## LITERATURE REVIEW

### Viscoelastic Flow Solvers

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 viscoelastic 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. A comprehensive treatment of the associated problematic is presented in Joseph (1990). It is well known that the flow of Newtonian fluids presents a host of mathematical difficulties. It turns out that all of these difficulties exist also for the flow of viscoelastic liquids especially at high Weissenberg number besides other sources of numerical difficulties. One of the sources of difficulties in viscoelastic liquids is the presence of stress boundary layers and singularities. Another source of difficulty stems from the convective behaviour and in the treatment of the stress field as a primary unknown, we refer to Keunings (1990) for more details about challenges in viscoelastic fluids simulations. To the best of our knowledge, this study is the first attempt to formulate the governing equations of viscoelastic liquids as a purely hyperbolic 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 systems. (ii) A major source of difficulty in numerical simulation for viscoelastic liquids is the change of type of the system of equations. Phelan et al. (1989) implemented a hyperbolic model for the incompressible flow regime only. Their results appear to be very sensitive to the grid discretisation and their solutions are limited to very coarse spatial discretisations. The mathematical and physical consequences of the elasticity of liquids as well as the asymptotic theories of constitutive models are well explained in Joseph (1990). In polymer processing operations, such as injection moluding and high-speed extrusion, pressure and flow rate may be large. Hence, compressibility effects within the viscoelastic regime may become important and influence resulting flow phenomena. Comprehensive literature reviews for viscoelastic incompressible flow may be found in Baaijens (1998). A variety of formulations have been applied over the last decade or so, such as finite volume (FV) methods (Oliveira and Pinho, 1998; Phillips and Williams, 1999), finite element (FE) method (Marchal and Crochet, 1986; Baloch et al., 1995; Guenette and Fortin, 1995; Matallah et al., 1998), spectral collocation methods (Fietier and Deville, 2003), and hybrid FE/FV methods (Wapperom and Webster, 1998; Aboubacar and Webster, 2001). These would include time-marching and steady-state approaches, leading to various options for the state-of-the-art (e.g., DEVSS as shown by Guenette and Fortin, 1995; Discontinuous-Galerkin and Galerkin-Least-Squares as shown by Baaijens, 1998, and others). The compressible viscoelastic domain is relatively unsheltered in the literature. Georgiou (2003) has addressed non-Newtonian inelastic (Carreau) fluid modelling for compressible flows, with particular interest in slip effects at the wall, relevant to time-dependent Poiseuille flow and extrudate-swell. Brujan (1999) has derived an equation of motion for bubble dynamics, incorporating the effects of compressibility and viscoelastic properties for an Oldroyd-B model fluid. The objective there has been to analyse the physics of cavitation. In Barrett and Gotts (2002), the equations of motion have been transformed into the Laplace domain to analyse a compressible dynamic viscoelastic hollow sphere problem. Hao and Pan (2007) proposed a finite element/operator-splitting method for simulating viscoelastic flow at high Weissenberg numbers. This scheme is stable when simulating lid-driven cavity Stokes flow at high-Weissenberg numbers. Guaily and Epstein (2010) proposed a unified purely hyperbolic model for Maxwell fluids. Misoulis and Hatzikiriakos (2009) re-examined the rheological constitutive equation proposed by Patil et al. (2006) to model the flow of polytetrafluoroethylene (PTFE) paste taking into account the significant compressibility of the paste and implementing the slip law based on the consistent normal-to-the-surface unit vector. The proposed model by Patil et al. (2006) was solved for different cases of extrusion rate and die geometry, with the emphasis being on the relative effects of compressibility and slip. Both of them found to be important and alter significantly the results for velocity, pressure, and other parameters, and of course on the flow rate.

### Least-Squares Finite Element Method (LSFEM)

Least-squares schemes for approximating the solution to differential equations were proposed some time ago as a particular variant of the method of weighted residuals. The basic idea is quite straightforward. Given a trial solution expansion with unknown coefficients and satisfying the boundary conditions, construct the corresponding residual for the differential equations. Next, minimise the integral mean square residual to generate an algebraic system. Finally, solve this algebraic system to determine the coefficients and hence the approximation. This approach is appealing because the resulting algebraic system is symmetric and positive definite for a first-order system of differential equations. A theoretical analysis of a class of least-squares methods for the approximate solution of elliptic differential equations was discussed by Varga (1971). Baker (1973) obtained error estimates for the least-squares finite element approximation of the Dirichlet problem for the Laplacian. When compared with the Galerkin finite element method, least-squares finite element methods generally lead to more stringent continuity requirements for the trial functions. Lynn and Arya (1973); and Zienkiewicz et al. (1974) demonstrated that it is possible to reduce the order of continuity requirements at the expense of introducing more unknowns, by first transforming the original differential equation into an equivalent system of first-order differential equations. Subsequently, least-squares finite element methods were applied to boundary-layer flow problems as shown by Lynn and Alani (1976). Fix and Gunzburger (1978) also studied a least-squares method for systems of first-order equations and applied it to the Tricomi equation. A least-squares finite element formulation for the Euler equations governing inviscid compressible flow was proposed by Fletcher (1979). An important feature of his formulation is the designation of groups of variables rather than single variables. Jiang and Chai (1980) applied the least-squares finite element method to a first-order quasi-linear system for compressible potential flow. A least-squares collocation finite element method was investigated by Kwok et al. (1977); and Carey et al. (1980). High-order elements were used for the second-order problem rather than a lower order system being introduced. More recently, least-squares finite element methods have received considerable attention in relation to transonic full potential flow calculations and numerical solution of the Navier–Stokes equations for incompressible viscous flow (Bristeau et al., 1979, 1985). Carey and Jiang (1987) developed a systematic procedure for constructing a least-squares finite element method for partial differential equations. The problem is first recast as a first-order system and the least-squares residual used to form a variational statement. Carey et al. (1998) described a least-squares mixed finite element method and supporting error estimates and briefly summarised some computational results for linear elliptic (steady diffusion) problems. The extension to the stationary Navier–Stokes problems for Newtonian, generalised Newtonian and viscoelastic fluids is then considered. Pontaza et al. (2004) used the approach described by Carey and Jiang (1987) to solve both the Euler and Navier–Stokes equations for the compressible regime. Bolton and Thatcher (2006) used the LSFEM to solve the Navier–Stokes equations in the form of stress and stream functions. Pontaza and Reddy (2006) have presented least-squares based finite element formulations, as an alternate approach to the well-known weak form Galerkin finite element formulations. Gerritsma et al. (2008) described a least-squares spectral element formulation in which time stepping was used to reach steady-state solutions. The time integration method provides artificial diffusion, which suppresses the oscillations in the vicinity of discontinuities. Ahmadi et al. (2009) presented a framework and derivations of 2D higher order global differentiability approximations for 2D distorted quadrilateral elements widely used by LSFEM. Guaily and Megahed (2010) used the LSFEM coupled with a directionally adaptive grid to capture shock waves in compressible inviscid flows.

The present article is organised as follows: the governing equations are presented in second section. In third section, the disadvantage of the standard formulation is explained, followed by our proposed remedy for this disadvantage. In fourth section, we formulate the governing equations in a matrix form. Then, in fifth section, we classify the resulting system of equations. In sixth section, the time integration procedure is explained then the LSFEM formulation is presented in seventh section followed by a description of the method used to describe the boundary conditions in eighth section. In nineth section, numerical results are presented for the sake of validation. tenth section consists of a summary and conclusions.