# Professor Timothy Phillips

Chair in Applied Mathematics

- PhillipsTN@cardiff.ac.uk
- +44 29208 74194
- Abacws, Room 5.04, Senghennydd Road, Cathays, Cardiff, CF24 4AG

- Available for postgraduate supervision

## Overview

### Research interests

My research interests are in non-Newtonian fluid mechanics and computational fluid dynamics and involves the mathematical modelling of complex fluids and the development of computational techniques for predicting complex flows. My research has been funded by the Engineering and Physical Sciences Research Council since 1985 and I have also received funding from the Leverhulme Trust, the Royal Society, EU and Industry. My most recent substantial EPSRC grant was a Portfolio Partnership Award in the area of Complex Fluids and Complex Flows in collaboration with the School of Engineering at Swansea. This award allowed me to develop mathematical modelling and numerical simulation tools for applications including lubrication and cavitation bubble dynamics. I am currently participating in the UK Consortium on Mesoscale Engineering Sciences (UKCOMES) funded by EPSRC.

My most recent work has been concerned with the development of mathematical models for compressible viscoelastic fluids. This has required an understanding of thermodynamics of complex systems. I also work on the immersed boundary method for fluid-structure interaction problems, numerical methods for multiphase flows, biomedical applications of bubble dynamics, modelling of groundwater flow using stochastic finite elements, proper generalized decomposition for elliptic systems and a range of numerical techniques for solving the Fokker-Planck equation. I have a particular interest in spectral element methods and the lattice Boltzmann method.

### Research group

### Administrative duties

- Member of School Research Committee
- Director of EDI
- Chair of School EDI Committee

## Publication

### 2023

- Leonenko, G. and Phillips, T. N. 2023. Transient numerical approximation of hyperbolic diffusions and beyond. Journal of Computational and Applied Mathematics 422, article number: 114893. (10.1016/j.cam.2022.114893)
- Doherty, W., Phillips, T. N. and Xie, Z. 2023. A stabilised finite element framework for viscoelastic multiphase flows using a conservative level-set method. Journal of Computational Physics 477, article number: 111936. (10.1016/j.jcp.2023.111936)
- Vargas, R. O., Gómez-López, A., Escandón, J. P., Mil-Martı́nez, R. and Phillips, T. 2023. Multiscale modeling of complex fluids under SAOS and LAOS using a combined FENE transient network model. Physica Scripta 98(2), article number: 25213. (10.1088/1402-4896/acb23b)

### 2022

- Fanelli, C., Kaouri, K., Phillips, T. N., Myers, T. G. and Font, F. 2022. Magnetic nanodrug delivery in non-Newtonian blood flows. Microfluidics and Nanofluidics 26(10), article number: 74. (10.1007/s10404-022-02576-6)

### 2021

- Mackay, A. T. and Phillips, T. N. 2021. Compressible and nonisothermal viscoelastic flow between eccentrically rotating cylinders. Theoretical and Computational Fluid Dynamics 35, pp. 735-756. (10.1007/s00162-021-00582-y)

### 2020

- Traverso, L. and Phillips, T. N. 2020. Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: random lognormal permeability. International Journal for Numerical Methods in Fluids 92(11), pp. 1626-1652. (10.1002/fld.4842)
- Cracco, M., Davies, C. and Phillips, T. N. 2020. Linear stability of the flow of a second order fluid past a wedge. Physics of Fluids 32, article number: 84102. (10.1063/5.0018300)

### 2019

- MacKay, A. T. and Phillips, T. N. 2019. On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework. Journal of Non-Newtonian Fluid Mechanics 266, pp. 59-71. (10.1016/j.jnnfm.2019.02.006)
- Phillips, T. N., Oldham, D. and Davies, J. H. 2019. Towards global SEM mantle convection simulations on Polyhedral-based grids. Journal of Computational and Applied Mathematics 348, pp. 48-57. (10.1016/j.cam.2018.08.042)

### 2018

- Jafari, A., Chitsaz, A., Nouri, R. and Phillips, T. N. 2018. Property preserving reformulation of constitutive laws for the conformation tensor. Theoretical and Computational Fluid Dynamics 32, pp. 789-803. (10.1007/s00162-018-0476-y)

### 2017

- Croft, T. L. D. and Phillips, T. N. 2017. Least-squares proper generalized decompositions for weakly coercive elliptic problems. SIAM Journal on Scientific Computing 39(4), pp. A1366-A1388. (10.1137/15M1049269)
- Kynch, R. and Phillips, T. N. 2017. A high resolution spectral element approximation of viscoelastic flows in axisymmetric geometries using a DEVSS-G/DG formulation. Journal of Non-Newtonian Fluid Mechanics 240, pp. 15-33. (10.1016/j.jnnfm.2016.12.008)

### 2016

- Walters, M. J. and Phillips, T. N. 2016. A non-singular boundary element method for modelling bubble dynamics in viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics 235, pp. 109-124. (10.1016/j.jnnfm.2016.07.012)
- Rowlatt, C. F. and Phillips, T. N. 2016. A spectral element formulation of the immersed boundary method for Newtonian fluids. Computer Methods in Applied Mechanics and Engineering 298, pp. 29-57. (10.1016/j.cma.2015.08.010)

### 2015

- Claus, S., Cantwell, C. D. and Phillips, T. N. 2015. Spectral/hp element methods for plane Newtonian extrudate swell. Computers & Fluids 116, pp. 105-117. (10.1016/j.compfluid.2015.04.002)
- Leonenko, G. M. and Phillips, T. N. 2015. Numerical approximation of high-dimensional Fokker-Planck equations with polynomial coefficients. Journal of computational and applied mathematics 273, pp. 296-312. (10.1016/j.cam.2014.05.024)

### 2014

- Traverso, L., Phillips, T. N. and Yang, Y. 2014. Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients. International Journal for Numerical Methods in Fluids 74(5), pp. 359-385. (10.1002/fld.3854)
- Phillips, T. N. and Rowlatt, C. F. 2014. The extended spectral element method for the approximation of discontinuous functions. Siam Journal on Numerical Analysis

