Timothy Phillips - Academia.edu (original) (raw)

Papers by Timothy Phillips

Physical Review E, Dec 11, 2008

In this reply to the comment by Lallemand and Luo, we defend our assertion that the alternative a... more In this reply to the comment by Lallemand and Luo, we defend our assertion that the alternative approach for the solution of the dispersion relation for a generalised lattice Boltzmann dispersion equation presented in [1] is mathematically transparent, elegant, and easily justified. Furthermore, the rigorous perturbation analysis used in [1] does not require the reciprocals of the relaxation parameters to be small.

Nucleation and Atmospheric Aerosols, 2007

Journal of Computational Physics, Mar 1, 1984

Iterative methods are considered for the solution of a coupled pair of second order elliptic part... more Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways-by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem.

SIAM journal on scientific and statistical computing, 1986

A compact difference scheme is described for treating the first-order system of PDE's which descr... more A compact difference scheme is described for treating the first-order system of PDE's which describe the equilibrium equations of an elastic body. An algebraic simplification enables the solution to be obtained by standard direct or iterative techniques.

Computational Rheology, 2002

International Journal for Numerical Methods in Engineering, 1996

The steady flow of a viscoelastic fluid past a sphere in a cylindrical tube is considered. A spec... more 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.

Physics of Fluids, Aug 1, 2020

The linear stability analysis of Rivlin-Ericksen fluids of second order is investigated for bound... more The linear stability analysis of Rivlin-Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.

The development of accurate and stable timedependent approximations to viscoelastic flow problems... more The development of accurate and stable timedependent approximations to viscoelastic flow problems remains a priority and challenge for researchers in this field. The mathematical statement of viscoelastic flow problems is invariably in terms of velocity, pressure and extra-stress, the so-called mixed formulation. This class of flow problems is characterized by a nonlinear rheological equation of state which relates the extra-stress tensor to the rate of deformation tensor. For high values of the Deborah or Weissenberg numbers it is the discretization of this equation which is the cause of so many numerical difficulties due to the dominating influence of the convective terms.

Journal of Rheology, Nov 1, 1993

The flow of an Oldroyd-B fluid through an undulating tube is considered. The effect of elasticity... more The flow of an Oldroyd-B fluid through an undulating tube is considered. The effect of elasticity and inertia on flow resistance is investigated numerically using a time-splitting technique. This technique can be used to solve both steady and transient viscoelastic flows. A pseudospectral method based on mixed Fourier–Chebyshev expansions is used to represent the flow variables in space. An approximation space for pressure is constructed which is compatible with that for the velocity. This is achieved by removing the spurious modes using a singular value decomposition. A projection method ensures that mass is conserved identically at the collocation points. Numerical results are presented in such a way as to highlight the inertial and elastic effects. To this end two sets of results are given: the first, inertialess viscoelastic flow; the second, flow at nonzero Reynolds number holding the Weissenberg number constant. The two cases are shown to have quite opposite effects upon the flow rate and resistance.

Journal of Computational and Applied Mathematics, Aug 1, 1994

The governing equations for Stokes flow are formulated in terms of a stream function and Airy str... more The governing equations for Stokes flow are formulated in terms of a stream function and Airy stress function. This formulation ensures that mass and momentum are conserved identically. In terms of these new variables, the equations of motion are written as a second-order elliptic system. These equations are embedded in biharmonic equations and the boundary conditions appropriate for this higher-order system are determined using a least-squares process. This technique is applied to the planar stick-slip problem. A numerical solution to the problem is obtained using a spectral domain decomposition method. An algebraic mapping is used to treat the flow domain without truncation. The coefficients in a singular expansion of the stream function about the stick-slip singularity are computed using a post-processing technique.

