Benjamin J. Walker

Lecturer in Applied Mathematics
University College London
benjamin.walker@ucl.ac.uk

VisualPDE.com

Interactive Mathematics

Some topics in mathematics can be difficult to teach in a way that engages people interactively. Andrew Krause, Adam Townsend, and I have put together VisualPDE.com to try to change this for partial differential equations (PDEs). VisualPDE simulates systems of equations at lightning speed, all in your browser, and is infinitely customisable. It's even solving a PDE right now - try clicking in the nearby square to interact with one of our examples.

Research interests

Solid mechanics and biological growth

Exploration of the role of solid mechanics in biology, from influencing the growth of tumours to controlling the development of vascular networks.

Filament mechanics

Efficient simulation of elastohydrodynamics in 2D and 3D, exploring the role and regulation of active moments in eukaryotic flagella, in addition to the responses of filaments to flow.

Microscale swimming

Investigation into the low Reynolds number swimming of flagellates, such as the boundary response of large-bodied Leishmania and the interactions between nearby spermatozoa.

Flagellar analysis

Automated processing and identification of flagella from videomicroscopy, enabling population-level waveform analysis without requiring costly researcher input. Publication and ImageJ code for flagella capture freely available here.

Slender-body theory

Incorporation of non-uniform body shapes and curvatures into slender-body theories via asymptotic methods, enabling simple and rapid flow field computation and further theoretical study using regularised singularities.

Research snapshot

Multiscale methods and microswimmer models

The effective hydrodynamic shapes of yawing spheroids.
The effects of rapid yawing on simple swimmer models and planar Jeffery's orbits.
Physical Review Fluids

If we take a minimal model of a microswimmer and include the effects of rapid side-to-side yawing, does the prediction of the model change? Here, we find out through a simple multiple-scales analysis.

An example shape-changing swimmer being rotated by a background flow.
Canonical orbits for rapidly deforming planar microswimmers in shear flow.
Physical Review Fluids

Suppose we have a swimmer that moves by rapidly changing its shape in the plane of a shear flow. Here, we show that, on average, the shape-deforming swimmer undergoes the classical orbit dynamics studied by G. B. Jeffery a century ago.

Closed trajectories in semi-periodic phase-space (angle-position).
Emergent rheotaxis of shape-changing swimmers in Poiseuille flow.
JFM Rapids and Focus on Fluids

Given a simple model of a swimmer being carried by the flow in a pipe, can we deduce its long-term behaviour? Here, we avoid numerical simulation and analytically determine the eventual behaviour of the swimmer through the computation of a single swimmer-dependent constant.

Phase portrait of the interaction of two freely rotating dipoles (angle-angle).
Systematic parameterisations of minimal models of microswimming.
Physical Review Fluids

When we use the simplest microswimmer models to develop intuition for complex, rapidly evolving swimmers, do we have to be careful with our choice of effective parameters? Here, through minimally complex examples and simple multi-timescale analysis, we see that we must be careful even when only seeking qualitative agreement.

Publications

 Google Scholar

    Preprints

  1. M. Butler, B. J. Walker, T. Montenegro-Johnson, P. Katsamba. Elastohydrodynamics of 3D chemically active filaments.
  2. E. F. Yeo, B. J. Walker, P. Pearce, M. P. Dalwadi. A shear-induced limit on bacterial surface adhesion in fluid flow.
  3. E. A. Gaffney, K. Ishimoto, B. J. Walker. Motility and rotation of multi-timescale microswimmers in linear background flows. arXiv.
  4. S. Fochler, B. J. Walker, R. J. Wheeler, E. Gluenz. Divergent Protein Kinase A contributes to the regulation of flagellar waveform of the motile flagellum of Leishmania mexicana. bioRxiv.

    2025

  5. H. Z. Ford, G. L. Celora, E. R. Westbrook, M. P. Dalwadi, B. J. Walker, H. Baumann, C. J. Weijer, P. Pearce, J. R. Chubb. Pattern formation along signaling gradients driven by active droplet behavior of cell swarms. Proceedings of the National Academy of Sciences.
  6. B. J. Walker, H. M. Byrne. On discretely structured growth models and their moments. Bulletin of Mathematical Biology.
  7. C. Fort, B. J. Walker, L. Baert, R. J. Wheeler. Proteins with proximal-distal asymmetries in axoneme localisation control flagellum beat frequency. Nature Communications.

    2024

  8. C. Moreau, B. J. Walker, R. N. Poon, D. Soto, D. I. Goldman, E. A. Gaffney, K. Y. Wan. Minimal design of a synthetic cilium. Physical Review Research.
  9. B. J. Walker. Interactive mathematics via VisualPDE. Mathematics Today.
  10. A. L. Krause, E. A. Gaffney, T. J. Jewell, V. Klika, B. J. Walker. Turing instabilities are not enough to ensure pattern formation. Bulletin of Mathematical Biology.
  11. M. P. Bambic, N. A. M. Araújo, B. J. Walker, D. R. Hewitt, Q. X. Pei, R. Ni, G. Volpe. Optimal face-to-face coupling for fast self-folding kirigami. Soft Matter.
  12. L. Koens, B. J. Walker. Viscous tubular-body theory for plane interfaces. Journal of Fluid Mechanics.
  13. M. P. Dalwadi, C. Moreau, E. A. Gaffney, B. J. Walker, K. Ishimoto. Generalised Jeffery's equations for rapidly spinning particles. Part 2: Helicoidal objects with chirality. Journal of Fluid Mechanics.
  14. M. P. Dalwadi, C. Moreau, E. A. Gaffney, K. Ishimoto, B. J. Walker. Generalised Jeffery's equations for rapidly spinning particles. Part 1: Spheroids. Journal of Fluid Mechanics.