### 2013

- Claus, S. and Phillips, T. N. 2013. Viscoelastic flow around a confined cylinder using spectral/hp element methods. Journal of Non-Newtonian Fluid Mechanics 200, pp. 131-146. (10.1016/j.jnnfm.2013.03.004)
- Lind, S. J. and Phillips, T. N. 2013. Bubble collapse in compressible fluids using a spectral element marker particle method. Part 2. Viscoelastic fluids. International Journal for Numerical Methods in Fluids 71(9), pp. 1103-1130. (10.1002/fld.3701)
- Lind, S. J. and Phillips, T. N. 2013. The effect of viscoelasticity on the dynamics of gas bubbles near free surfaces. Physics of Fluids 25(2), pp. 022104-022135. (10.1063/1.4790512)
- Traverso, L., Phillips, T. N. and Yang, Y. 2013. Mixed finite element methods for groundwater flow in heterogeneous aquifers. Computers and Fluids 88, pp. 60-80. (10.1016/j.compfluid.2013.08.018)

### 2012

- Lind, S. J. and Phillips, T. N. 2012. Bubble collapse in compressible fluids using a spectral element marker particle method. Part 1. Newtonian fluids. International Journal for Numerical Methods in Fluids 70(9), pp. 1167-1187. (10.1002/fld.2737)
- Lind, S. J. and Phillips, T. N. 2012. The effect of viscoelasticity on the dynamics of two gas bubbles near a rigid boundary. IMA Journal of Applied Mathematics 77(5), pp. 652-677. (10.1093/imamat/hxs041)
- Leonenko, G. M. and Phillips, T. N. 2012. High-order approximation of Pearson diffusion processes. Journal of Computational and Applied Mathematics 236(11), pp. 2853-2868. (10.1016/j.cam.2012.01.022)
- Oldham, D. N., Davies, J. H. and Phillips, T. N. 2012. Generic polyhedron grid generation for solving partial differential equations on spherical surfaces. Computers & Geosciences 39, pp. 11-17. (10.1016/j.cageo.2011.06.004)
- Bollada, P. C. and Phillips, T. N. 2012. On the mathematical modelling of a compressible viscoelastic fluid. Archive for Rational Mechanics and Analysis 205(1), pp. 1-26. (10.1007/s00205-012-0496-5)

### 2011

- Phillips, T. N. and Roberts, G. W. 2011. Lattice Boltzmann models for non-Newtonian flows. IMA Journal of Applied Mathematics 76(5), pp. 790-816. (10.1093/imamat/hxr003)
- Lind, S. J. and Phillips, T. N. 2011. The influence of viscoelasticity on the collapse of cavitation bubbles near a rigid boundary. Theoretical and Computational Fluid Dynamics 26(1-4), pp. 245-277. (10.1007/s00162-011-0227-9)
- Russo, G. and Phillips, T. N. 2011. Spectral element predictions of die-swell for Oldroyd-B fluids. Computers & Fluids 43(1), pp. 107-118. (10.1016/j.compfluid.2010.09.028)
- Lozinski, A., Owens, R. G. and Phillips, T. N. 2011. The Langevin and Fokker-Planck equations in polymer rheology. In: Glowinski, R. and Xu, J. eds. Handbook of Numerical Analysis - Special volume: Numerical Methods for Non-Newtonian Fluids., Vol. 16. Amsterdam: Elsevier, pp. 211-303., (10.1016/B978-0-444-53047-9.00002-2)
- Leonenko, G. M. and Phillips, T. N. 2011. The prediction of plane Couette flow for a FENE fluid using a reduced basis approximation of the Fokker-Planck equation. International Journal for Multiscale Computational Engineering 9(1), pp. 73-88. (10.1615/IntJMultCompEng.v9.i1.60)

### 2010

- Russo, G. and Phillips, T. N. 2010. Numerical simulation of steady planar die swell for a Newtonian fluid using the spectral element method. Computers & Fluids 39(5), pp. 780-792. (10.1016/j.compfluid.2009.12.006)
- Lind, S. J. and Phillips, T. N. 2010. The effect of viscoelasticity on a rising gas bubble. Journal of non-Newtonian fluid mechanics 165(15-16), pp. 852-865. (10.1016/j.jnnfm.2010.04.002)
- Lind, S. J. and Phillips, T. N. 2010. Spherical bubble collapse in viscoelastic fluids. Journal of non-Newtonian fluid mechanics 165(1-2), pp. 56-64. (10.1016/j.jnnfm.2009.09.002)
- Russo, G. and Phillips, T. N. 2010. Numerical prediction of extrudate swell of branched polymer melts. Rheologica Acta 49(6), pp. 657-676. (10.1007/s00397-009-0426-0)
- Yokoi, K., Kay, P. J., Bowen, P. J. and Phillips, T. N. 2010. Numerical Studies of Multiple Drop Impacts onto a Flat Substrate. Presented at: 23rd European Conference on Liquid Atomisation and Spray Systems (ILASS), Brno, Czech Republic.

### 2009

- Bollada, P. C. and Phillips, T. N. 2009. A modified deformation field method for integral constitutive models. Journal of Non-Newtonian Fluid Mechanics 163(1-3), pp. 78-87. (10.1016/j.jnnfm.2009.07.001)
- Leonenko, G. M. and Phillips, T. N. 2009. On the solution of the Fokker-Planck equation using a high-order reduced basis approximation. Computer Methods in Applied Mechanics and Engineering 199(1-4), pp. 158-168. (10.1016/j.cma.2009.09.028)
- Bierbrauer, F., Bollada, P. C. and Phillips, T. N. 2009. A consistent reflected image particle approach to the treatment of boundary conditions in smoothed particle hydrodynamics. Computer Methods in Applied Mechanics and Engineering 198(41-44), pp. 3400-3410. (10.1016/j.cma.2009.06.014)
- Inkson, N., Phillips, T. N. and van Os, R. G. M. 2009. Numerical simulation of flow past a cylinder using models of XPP type. Journal of non-Newtonian fluid mechanics 156(1-2), pp. 7-20. (10.1016/j.jnnfm.2008.06.004)
- Vargas, R. O., Manero, O. and Phillips, T. N. 2009. Viscoelastic flow past confined objects using a micro-macro approach. Rheologica Acta 48(4), pp. 373-395. (10.1007/s00397-008-0323-y)

