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.

### Benjamin J. Walker

University of Bath

Lecturer in Applied Mathematics & 1851 Research Fellow

Deputy Director of the Institute for Mathematical Innovation

bjw43@bath.ac.uk

University College London

Honorary Lecturer

benjamin.walker@ucl.ac.uk

VisualPDE.com

## Research interests

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

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.

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

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.

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

*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.

*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.

*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.

*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-
H. Z. Ford, G. L. Celora, E. R. Westbrook, M. P. Dalwadi, . Pattern formation along signaling gradients driven by active droplet behaviour of cell groups.**B. J. Walker**, H. Baumann, C. J. Weijer, P. Pearce, J. R. Chubb*bioRxiv*. -
C. Moreau, . State-dependent stiffness enhances wave propagation along elastic filaments.**B. J. Walker**, R. N. Poon, D. Soto, D. I. Goldman, E. A. Gaffney, K. Y. Wan*arXiv*. -
S. Fochler, . Divergent Protein Kinase A contributes to the regulation of flagellar waveform of the motile flagellum of Leishmania mexicana.**B. J. Walker**, R. J. Wheeler, E. Gluenz*bioRxiv*.### 2024

A. L. Krause, E. A. Gaffney, T. J. Jewell, V. Klika, . Turing instabilities are not enough to ensure pattern formation.**B. J. Walker***Bulletin of Mathematical Biology*.*Full text*.M. P. Bambic, N. A. M. Araújo, . Optimal face-to-face coupling for fast self-folding kirigami.**B. J. Walker**, D. R. Hewitt, Q. X. Pei, R. Ni, G. Volpe (2024)*Soft Matter*.*Full text*.L. Koens, . Viscous tubular-body theory for plane interfaces (2024).**B. J. Walker***Journal of Fluid Mechanics*.*Full text*.M. P. Dalwadi, C. Moreau, E. A. Gaffney, Generalised Jeffery's equations for rapidly spinning particles. Part 2: Helicoidal objects with chirality.**B. J. Walker**, K. Ishimoto (2024).*Journal of Fluid Mechanics*.*Full text*.M. P. Dalwadi, C. Moreau, E. A. Gaffney, K. Ishimoto, Generalised Jeffery's equations for rapidly spinning particles. Part 1: Spheroids.**B. J. Walker**(2024).*Journal of Fluid Mechanics*.*Full text*.### 2023

VisualPDE: rapid interactive simulations of partial differential equations.**B. J. Walker**, A. K. Townsend, A. K. Chudasama, A. L. Krause (2023).*Bulletin of Mathematical Biology*.*Full text*. Systematic parameterisations of minimal models of microswimming.**B. J. Walker**, K. Ishimoto, E. A. Gaffney (2023).*Physical Review Fluids*. Full text. Minimal morphoelastic models of solid tumour spheroids: a tutorial.**B. J. Walker**, G. L. Celora, A. Goriely, D. E. Moulton, H. M. Byrne (2023).*Bulletin of Mathematical Biology*.*Full text*. A hydrodynamic slender-body theory for local rotation at zero Reynolds number.**B. J. Walker**, K. Ishimoto, E. A. Gaffney (2023).*Physical Review Fluids*. Full text.A. L. Krause, E. A. Gaffney, Concentration-dependent domain evolution in reaction-diffusion systems.**B. J. Walker**(2023).*Bulletin of Mathematical Biology*.*Full text*.### 2022

Modelling mechanically dominated vasculature development.**B. J. Walker**, A. T. Dawes (2022).*Frontiers in Systems Biology*. Full text. Emergent rheotaxis of shape-changing swimmers in Poiseuille flow.**B. J. Walker**, K. Ishimoto, C. Moreau, E. A. Gaffney, M. P. Dalwadi (2022).*Journal of Fluid Mechanics.*Full text.-
The control of particles in the Stokes limit.**B. J. Walker**, K. Ishimoto, E. A. Gaffney, C. Moreau (2022).*Journal of Fluid Mechanics*. Full text. -
E. A. Gaffney, M. P. Dalwadi, C. Moreau, K. Ishimoto, Canonical orbits for rapidly deforming planar microswimmers in shear flow.**B. J. Walker**(2022).*Physical Review Fluids*. Full text. -
The effects of rapid yawing on simple swimmer models and planar Jeffery's orbits.**B. J. Walker**, K. Ishimoto, E. A. Gaffney, C. Moreau, M. P. Dalwadi (2022).*Physical Review Fluids*. Full text. C. Moreau, K. Ishimoto, E. A. Gaffney, Control and controllability of microswimmers by a shearing flow.**B. J. Walker**(2021).*Royal Society Open Science*.-
E. A. Gaffney, K. Ishimoto, Modelling motility: the mathematics of spermatozoa.**B. J. Walker**(2021).*Frontiers in Cell and Developmental Biology*. Full text. -
Regularised non-uniform segments and efficient no-slip elastohydrodynamics.**B. J. Walker**, E. A. Gaffney (2021).*Journal of Fluid Mechanics*. Full text. -
L. S. Kimpton, A morphoelastic shell model of the eye.**B. J. Walker**, C. L. Hall, B. Bintu, D. Crosby, H. M. Byrne, A. Goriely (2021).*Journal of Elasticity*. Efficient simulation of filament elastohydrodynamics in three dimensions.**B. J. Walker**, K. Ishimoto, E. A. Gaffney (2020).*Physical Review Fluids*. Full text.-
K. Ishimoto, E. A. Gaffney, Regularized representation of bacterial hydrodynamics.**B. J. Walker**(2020).*Physical Review Fluids*. Full text. -
A regularised slender-body theory of non-uniform filaments.**B. J. Walker**, M. P. Curtis, K. Ishimoto, E. A. Gaffney (2020).*Journal of Fluid Mechanics*. Full text. Computer-assisted beat-pattern analysis and the flagellar waveforms of bovine spermatozoa.**B. J. Walker**, S. Phuyal, K. Ishimoto, C. Tung, E. A. Gaffney (2020).*Royal Society Open Science*. Full text. Filament mechanics in a half-space via regularised Stokeslet segments.**B. J. Walker**, K. Ishimoto, H. Gadêlha, E. A. Gaffney (2019).*Journal of Fluid Mechanics*. Full text. The pairwise hydrodynamic interactions of synchronized spermatozoa.**B. J. Walker**, K. Ishimoto, E. A. Gaffney (2019).*Physical Review Fluids*. Full text. High-speed multifocal plane fluorescence microscopy for three-dimensional visualisation of beating flagella.**B. J. Walker**, R. J. Wheeler (2019).*Journal of Cell Science*. Full text. Automated identification of flagella from videomicroscopy via the medial axis transform.**B. J. Walker**, K. Ishimoto, R. J. Wheeler (2019).*Scientific Reports*. Full text. Boundary behaviours of**B. J. Walker**, R. J. Wheeler, K. Ishimoto, E. A. Gaffney (2019).*Leishmania mexicana*: A hydrodynamic simulation study.*Journal of Theoretical Biology*. Full text. Response of monoflagellate pullers to a shearing flow: A simulation study of microswimmer guidance.**B. J. Walker**, R. J. Wheeler, K. Ishimoto, E. A. Gaffney (2018).*Physical Review E*. Full text.