    2023

  15. B. J. Walker, A. K. Townsend, A. K. Chudasama, A. L. Krause. VisualPDE: rapid interactive simulations of partial differential equations. Bulletin of Mathematical Biology.
  16. B. J. Walker, K. Ishimoto, E. A. Gaffney. Systematic parameterisations of minimal models of microswimming. Physical Review Fluids.
  17. B. J. Walker, G. L. Celora, A. Goriely, D. E. Moulton, H. M. Byrne. Minimal morphoelastic models of solid tumour spheroids: a tutorial. Bulletin of Mathematical Biology.
  18. B. J. Walker, K. Ishimoto, E. A. Gaffney. A hydrodynamic slender-body theory for local rotation at zero Reynolds number. Physical Review Fluids.
  19. A. L. Krause, E. A. Gaffney, B. J. Walker. Concentration-dependent domain evolution in reaction-diffusion systems. Bulletin of Mathematical Biology.

    2022

  20. B. J. Walker, A. T. Dawes. Modelling mechanically dominated vasculature development. Frontiers in Systems Biology.
  21. B. J. Walker, K. Ishimoto, C. Moreau, E. A. Gaffney, M. P. Dalwadi. Emergent rheotaxis of shape-changing swimmers in Poiseuille flow. Journal of Fluid Mechanics.
  22. B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau. The control of particles in the Stokes limit. Journal of Fluid Mechanics.
  23. E. A. Gaffney, M. P. Dalwadi, C. Moreau, K. Ishimoto, B. J. Walker. Canonical orbits for rapidly deforming planar microswimmers in shear flow. Physical Review Fluids.
  24. B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau, M. P. Dalwadi. The effects of rapid yawing on simple swimmer models and planar Jeffery's orbits. Physical Review Fluids.

    25. 2021

  25. C. Moreau, K. Ishimoto, E. A. Gaffney, B. J. Walker. Control and controllability of microswimmers by a shearing flow. Royal Society Open Science.
  26. E. A. Gaffney, K. Ishimoto, B. J. Walker. Modelling motility: the mathematics of spermatozoa. Frontiers in Cell and Developmental Biology.
  27. B. J. Walker, E. A. Gaffney. Regularised non-uniform segments and efficient no-slip elastohydrodynamics. Journal of Fluid Mechanics.
  28. L. S. Kimpton, B. J. Walker, C. L. Hall, B. Bintu, D. Crosby, H. M. Byrne, A. Goriely. A morphoelastic shell model of the eye. Journal of Elasticity.

    29. 2020

  29. B. J. Walker, K. Ishimoto, E. A. Gaffney. Efficient simulation of filament elastohydrodynamics in three dimensions. Physical Review Fluids.
  30. K. Ishimoto, E. A. Gaffney, B. J. Walker. Regularized representation of bacterial hydrodynamics. Physical Review Fluids.
  31. B. J. Walker, M. P. Curtis, K. Ishimoto, E. A. Gaffney. A regularised slender-body theory of non-uniform filaments. Journal of Fluid Mechanics.
  32. B. J. Walker, S. Phuyal, K. Ishimoto, C. Tung, E. A. Gaffney. Computer-assisted beat-pattern analysis and the flagellar waveforms of bovine spermatozoa. Royal Society Open Science.

    33. 2019

  33. B. J. Walker, K. Ishimoto, H. Gadêlha, E. A. Gaffney. Filament mechanics in a half-space via regularised Stokeslet segments. Journal of Fluid Mechanics.
  34. B. J. Walker, K. Ishimoto, E. A. Gaffney. The pairwise hydrodynamic interactions of synchronized spermatozoa. Physical Review Fluids.
  35. B. J. Walker, R. J. Wheeler. High-speed multifocal plane fluorescence microscopy for three-dimensional visualisation of beating flagella. Journal of Cell Science.
  36. B. J. Walker, K. Ishimoto, R. J. Wheeler. Automated identification of flagella from videomicroscopy via the medial axis transform. Scientific Reports.
  37. B. J. Walker, R. J. Wheeler, K. Ishimoto, E. A. Gaffney. Boundary behaviours of Leishmania mexicana: A hydrodynamic simulation study. Journal of Theoretical Biology.

    38. 2018

  38. B. J. Walker, R. J. Wheeler, K. Ishimoto, E. A. Gaffney. Response of monoflagellate pullers to a shearing flow: A simulation study of microswimmer guidance. Physical Review E.