### 2008

- Bollada, P. C. and Phillips, T. 2008. An anisothermal, compressible, piezoviscous model for journal-bearing lubrication. International Journal for Numerical Methods in Fluids 58(1), pp. 27-55. (10.1002/fld.1705)
- Bierbrauer, F. and Phillips, T. N. 2008. Secondary atomisation: simulation of droplet break-up in disturbed flow fields. Proceedings of the 22nd European Conference on Liquid Atomisation and Spray Systems, Como, Italy, 2008, Paper ILASS08-A029
- Bierbrauer, F. and Phillips, T. N. 2008. Modelling spray impingement onto flat, rigid walls using an Eulerian-Lagrangian method. Proceedings of the 22nd European Confrerence on Liquid Atomisation and Spray Systems, Como, Italy, 2008 Paper ILASS08-030
- Gerritsma, M. I. and Phillips, T. N. 2008. On the characteristics and compatibility equations for the UCM model fluid. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM) 88(7), pp. 523-539. (10.1002/zamm.200700058)
- Reis, T. and Phillips, T. N. 2008. Numerical validation of a consistent axisymmetric lattice Boltzmann model. Physical Review E 77(2), article number: 26703. (10.1103/PhysRevE.77.026703)
- Reis, T. and Phillips, T. N. 2008. Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation. Physical Review -Series e- 77(2), article number: 26702. (10.1103/PhysRevE.77.026702)
- Bollada, P. C. and Phillips, T. N. 2008. A physical decomposition of the stress tensor for complex flows. Rheologica Acta 47(7), pp. 719-725. (10.1007/s00397-007-0256-x)
- Bierbrauer, F. and Phillips, T. N. 2008. The numerical prediction of droplet deformation and break-up using the Godunov marker-particle projection scheme. International Journal for Numerical Methods in Fluids 56(8), pp. 1155-1160. (10.1002/fld.1710)
- Gwynllyw, D. R. and Phillips, T. N. 2008. The influence of Oldroyd-B and PTT lubricants on moving journal bearing systems. Journal of Non-Newtonian Fluid Mechanics 150(2-3), pp. 196-210. (10.1016/j.jnnfm.2007.10.014)
- Leonenko, G. M. and Phillips, T. N. 2008. Reduced basis function method for solving the 2D Fokker-Planck equation. Presented at: SASOR 2008: 2nd Southern African Conference of Rheology, Cape Town, South Africa, 6-8 October 2008.

### 2007

- Bollada, P. C. and Phillips, T. N. 2007. On the effects of a compressible viscous lubricant on the load-bearing capacity of a journal bearing. International Journal for Numerical Methods in Fluids 55(11), pp. 1091-1120. (10.1002/fld.1545)
- Reis, T. and Phillips, T. N. 2007. Modified lattice Boltzmann model for axisymmetric flows. Physical Review E 75(5), article number: 56703. (10.1103/PhysRevE.75.056703)
- van Os, R. G. M. and Phillips, T. N. 2007. The choice of spectral element basis functions in domains with an axis of symmetry. Journal of computational and applied mathematics 201(1), pp. 217-229. (10.1016/j.cam.2006.02.016)
- Aguayo, J. P., Phillips, P. M., Phillips, T. N., Tamaddon-Jahromi, H. R., Snigerev, B. A. and Webster, M. F. 2007. The numerical prediction of planar viscoelastic contraction flows using the pom-pom model and higher-order finite volume schemes. Journal of Computational Physics 220(2), pp. 586-611. (10.1016/j.jcp.2006.05.039)
- Inkson, N. and Phillips, T. N. 2007. Unphysical phenomena associated with the extended pom-pom model in steady flow. Journal of non-Newtonian fluid mechanics 145(2-3), pp. 92-101. (10.1016/j.jnnfm.2007.05.002)
- Reis, T. and Phillips, T. N. 2007. Lattice Boltzmann model for simulating immiscible two-phase flows. Journal of Physics A: Mathematical and Theoretical 40(14), article number: 4033. (10.1088/1751-8113/40/14/018)
- Mavron, V. C. and Phillips, T. N. 2007. Elements of mathematics for economics and finance. London: Springer.

### 2006

- Binding, D. M., Phillips, P. M. and Phillips, T. N. 2006. Contraction/expansion flows: the pressure drop and related issues. Journal of non-Newtonian fluid mechanics 137(1-3), pp. 31-38. (10.1016/j.jnnfm.2006.03.006)
- Ervin, V. J. and Phillips, T. N. 2006. Residual a posteriori error estimator for a three-field model of a generalized Stokes problem. Computer Methods in Applied Mechanics and Engineering 195(19-22), pp. 2599-2610. (10.1016/j.cma.2005.05.019)
- Phillips, T. N. and Rh, G. D. 2006. On the enforcement of the zero mean pressure condition in the spectral element approximation of the Stokes problem. Computer Methods in Applied Mechanics and Engineering 195(9-12), pp. 1027-1049. (10.1016/j.cma.2005.04.004)
- Bowen, P. J., Phillips, T. N. and Zheng, Y. 2006. Direct numerical simulation of non-Newtonian droplet deformation. Presented at: 10th International Conference on Liquid Atomization and Spray Systems (ICLASS-2006), Kyoto, Japan, 27 August - 1 September 2006.

