Robert G. Owens

An energy estimate for the Oldroyd B model: theory and applications

Abstract In this paper, we present energy estimates for the stresses and velocity components in a general setting, for both inertial and inertialess flows of an Oldroyd B fluid. Our results apply to flows in bounded domains in any number of dimensions, subject to Dirichlet and possibly inflow boundary conditions. A novel numerical scheme is introduced and shown to be superior to a conventional Galerkin discretization of the Oldroyd B equations. In particular, the new scheme respects the derived energy estimates and guarantees positive definiteness of the stress tensor τ +((1−β)/We) I at all times, β being a solvent-to-total viscosity ratio and We a Weissenberg number. Numerical results for the planar viscoelastic Poiseuille problem illustrate some differences between the new and conventional schemes and reveal that the conventional scheme may lead to violation of the theoretical energy bounds in certain circumstances.

A locally-upwinded spectral technique (LUST) for viscoelastic flows

Abstract In this paper, we develop an SUPG spectral element scheme suitable for computations of viscoelastic flows at high Deborah numbers. The novelty of the scheme lies in the derivation of the upwinding factors used in the perturbed test tensors of the weak form of the equations. These factors are related to the Deborah number and to the local mesh spacing and have been derived using the superconsistency ideas of Funaro [SIAM J. Numer. Anal. 30 (1993) 1664–1676; J. Sci. Comp. 12 (1997) 385–394; Comput. Math. Appl. 33 (1997) 95–103]. Results of applying the method to the benchmark problem of flow past a single confined cylinder demonstrate the superior stability and accuracy properties of the new scheme when compared with the SUPG spectral element method used previously by the authors [J. Non-Newtonian Fluid Mech. 95 (2000) 1–33; Comput. Meth. Appl. Mech. Engrg. 190 (2001) 3999–4018]. The new method gives excellent agreement with reference results in the literature.

Fokker–Planck simulations of fast flows of melts and concentrated polymer solutions in complex geometries

In 1999, Ottinger introduced a thermodynamically admissible reptation model incorporating chain stretching, anisotropic tube cross sections, double reptation, and the convective constraint release mechanism. In this paper, we describe and use a new high-order Fokker–Planck-based numerical method for the simulation of the Ottinger model in complex geometries. Evidence, in the case of startup homogeneous flows, of the significant CPU time advantage (for comparable levels of accuracy) of our method over a stochastic simulation [Fang et al. (2000)], is presented. For the confined cylinder benchmark problem, differences in the drag behavior observed between the Ottinger model and those of Doi and Edwards (1978a, 1978b, 1978c) and Mead et al. (1998) are explained in terms of double reptation and the differing relaxation spectra.

A new spectral element method for the reliable computation of viscoelastic flow

Abstract A new stabilised spectral element method has been developed by the authors for the accurate integration of the mixed elliptic–hyperbolic system of partial differential equations governing certain viscoelastic flows. The method is illustrated by solving the benchmark problem of the flow of an Oldroyd-B and a PTT fluid past a cylinder in a channel. Results are presented to demonstrate the advantages of the proposed method over traditional Galerkin-type procedures in terms of accuracy and stability.

Rheological models for blood

Rheology is the science of the deformation and flow of materials. It deals with the theoretical concepts of kinematics, conservation laws and constitutive relations, describing the interrelation between force, deformation and flow. The experimental determination of the rheological behaviour of materials is called rheometry. The object of haemorheology is the application of rheology to the study of flow properties of blood and its formed elements, and the coupling of blood and the blood vessels in living organisms. This field involves the investigation of the macroscopic behaviour of blood determined in rheometric experiments, its microscopic properties in vitro and in vivo and studies of the interactions among blood cellular components and between these components and the endothelial cells that line blood vessels.

A non-homogeneous constitutive model for human blood. Part 1. Model derivation and steady flow

Reference GEOLEP-ARTICLE-2008-011doi:10.1017/S002211200800428XView record in Web of Science Record created on 2008-02-21, modified on 2016-08-08

How accurate is your solution?: Error indicators for viscoelastic flow calculations

Abstract The present work is an extension of earlier results of Owens [R.G. Owens, Comput. Methods Appl. Mech. Eng. 164 (1998) 375–395] and is an attempt to provide some theoretical undergirding to the question of appropriate error indicators for numerical solutions of flows of viscoelastic fluids having a differential constitutive equation. In particular, it is shown that the local elemental residual for the elastic stresses only accounts for the so-called stress cell error and as a consequence is, in general, an inadequate measure of the local error. An improved error indicator is then proposed which takes account of the transmitted error. Acknowledging that the stress errors form the major contribution to our error indicator for sufficiently large Deborah numbers we have devised a method of computing the error indicator on an element-by-element basis. Numerical results are presented to show how the approximate error and the ‘exact’ error obtained by calculating the difference between the numerical solution and a reference calculation decrease with mesh refinement. As an illustration of the use of our error indicator we proceed to describe an adaptive spectral element method for flow of an Oldroyd-B fluid past a sphere in a tube. The use of consistent upwinding is shown to result in greater accuracy and stability than is possible with a Galerkin approach. The results are compared with those in the literature.

Spectral approximations on the triangle

In this paper we describe a new family of polynomials which are eigenfunctions of a singular Sturm–Liouville problem on the triangle T 2 ={(x,y):x≥0,y≥0,x+y1}. The polynomials are shown to be orthogonal over T2 with respect to a unit weight function, and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in T2. The zeros of the polynomials may be used in cubature formulae on T2.

The Langevin and Fokker–Planck Equations in Polymer Rheology

Publisher Summary This chapter discusses the applications of Langevin and Fokker–Planck equations in polymer rheology. It presents the stochastic simulation techniques for solving the Langevin equation. It introduces the stochastic differential equations for dilute polymer solutions modeled by dumbbells. Micro-macro techniques for simulating flows of polymeric fluids are discussed in the chapter. These methods are based on coupling macroscopic techniques for solving the conservation equations with microscopic methods for determining the polymeric stress in the fluid. Some of the early attempts to reduce the statistical error in the stochastic simulations without increasing the number of realizations are described in the chapter. Some of the major advances in the development and implementation of micro-macro techniques presented, such as the method of Brownian configuration fields of Hulsen, van Heel, and van den Brule. The chapter also describes efficient implicit schemes for micro-macro simulations developed by Laso, Ramirez, and Picasso. These schemes give rise to a large nonlinear system of algebraic equations for both the macroscopic and microscopic degrees of freedom at each time step with efficiency being achieved using size reduction techniques. A brief account of the solution of stochastic differential equations for linear polymer melts based on the Doi–Edwards model is discussed in the chapter. The deterministic numerical methods based on the Fokker–Planck equation for several kinetic theory models of polymer fluids are discussed in the chapter.

STEADY VISCOELASTIC FLOW PAST A SPHERE USING SPECTRAL ELEMENTS

The steady flow of a viscoelastic fluid past a sphere in a cylindrical tube is considered. A spectral element method is used to solve the system of coupled non-linear partial differential equations governing the flow. The spectral element method combines the flexibility of the traditional finite element method with the accuracy of spectral methods. A time-splitting algorithm is used to determine the solution to the steady problem. Results are presented for the Oldroyd B model. These show excellent agreement with the literature. The results converge with mesh refinement. A limiting Deborah number of approximately 0⋅6 is found, irrespective of the spatial resolution.