Journal of Non-newtonian Fluid Mechanics, Sep 1, 2007

ABSTRACT The extended pom-pom (XPP) model was first introduced to eradicate perceived deficiencie... more ABSTRACT The extended pom-pom (XPP) model was first introduced to eradicate perceived deficiencies of the original pom-pom model such as the lack of a second normal stress difference and a discontinuity in the derivative of the extensional viscosity. However, in this paper it is shown that the XPP model itself possesses some disconcerting attributes. In simple steady shear and uniaxial extensional flow, multiple solutions are found. Furthermore, the extended pom-pom model can predict positive as well as negative values for the second normal stress difference. Experimentally, the second normal stress difference is negative for polymer melts.

Handbook of Numerical Analysis, 2011

Publisher Summary This chapter discusses the applications of Langevin and Fokker–Planck equations... more 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.

Computers & Fluids, Nov 1, 1997

A muli-domain spectral collocation method is developed for the approximation of the Stokes proble... more A muli-domain spectral collocation method is developed for the approximation of the Stokes problem in two dimensions. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well-posed. The collocation scheme possesses the property that the continuity equation is satisfied at all the collocation points. In addition, for problems defined in domains which are rectangularly decomposable, the scheme yields velocity approximations that are globally divergence-free.

Journal of Non-newtonian Fluid Mechanics, Sep 1, 2004

The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom... more The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom model [J. Rheol. 42 (1998) 81]. The pom-pom model is a coarse-grained molecular model that was developed for describing branched polymers. A description of the configuration distribution is given in terms of the orientation and stretch of the pom-pom molecule. The variant of

Journal of Non-newtonian Fluid Mechanics, Sep 1, 2005

ABSTRACT

Rheologica Acta, May 12, 2010

It was jointly organized by the British Society of Rheology (BSR), the Wales Institute of Non-New... more It was jointly organized by the British Society of Rheology (BSR), the Wales Institute of Non-Newtonian Fluid Mechanics (INNFM), and Cardiff University, under the auspices of the European Society of Rheology (ESR). The meeting attracted about 300 participants from 33 countries. The scientific program consisted of five parallel sessions with the Weissenberg Award lecture, two BSR Gold Medal lectures and two plenary and 11 keynote presentations; 160 contributed oral presentations, and 116 contributed poster presentations. The keynote and contributed oral and poster presentations focused on either mainstream or emerging areas of rheology as follows (the symposia chairs are given in parentheses): (1) Blends, Copolymers and Nanocomposites (João Maia and Paula Moldenaers); (2) Process Modelling (Jae Hyun and Don Baird); (3) Rheometry and Beyond: Advanced Experimental Methods (Gareth McKinley and Susan Muller); (4) Interfacial Phenomena, Surfactants and Foams (Simon Cox and Gerry Fuller); (5) Suspensions and Colloids (Richard Buscall and Geoff Maitland); (6) Viscoplasticity, Solids Rheology (Corneliu Balan, Igor

Journal of Non-newtonian Fluid Mechanics, 2009

We present stable and accurate spectral element methods for predicting the steady-state flow of b... more We present stable and accurate spectral element methods for predicting the steady-state flow of branched polymer melts past a confined cylinder. The fluid is modelled using a modification of the pom-pom model known as the single eXtended Pom-Pom (XPP) model, where we have included a multi-mode model of a commercial low-density polyethylene. We have analyzed the XPP model and found interesting multiple solutions for certain choices of the parameters which indicate possible problems with the model. The operator-integration-factor-splitting technique is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

We numerically study multiple droplet impingements. The numerical method consists of a CLSVOF (co... more We numerically study multiple droplet impingements. The numerical method consists of a CLSVOF (coupled level set and volume-of-fluid) framework, CIP-CSL (constraint interpolation profile-Conservative Semi-Lagrange) method, VSIAM3 (volume/surface integrated average based multi-moment method) and a CSF (continuum surface force) model. The numerical framework can robustly simulate multiple droplet impingements.

Journal of Computational and Applied Mathematics, Apr 1, 2007

New spectral element basis functions are constructed for problems possessing an axis of symmetry.... more New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.