### 2005

- van Os, R. G. M. and Phillips, T. N. 2005. Efficient and stable spectral element methods for predicting the flow of an XPP fluid past a cylinder. Journal of non-Newtonian fluid mechanics 129(3), pp. 143-162. (10.1016/j.jnnfm.2005.06.004)
- Gwynllyw, D. R. and Phillips, T. N. 2005. Some issues regarding spectral element meshes for moving journal bearing systems. International Journal for Numerical Methods in Fluids 48(4), pp. 423-454. (10.1002/fld.932)

### 2004

- Van Os, R. G. M. and Phillips, T. 2004. Spectral element methods for transient viscoelastic flow problems. Journal of Computational Physics 201(1), pp. 286-314. (10.1016/j.jcp.2004.05.016)
- van Os, R. G. M. and Phillips, T. N. 2004. The prediction of complex flows of polymer melts using spectral elements. Journal of non-Newtonian fluid mechanics 122(1-3), pp. 287-301. (10.1016/j.jnnfm.2003.12.009)
- Momeni-Masuleh, S. H. and Phillips, T. N. 2004. Viscoelastic flow in an undulating tube using spectral methods. Computers & Fluids 33(8), pp. 1075-1095. (10.1016/j.compfluid.2003.09.002)
- Aboubacar, M., Phillips, T. N., Tamaddon-Jahromi, H. R., Snigerev, B. A. and Webster, M. F. 2004. High-order finite volume methods for viscoelastic flow problems. Journal of Computational Physics 199(1), pp. 16-40. (10.1016/j.jcp.2004.01.033)

### 2003

- Zheng, Y., Phillips, T. N. and Bowen, P. J. 2003. Development of DNS Spectral Element Methods for Study of Newtonian Droplet-Gas Systems. Presented at: 9th International Conference on Liquid Atomisation and Spray Systems (ICLASS), Sorrento, Italy.

### 2002

- Phillips, T. N. and Williams, A. J. 2002. Comparison of creeping and inertial flow of an Oldroyd B fluid through planar and axisymmetric contractions. Journal of non-Newtonian fluid mechanics 108(1-3), pp. 25-47. (10.1016/S0377-0257(02)00123-4)
- Owens, R. G. and Phillips, T. N. 2002. Computational rheology. London: Imperial College Press.

### 2001

- Phillips, T. N. and Williams, A. J. 2001. Conservative semi-Lagrangian finite volume schemes. Numerical Methods for Partial Differential Equations 17(4), pp. 403-425. (10.1002/num.1019)
- Gerritsma, M. I. and Phillips, T. N. 2001. On the use of characteristic variables in viscoelastic flow problems. IMA Journal of Applied Mathematics 66(2), pp. 127-147. (10.1093/imamat/66.2.127)
- Phillips, T. N. and Williams, A. J. 2001. A semi-Lagrangian finite volume method for Newtonian contraction flows. SIAM Journal on Scientific Computing 22(6), pp. 2152-2177. (10.1137/S1064827599365288)

