- act solution to Stokes’ first problem for Maxwell fluids as well as for Oldroyd B-type fluids. Jordan and Puri (2005) re-examined Stokes’ first problem for Maxwell fluids and presented a collection of new small-time results that are used to carry out an in-depth analytical study of this problem, the main emphasis there being on obtaining energy results. In their work, results for Maxwell flu- ids are compared with those of viscous Newtonian fluids and it is found that the presence of fluid elasticity counteracts viscous effects, that the Maxwell and viscous Newtonian kinetic energy profiles intersect, and that the point of intersection is both greater than and proportional to the Maxwell relaxation time. Jordan et al. 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.12.004 * Corresponding author. Address: 2500 University Drive NW, MEB, Calgary, AB, Canada T2N 1N4. Tel.: +1 403 891 6267; fax: +1 403 282 8406. E-mail addresses: agguaily@ucalgary.ca (A. Guaily), mepstein@ucalgary.ca (M. Epstein). Mechanics Research Communications 37 (2010) 158–163 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom