Dr Warren Smith BSc MSc DPhil

School of Mathematics

Contact details

School of Mathematics
Watson Building
University of Birmingham
B15 2TT

Warren has published over 30 research papers in scientific journals in the fields of mathematical modelling and perturbation theory. He has received grants from the EPSRC and has collaborated with research groups in the Netherlands, United States and world-leading multinationals.

Warren has pioneered the application of strongly nonlinear analysis to the Navier-Stokes equations. Original results have resulted for the asymptotic structure of free space, the viscous decay rate of the oscillations of viscous drops and the radiative decay rate of spherical bubbles.


  • PG Cert in Learning and Teaching, 2003
  • DPhil in Mathematical Modelling, 1992
  • MSc in Mathematical Modelling and Numerical Analysis, 1990
  • BSc (Hons) in Mathematics, 1989


Warren Smith obtained a first class BSc (Hons) in Mathematics from Warwick University in 1989. He then moved to Exeter College, Oxford to attend the taught MSc in Mathematical Modelling and Numerical Analysis and study for a DPhil modelling the latest design of Rolls-Royce gas journal bearing. His doctorate was followed by a short period in scientific consultancy, an EPSRC-funded Postdoctoral Research Assistant in the University of Nottingham and an EU-funded ECMI Research Fellowship in TU Eindhoven. In 2001, he accepted a Lectureship at the University of Birmingham in which he remains apart from a three-month fellowship at Cornell University.

Warren acts as a reviewer for more than ten leading scientific journals, the research councils and AMS Mathematical Reviews. He has edited a special issue of the Journal of Engineering Mathematics on Practical Asymptotics, been invited to contribute to a special issue of the Journal of Engineering Mathematics on Industrial Applied Mathematics, delivered invited lectures at international conferences and organised a study group for mathematics in industry in the Netherlands.


Semester 1

LH/LM Methods in Partial Differential Equations

LH/LM Continuum Mechanics

Postgraduate supervision

  • Mathematical modelling of the distensible tube wave energy converter 
  • Strongly nonlinear analysis applied to the Navier-Stokes equations


Research Themes

  • Perturbation theory
  • Strongly nonlinear analysis
  • Acoustic cavitation

Research Activity

Warren has pioneered the application of strongly nonlinear analysis to the Navier-Stokes equations. This analysis has succeeded in demonstrating that viscous effects always represent a singular perturbation for rotational flows even in the absence of interior/boundary layers.

Throughout his academic career, Warren has sought after multi-scale industrial topics. These allow the application of regular perturbation theory (for example, in laser percussion drilling) or singular perturbation theory (for example, in acoustic cavitation). The European Study Groups with Industry (ESGI), links with ECMI and interdisciplinary collaboration have provided sources for these projects. During his career, he has collaborated with mechanical engineers (Brunel, Oxford), electrical engineers (Nottingham), chemical engineers (Cornell, Eindhoven), physicists (Eindhoven) and biomedical engineers (Birmingham).


Recent publications


Corbett, C, Wang, Q, Liu, W, Smith, W & Walmsley, D 2023, 'Cleaning effects due to shape oscillation of bubbles over a rigid boundary', Physics of Fluids, vol. 35, no. 12, 123335. https://doi.org/10.1063/5.0173730

Yu, Y, Smith, W, Wang, Q & Walmsley, AD 2023, 'Numerical investigation of cavitation in periodontal Pockets: Insights for enhancing cleaning efficiency', Ultrasonics Sonochemistry, vol. 100, 106625. https://doi.org/10.1016/j.ultsonch.2023.106625

O'Brien, EN, Mahmud, M, Smith, WR, Wang, QX & Phillips, TN 2023, 'The evolution of a three-dimensional microbubble at a corner in a Maxwell fluid', Physics of Fluids, vol. 35, no. 10, 103120. https://doi.org/10.1063/5.0170820

Yu, Y, Smith, W, Wang, Q & Leighton, TG 2023, 'The mechanisms of jetting, vortex sheet and vortex ring development in asymmetric bubble dynamics', Physics of Fluids.

Smith, W & Wang, Q 2023, 'The role of surface tension on the growth of bubbles by rectified diffusion', Ultrasonics Sonochemistry, vol. 98, 106473. https://doi.org/10.1016/j.ultsonch.2023.106473

Smith, W & Wang, Q 2022, 'A theoretical model for the growth of spherical bubbles by rectified diffusion', Journal of Fluid Mechanics, vol. 939, A28. https://doi.org/10.1017/jfm.2022.218

Smith, W & Wang, Q 2022, 'A tractable mathematical model for rectified diffusion', Journal of Fluid Mechanics, vol. 951, A12. https://doi.org/10.1017/jfm.2022.849

Yu, Y, Mahmud, M, Vyas, N, Smith, WR, Wang, Q & Damien Walmsley, A 2022, 'Cavitation in a periodontal pocket by an ultrasonic dental scaler: A numerical investigation', Ultrasonics Sonochemistry, vol. 90, 106178. https://doi.org/10.1016/j.ultsonch.2022.106178

Wang, Q, Liu, W, Corbett, C & Smith, W 2022, 'Microbubble dynamics in a viscous compressible liquid subject to ultrasound', Physics of Fluids, vol. 34, no. 1, 012105. https://doi.org/10.1063/5.0077091

Smith, W & Wang, Q 2021, 'The pitfalls of investigating rotational flows with the Euler equations', Journal of Fluid Mechanics, vol. 927, A42. https://doi.org/10.1017/jfm.2021.805

Smith, W & Wang, Q 2021, 'The radiated acoustic pressure and time scales of a spherical bubble', Fluid Dynamics Research, vol. 53, no. 1, 015502. https://doi.org/10.1088/1873-7005/abd1d0

Wang, Q, Mahmud, M, Smith, W & Walmsley, D 2020, 'Numerical investigation of bubble dynamics at a corner', Physics of Fluids, vol. 32, no. 5, 053306 . https://doi.org/10.1063/1.5140740

Manmi, K, Wu, WB, Vyas, N, Smith, W, Wang, Q & Walmsley, D 2020, 'Numerical investigation of cavitation generated by an ultrasonic dental scaler tip vibrating in a compressible liquid', Ultrasonics Sonochemistry, vol. 63, 104963. https://doi.org/10.1016/j.ultsonch.2020.104963

Smith, W & Wang, Q 2018, 'Radiative decay of the nonlinear oscillations of an adiabatic spherical bubble at small Mach number', Journal of Fluid Mechanics, vol. 837, pp. 1-18. https://doi.org/10.1017/jfm.2017.658, https://doi.org/10.1017/jfm.2017.658

Smith, W & Wissink, J 2017, 'Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow', European Journal of Applied Mathematics. https://doi.org/10.1017/S0956792517000213

View all publications in research portal