### Articles

- Leonenko, G. and Phillips, T. N. 2023. Transient numerical approximation of hyperbolic diffusions and beyond. Journal of Computational and Applied Mathematics 422, article number: 114893. (10.1016/j.cam.2022.114893)
- Doherty, W., Phillips, T. N. and Xie, Z. 2023. A stabilised finite element framework for viscoelastic multiphase flows using a conservative level-set method. Journal of Computational Physics 477, article number: 111936. (10.1016/j.jcp.2023.111936)
- Vargas, R. O., Gómez-López, A., Escandón, J. P., Mil-Martı́nez, R. and Phillips, T. 2023. Multiscale modeling of complex fluids under SAOS and LAOS using a combined FENE transient network model. Physica Scripta 98(2), article number: 25213. (10.1088/1402-4896/acb23b)
- Fanelli, C., Kaouri, K., Phillips, T. N., Myers, T. G. and Font, F. 2022. Magnetic nanodrug delivery in non-Newtonian blood flows. Microfluidics and Nanofluidics 26(10), article number: 74. (10.1007/s10404-022-02576-6)
- Mackay, A. T. and Phillips, T. N. 2021. Compressible and nonisothermal viscoelastic flow between eccentrically rotating cylinders. Theoretical and Computational Fluid Dynamics 35, pp. 735-756. (10.1007/s00162-021-00582-y)
- Traverso, L. and Phillips, T. N. 2020. Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: random lognormal permeability. International Journal for Numerical Methods in Fluids 92(11), pp. 1626-1652. (10.1002/fld.4842)
- Cracco, M., Davies, C. and Phillips, T. N. 2020. Linear stability of the flow of a second order fluid past a wedge. Physics of Fluids 32, article number: 84102. (10.1063/5.0018300)
- MacKay, A. T. and Phillips, T. N. 2019. On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework. Journal of Non-Newtonian Fluid Mechanics 266, pp. 59-71. (10.1016/j.jnnfm.2019.02.006)
- Phillips, T. N., Oldham, D. and Davies, J. H. 2019. Towards global SEM mantle convection simulations on Polyhedral-based grids. Journal of Computational and Applied Mathematics 348, pp. 48-57. (10.1016/j.cam.2018.08.042)
- Jafari, A., Chitsaz, A., Nouri, R. and Phillips, T. N. 2018. Property preserving reformulation of constitutive laws for the conformation tensor. Theoretical and Computational Fluid Dynamics 32, pp. 789-803. (10.1007/s00162-018-0476-y)
- Croft, T. L. D. and Phillips, T. N. 2017. Least-squares proper generalized decompositions for weakly coercive elliptic problems. SIAM Journal on Scientific Computing 39(4), pp. A1366-A1388. (10.1137/15M1049269)
- Kynch, R. and Phillips, T. N. 2017. A high resolution spectral element approximation of viscoelastic flows in axisymmetric geometries using a DEVSS-G/DG formulation. Journal of Non-Newtonian Fluid Mechanics 240, pp. 15-33. (10.1016/j.jnnfm.2016.12.008)
- Walters, M. J. and Phillips, T. N. 2016. A non-singular boundary element method for modelling bubble dynamics in viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics 235, pp. 109-124. (10.1016/j.jnnfm.2016.07.012)
- Rowlatt, C. F. and Phillips, T. N. 2016. A spectral element formulation of the immersed boundary method for Newtonian fluids. Computer Methods in Applied Mechanics and Engineering 298, pp. 29-57. (10.1016/j.cma.2015.08.010)
- Claus, S., Cantwell, C. D. and Phillips, T. N. 2015. Spectral/hp element methods for plane Newtonian extrudate swell. Computers & Fluids 116, pp. 105-117. (10.1016/j.compfluid.2015.04.002)
- Leonenko, G. M. and Phillips, T. N. 2015. Numerical approximation of high-dimensional Fokker-Planck equations with polynomial coefficients. Journal of computational and applied mathematics 273, pp. 296-312. (10.1016/j.cam.2014.05.024)
- Traverso, L., Phillips, T. N. and Yang, Y. 2014. Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients. International Journal for Numerical Methods in Fluids 74(5), pp. 359-385. (10.1002/fld.3854)
- Phillips, T. N. and Rowlatt, C. F. 2014. The extended spectral element method for the approximation of discontinuous functions. Siam Journal on Numerical Analysis
- Claus, S. and Phillips, T. N. 2013. Viscoelastic flow around a confined cylinder using spectral/hp element methods. Journal of Non-Newtonian Fluid Mechanics 200, pp. 131-146. (10.1016/j.jnnfm.2013.03.004)
- Lind, S. J. and Phillips, T. N. 2013. Bubble collapse in compressible fluids using a spectral element marker particle method. Part 2. Viscoelastic fluids. International Journal for Numerical Methods in Fluids 71(9), pp. 1103-1130. (10.1002/fld.3701)
- Lind, S. J. and Phillips, T. N. 2013. The effect of viscoelasticity on the dynamics of gas bubbles near free surfaces. Physics of Fluids 25(2), pp. 022104-022135. (10.1063/1.4790512)
- Traverso, L., Phillips, T. N. and Yang, Y. 2013. Mixed finite element methods for groundwater flow in heterogeneous aquifers. Computers and Fluids 88, pp. 60-80. (10.1016/j.compfluid.2013.08.018)
- Lind, S. J. and Phillips, T. N. 2012. Bubble collapse in compressible fluids using a spectral element marker particle method. Part 1. Newtonian fluids. International Journal for Numerical Methods in Fluids 70(9), pp. 1167-1187. (10.1002/fld.2737)
- Lind, S. J. and Phillips, T. N. 2012. The effect of viscoelasticity on the dynamics of two gas bubbles near a rigid boundary. IMA Journal of Applied Mathematics 77(5), pp. 652-677. (10.1093/imamat/hxs041)
- Leonenko, G. M. and Phillips, T. N. 2012. High-order approximation of Pearson diffusion processes. Journal of Computational and Applied Mathematics 236(11), pp. 2853-2868. (10.1016/j.cam.2012.01.022)
- Oldham, D. N., Davies, J. H. and Phillips, T. N. 2012. Generic polyhedron grid generation for solving partial differential equations on spherical surfaces. Computers & Geosciences 39, pp. 11-17. (10.1016/j.cageo.2011.06.004)
- Bollada, P. C. and Phillips, T. N. 2012. On the mathematical modelling of a compressible viscoelastic fluid. Archive for Rational Mechanics and Analysis 205(1), pp. 1-26. (10.1007/s00205-012-0496-5)
- Phillips, T. N. and Roberts, G. W. 2011. Lattice Boltzmann models for non-Newtonian flows. IMA Journal of Applied Mathematics 76(5), pp. 790-816. (10.1093/imamat/hxr003)
- Lind, S. J. and Phillips, T. N. 2011. The influence of viscoelasticity on the collapse of cavitation bubbles near a rigid boundary. Theoretical and Computational Fluid Dynamics 26(1-4), pp. 245-277. (10.1007/s00162-011-0227-9)
- Russo, G. and Phillips, T. N. 2011. Spectral element predictions of die-swell for Oldroyd-B fluids. Computers & Fluids 43(1), pp. 107-118. (10.1016/j.compfluid.2010.09.028)
- Leonenko, G. M. and Phillips, T. N. 2011. The prediction of plane Couette flow for a FENE fluid using a reduced basis approximation of the Fokker-Planck equation. International Journal for Multiscale Computational Engineering 9(1), pp. 73-88. (10.1615/IntJMultCompEng.v9.i1.60)
- Russo, G. and Phillips, T. N. 2010. Numerical simulation of steady planar die swell for a Newtonian fluid using the spectral element method. Computers & Fluids 39(5), pp. 780-792. (10.1016/j.compfluid.2009.12.006)
- Lind, S. J. and Phillips, T. N. 2010. The effect of viscoelasticity on a rising gas bubble. Journal of non-Newtonian fluid mechanics 165(15-16), pp. 852-865. (10.1016/j.jnnfm.2010.04.002)
- Lind, S. J. and Phillips, T. N. 2010. Spherical bubble collapse in viscoelastic fluids. Journal of non-Newtonian fluid mechanics 165(1-2), pp. 56-64. (10.1016/j.jnnfm.2009.09.002)
- Russo, G. and Phillips, T. N. 2010. Numerical prediction of extrudate swell of branched polymer melts. Rheologica Acta 49(6), pp. 657-676. (10.1007/s00397-009-0426-0)
- Bollada, P. C. and Phillips, T. N. 2009. A modified deformation field method for integral constitutive models. Journal of Non-Newtonian Fluid Mechanics 163(1-3), pp. 78-87. (10.1016/j.jnnfm.2009.07.001)
- Leonenko, G. M. and Phillips, T. N. 2009. On the solution of the Fokker-Planck equation using a high-order reduced basis approximation. Computer Methods in Applied Mechanics and Engineering 199(1-4), pp. 158-168. (10.1016/j.cma.2009.09.028)
- Bierbrauer, F., Bollada, P. C. and Phillips, T. N. 2009. A consistent reflected image particle approach to the treatment of boundary conditions in smoothed particle hydrodynamics. Computer Methods in Applied Mechanics and Engineering 198(41-44), pp. 3400-3410. (10.1016/j.cma.2009.06.014)
- Inkson, N., Phillips, T. N. and van Os, R. G. M. 2009. Numerical simulation of flow past a cylinder using models of XPP type. Journal of non-Newtonian fluid mechanics 156(1-2), pp. 7-20. (10.1016/j.jnnfm.2008.06.004)
- Vargas, R. O., Manero, O. and Phillips, T. N. 2009. Viscoelastic flow past confined objects using a micro-macro approach. Rheologica Acta 48(4), pp. 373-395. (10.1007/s00397-008-0323-y)
- Bollada, P. C. and Phillips, T. 2008. An anisothermal, compressible, piezoviscous model for journal-bearing lubrication. International Journal for Numerical Methods in Fluids 58(1), pp. 27-55. (10.1002/fld.1705)
- Bierbrauer, F. and Phillips, T. N. 2008. Secondary atomisation: simulation of droplet break-up in disturbed flow fields. Proceedings of the 22nd European Conference on Liquid Atomisation and Spray Systems, Como, Italy, 2008, Paper ILASS08-A029
- Bierbrauer, F. and Phillips, T. N. 2008. Modelling spray impingement onto flat, rigid walls using an Eulerian-Lagrangian method. Proceedings of the 22nd European Confrerence on Liquid Atomisation and Spray Systems, Como, Italy, 2008 Paper ILASS08-030
- Gerritsma, M. I. and Phillips, T. N. 2008. On the characteristics and compatibility equations for the UCM model fluid. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM) 88(7), pp. 523-539. (10.1002/zamm.200700058)
- Reis, T. and Phillips, T. N. 2008. Numerical validation of a consistent axisymmetric lattice Boltzmann model. Physical Review E 77(2), article number: 26703. (10.1103/PhysRevE.77.026703)
- Reis, T. and Phillips, T. N. 2008. Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation. Physical Review -Series e- 77(2), article number: 26702. (10.1103/PhysRevE.77.026702)
- Bollada, P. C. and Phillips, T. N. 2008. A physical decomposition of the stress tensor for complex flows. Rheologica Acta 47(7), pp. 719-725. (10.1007/s00397-007-0256-x)
- Bierbrauer, F. and Phillips, T. N. 2008. The numerical prediction of droplet deformation and break-up using the Godunov marker-particle projection scheme. International Journal for Numerical Methods in Fluids 56(8), pp. 1155-1160. (10.1002/fld.1710)
- Gwynllyw, D. R. and Phillips, T. N. 2008. The influence of Oldroyd-B and PTT lubricants on moving journal bearing systems. Journal of Non-Newtonian Fluid Mechanics 150(2-3), pp. 196-210. (10.1016/j.jnnfm.2007.10.014)
- Bollada, P. C. and Phillips, T. N. 2007. On the effects of a compressible viscous lubricant on the load-bearing capacity of a journal bearing. International Journal for Numerical Methods in Fluids 55(11), pp. 1091-1120. (10.1002/fld.1545)
- Reis, T. and Phillips, T. N. 2007. Modified lattice Boltzmann model for axisymmetric flows. Physical Review E 75(5), article number: 56703. (10.1103/PhysRevE.75.056703)
- van Os, R. G. M. and Phillips, T. N. 2007. The choice of spectral element basis functions in domains with an axis of symmetry. Journal of computational and applied mathematics 201(1), pp. 217-229. (10.1016/j.cam.2006.02.016)
- Aguayo, J. P., Phillips, P. M., Phillips, T. N., Tamaddon-Jahromi, H. R., Snigerev, B. A. and Webster, M. F. 2007. The numerical prediction of planar viscoelastic contraction flows using the pom-pom model and higher-order finite volume schemes. Journal of Computational Physics 220(2), pp. 586-611. (10.1016/j.jcp.2006.05.039)
- Inkson, N. and Phillips, T. N. 2007. Unphysical phenomena associated with the extended pom-pom model in steady flow. Journal of non-Newtonian fluid mechanics 145(2-3), pp. 92-101. (10.1016/j.jnnfm.2007.05.002)
- Reis, T. and Phillips, T. N. 2007. Lattice Boltzmann model for simulating immiscible two-phase flows. Journal of Physics A: Mathematical and Theoretical 40(14), article number: 4033. (10.1088/1751-8113/40/14/018)
- Binding, D. M., Phillips, P. M. and Phillips, T. N. 2006. Contraction/expansion flows: the pressure drop and related issues. Journal of non-Newtonian fluid mechanics 137(1-3), pp. 31-38. (10.1016/j.jnnfm.2006.03.006)
- Ervin, V. J. and Phillips, T. N. 2006. Residual a posteriori error estimator for a three-field model of a generalized Stokes problem. Computer Methods in Applied Mechanics and Engineering 195(19-22), pp. 2599-2610. (10.1016/j.cma.2005.05.019)
- Phillips, T. N. and Rh, G. D. 2006. On the enforcement of the zero mean pressure condition in the spectral element approximation of the Stokes problem. Computer Methods in Applied Mechanics and Engineering 195(9-12), pp. 1027-1049. (10.1016/j.cma.2005.04.004)
- van Os, R. G. M. and Phillips, T. N. 2005. Efficient and stable spectral element methods for predicting the flow of an XPP fluid past a cylinder. Journal of non-Newtonian fluid mechanics 129(3), pp. 143-162. (10.1016/j.jnnfm.2005.06.004)
- Gwynllyw, D. R. and Phillips, T. N. 2005. Some issues regarding spectral element meshes for moving journal bearing systems. International Journal for Numerical Methods in Fluids 48(4), pp. 423-454. (10.1002/fld.932)
- Van Os, R. G. M. and Phillips, T. 2004. Spectral element methods for transient viscoelastic flow problems. Journal of Computational Physics 201(1), pp. 286-314. (10.1016/j.jcp.2004.05.016)
- van Os, R. G. M. and Phillips, T. N. 2004. The prediction of complex flows of polymer melts using spectral elements. Journal of non-Newtonian fluid mechanics 122(1-3), pp. 287-301. (10.1016/j.jnnfm.2003.12.009)
- Momeni-Masuleh, S. H. and Phillips, T. N. 2004. Viscoelastic flow in an undulating tube using spectral methods. Computers & Fluids 33(8), pp. 1075-1095. (10.1016/j.compfluid.2003.09.002)
- Aboubacar, M., Phillips, T. N., Tamaddon-Jahromi, H. R., Snigerev, B. A. and Webster, M. F. 2004. High-order finite volume methods for viscoelastic flow problems. Journal of Computational Physics 199(1), pp. 16-40. (10.1016/j.jcp.2004.01.033)
- Phillips, T. N. and Williams, A. J. 2002. Comparison of creeping and inertial flow of an Oldroyd B fluid through planar and axisymmetric contractions. Journal of non-Newtonian fluid mechanics 108(1-3), pp. 25-47. (10.1016/S0377-0257(02)00123-4)
- Phillips, T. N. and Williams, A. J. 2001. Conservative semi-Lagrangian finite volume schemes. Numerical Methods for Partial Differential Equations 17(4), pp. 403-425. (10.1002/num.1019)
- Gerritsma, M. I. and Phillips, T. N. 2001. On the use of characteristic variables in viscoelastic flow problems. IMA Journal of Applied Mathematics 66(2), pp. 127-147. (10.1093/imamat/66.2.127)
- Phillips, T. N. and Williams, A. J. 2001. A semi-Lagrangian finite volume method for Newtonian contraction flows. SIAM Journal on Scientific Computing 22(6), pp. 2152-2177. (10.1137/S1064827599365288)

### Book sections

- Lozinski, A., Owens, R. G. and Phillips, T. N. 2011. The Langevin and Fokker-Planck equations in polymer rheology. In: Glowinski, R. and Xu, J. eds. Handbook of Numerical Analysis - Special volume: Numerical Methods for Non-Newtonian Fluids., Vol. 16. Amsterdam: Elsevier, pp. 211-303., (10.1016/B978-0-444-53047-9.00002-2)

### Books

- Mavron, V. C. and Phillips, T. N. 2007. Elements of mathematics for economics and finance. London: Springer.
- Owens, R. G. and Phillips, T. N. 2002. Computational rheology. London: Imperial College Press.

### Conferences

- Yokoi, K., Kay, P. J., Bowen, P. J. and Phillips, T. N. 2010. Numerical Studies of Multiple Drop Impacts onto a Flat Substrate. Presented at: 23rd European Conference on Liquid Atomisation and Spray Systems (ILASS), Brno, Czech Republic.
- Leonenko, G. M. and Phillips, T. N. 2008. Reduced basis function method for solving the 2D Fokker-Planck equation. Presented at: SASOR 2008: 2nd Southern African Conference of Rheology, Cape Town, South Africa, 6-8 October 2008.
- Bowen, P. J., Phillips, T. N. and Zheng, Y. 2006. Direct numerical simulation of non-Newtonian droplet deformation. Presented at: 10th International Conference on Liquid Atomization and Spray Systems (ICLASS-2006), Kyoto, Japan, 27 August - 1 September 2006.
- Zheng, Y., Phillips, T. N. and Bowen, P. J. 2003. Development of DNS Spectral Element Methods for Study of Newtonian Droplet-Gas Systems. Presented at: 9th International Conference on Liquid Atomisation and Spray Systems (ICLASS), Sorrento, Italy.

- Lind, S. J. and Phillips, T. N. 2013. The effect of viscoelasticity on the dynamics of gas bubbles near free surfaces. Physics of Fluids 25(2), pp. 022104-022135. (10.1063/1.4790512)
- Bollada, P. C. and Phillips, T. N. 2012. On the mathematical modelling of a compressible viscoelastic fluid. Archive for Rational Mechanics and Analysis 205(1), pp. 1-26. (10.1007/s00205-012-0496-5)
- Gerritsma, M. I. and Phillips, T. N. 2008. On the characteristics and compatibility equations for the UCM model fluid. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM) 88(7), pp. 523-539. (10.1002/zamm.200700058)
- Reis, T. and Phillips, T. N. 2008. Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation. Physical Review -Series e- 77(2), article number: 26702. (10.1103/PhysRevE.77.026702)

## Research

### External funding since 2000

- Type-Dependent Computations for Viscoelastic Flows. EPSRC £179,104.
- Masters Training Package in Complex Fluids: Modelling, Computation and Industrial Applications. EPSRC £424,542.
- Direct Numerical Simulation of Droplet-Gas Systems for Power Generation Applications. EPSRC £74,597.
- The Prediction and Validation of Contraction Flows of Complex Fluids. EPSRC £148,161 (with D. M. Binding).
- Portfolio Partnership in Complex Fluids and Complex Flows. EPSRC £3.1M (with UW Swansea).
- Modelling of Mantle Convection. Leverhulme Trust £108,726 (with J. H. Davies).
- UK Consortium On Mesoscale Engineering Sciences (UKCOMES). EPSRC £397,424.

### Major conference talks since 2010

- High-order approximations of viscoelastic flow problems. Invited speaker at Symposium on High Accuracy Flow Simulations, Lausanne, Switzerland, February 2010.
- Mathematical and computational modelling of compressible viscoelastic fluids. 6th Annual European Rheology Conference, Goteborg, Sweden, April 2010.
- Mathematical and computational modelling of compressible viscoelastic fluids. Invited speaker at Computational Challenges in Partial Differential Equations. INI/WIMCS Meeting, Swansea, April 2011.
- High-order approximations of viscoelastic flows. Invited plenary speaker at the Institute of Non-Newtonian Fluid Mechanics Spring Meeting, Portmeirion, April 2011.
- The influence of viscoelasticity on cavitation bubble dynamics. Invited speaker at Bubble Dynamics Workshop, Birmingham, July 2011.
- Bubble collapse in compressible viscoelastic fluids using a spectral element marker particle method. 17th International Workshop on Numerical Methods for Non-Newtonian Flows, Blois, France, March 2012.
- The effect of viscoelasticity on the dynamics of bubbles near rigid and free surfaces. Invited speaker at Bubble Dynamics Workshop, Birmingham, April 2012.
- The effect of viscoelasticity on the dynamics of gas bubbles near free surfaces. XVIth International Congress on Rheology, Lisbon, Portugal, August 2012.
- Bubble collapse in compressible viscoelastic fluids using a spectral element marker particle method. BSR Midwinter Meeting, Aberystwyth, December 2012.
- Modelling the dynamics of gas bubbles near free surfaces. BSR Annual Award Lecture. Advances in Non-Newtonian Fluid Mechanics, Royal Society International Conference Centre, Chicheley Hall, March 2013.
- A least-squares proper generalised decomposition algorithm for the Stokes problem, Second International Workshop on “Reduced Basis, POD and PGD model Reduction Techniques”, Blois, France, November 2013.
- A non-singular boundary element method for modelling the dynamics of cavitation bubbles, INNFM Conference on Rheology of Structured Fluids, Ruthin, Wales, March 2015.
- The immersed boundary method for fluid structure interaction (FSI) problems with an Oldroyd-B fluid. Annual European Rheology Conference, Nantes, France, April 2015.
- The immersed boundary method for fluid-structure interaction (FSI) problems with an Oldroyd-B fluid. XVIIth International Congress on Rheology, Kyoto, Japan, August 2016.
The extended spectral element method for the approximation of discontinuous functions and fluid-structure interaction problems. 20th IMACS World Congress, Xiamen, China, December 2016.

### Conferences organized

- INNFM/BSR Conference on Complex Flows of Complex Fluids, Liverpool, March 2008.
- Annual European Rheology Conference, Cardiff, April 2009.
- Advances in Non-Newtonian Fluid Mechanics, Kavli Royal Society International Centre, Chicheley Hall, March 2013.
- British Applied Mathematics Colloquium, Cardiff, April 2014.

### Professional responsibilities

- President, British Society of Rheology, 2017-19
- Council Member, British Society of Rheology.
- Editorial Board Member. Theoretical and Computational Fluid Mechanics, 2012-
- Editorial Board Member. Journal of Non-Newtonian Fluid Mechanics, 2017-
- External examiner for the undergraduate degree schemes in Mathematics at University of Reading, 2016-.
- Member, EPSRC College, 2000-
- Chair, MAGIC Programme Committee, 2017-
- Council Member, Learned Society of Wales, 2017-2020

## Teaching

### Undergraduate

- MA3304 Methods of Applied Mathematics

## Biography

### Biography

- BA – Mathematics, University of Oxford, 1979.
- MSc – Mathematics (mathematical modelling and numerical analysis), University of Oxford, 1980.
- DPhil – Mathematics, University of Oxford, 1983.
- MA – University of Oxford, 1987.
- DSc – University of Oxford, 2004.

After graduating from the University of Oxford, I was appointed to a research fellowship at ICASE (NASA Langley, USA), prior to taking up an academic position at Aberystwyth in 1984. I was appointed to a Professorial Fellowship at Cardiff University in 2004. I have held visiting professorships at the University of Delaware (Chemical Engineering) and the Ecole Polytechnique Fédérale de Lausanne (Mechanical Engineering).

I have published over 100 papers in international journals and over 30 papers in peer-reviewed conference proceedings. I co-authored one of the leading research monographs in my field - Computational Rheology - which was published by Imperial College Press in 2002. I have also published an undergraduate text – Elements of Mathematics for Economics and Finance – which was published by Springer in 2007. I am in the process of producing a second edtion of this book which is due towards the end of 2023. I have considerable experience in supervising young researchers (15 PDRAs and 24 PhD students).

### Honours and awards

- DSc University of Oxford, 2004
- Fellow, Learned Society of Wales, Elected 2012
- Annual Award, British Society of Rheology, 2012
- Fellow, Institute of Mathematics and its Applications, Elected 2010

### Professional memberships

- Fellow of the Institute of Mathematics and its Applications (IMA)
- Fellow of the Learned Society of Wales
- Member of the Society for Industrial and Applied Mathematics (SIAM)
- Member of the British Society of Rheology

### Academic positions

- 2004 - present: Professor, School of Mathematics, Cardiff University
- 2000 - 2004: Professor, Department of Mathematics, University of Wales, Aberystwyth
- 1996 - 2000: Reader, Department of Mathematics, University of Wales, Aberystwyth
- 1992 - 1996: Senior Lecturer, Department of Mathematics, University of Wales, Aberystwyth
- 1984 - 1992: New Blood Lecturer in Computational Fluid Dynamics, Department of Applied Mathematics, University College of Wales, Aberystwyth
- 1982 - 1984: Research Scientist, Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Virginia, USA

### Committees and reviewing

External committees

- Chair of MAGIC Programme Committee
- Member of MAGIC Management Committee
- Member of MAGIC Academic Steering Committee
- Council member, Institute of Mathematics and its Applications

MAGIC is one of 6 EPSRC funded Taught Course Centres (TCC). The MAGIC group runs a wide range of postgraduate-level lecture courses in mathematics:

http://maths-magic.ac.uk/index.php

Editorial Board, Theoretical and Computational Fluid Dynamics

Editorial Board, Journal of Non-Newtonian Fluid Mechanics

## Supervisions

I am interested in supervising PhD students in the areas of:

- numerical methods for non-Newtonian flows
- dynamics of cavitation bubbles
- multiphase flows
- extended spectral element method
- immersed boundary method
- lattice Boltzmann method

### Current supervision

## Mr Abhishek Chakraborty

Research student

## Mr Will Doherty

Research student

## Miss Layla Sadeghi Namaghi

Research student

## Mr Philippe Hergibo

Research student

## Related news

### Top architects chosen for £23m maths and computer science centre

26 September 2017

### School of Mathematics accreditation success

15 August 2017