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
bjw43@bath.ac.uk
University College London
Honorary Lecturer
benjamin.walker@ucl.ac.uk
VisualPDE.com
Multiscale methods and microswimmer models
Physical Review Fluids
If we take a minimal model of a microswimmer and include the effects of rapid sidetoside yawing, does the prediction of the model change? Here, we find out through a simple multiplescales analysis. Image: the effective hydrodynamic shapes of yawing spheroids.
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 shapedeforming swimmer undergoes the classical orbit dynamics studied by G. B. Jeffery a century ago. Image: an example shapechanging swimmer being rotated by a background 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 longterm behaviour? Here, we avoid numerical simulation and analytically determine the eventual behaviour of the swimmer through the computation of a single swimmerdependent constant. Image: closed trajectories in semiperiodic phasespace (angleposition).
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 multitimescale analysis, we see that we must be careful even when only seeking qualitative agreement. Image: phase portrait of the interaction of two freely rotating dipoles (angleangle).
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 largebodied Leishmania and the interactions between nearby spermatozoa.
Automated processing and identification of flagella from videomicroscopy, enabling populationlevel waveform analysis without requiring costly researcher input. Publication and ImageJ code for flagella capture freely available here.
Incorporation of nonuniform body shapes and curvatures into slenderbody theories via asymptotic methods, enabling simple and rapid flow field computation and further theoretical study using regularised singularities.
Publications

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. 
A. L. Krause, E. A. Gaffney, T. J. Jewell, V. Klika, B. J. Walker . Turing instabilities are not enough to ensure pattern formation. arXiv. 
M. P. Bambic, N. A. M. Araújo, B. J. Walker, D. R. Hewitt, Q. X. Pei, R. Ni, G. Volpe . Optimal facetoface coupling for fast selffolding kirigami. arXiv. 
L. Koens, B. J. Walker . Viscous tubularbody theory for plane interfaces. arXiv. 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, in press. arXiv.
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, in press. arXiv. B. J. Walker, A. K. Townsend, A. K. Chudasama, A. L. Krause (2023). VisualPDE: rapid interactive simulations of partial differential equations. Bulletin of Mathematical Biology. Full text.B. J. Walker, K. Ishimoto, E. A. Gaffney (2023). Systematic parameterisations of minimal models of microswimming. Physical Review Fluids. Full text.B. J. Walker, G. L. Celora, A. Goriely, D. E. Moulton, H. M. Byrne (2023). Minimal morphoelastic models of solid tumour spheroids: a tutorial. Bulletin of Mathematical Biology. Full text.
B. J. Walker, K. Ishimoto, E. A. Gaffney (2023). A hydrodynamic slenderbody theory for local rotation at zero Reynolds number. Physical Review Fluids. Full text. 
A. L. Krause, E. A. Gaffney, B. J. Walker (2023). Concentrationdependent domain evolution in reactiondiffusion systems. Bulletin of Mathematical Biology. Full text. 
B. J. Walker, A. T. Dawes (2022). Modelling mechanically dominated vasculature development. Frontiers in Systems Biology. Full text. 
B. J. Walker, K. Ishimoto, C. Moreau, E. A. Gaffney, M. P. Dalwadi (2022). Emergent rheotaxis of shapechanging swimmers in Poiseuille flow. Journal of Fluid Mechanics. Full text. 
B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau (2022). The control of particles in the Stokes limit. Journal of Fluid Mechanics. Full text. 
E. A. Gaffney, M. P. Dalwadi, C. Moreau, K. Ishimoto, B. J. Walker (2022). Canonical orbits for rapidly deforming planar microswimmers in shear flow. Physical Review Fluids. Full text. 
B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau, M. P. Dalwadi (2022). The effects of rapid yawing on simple swimmer models and planar Jeffery's orbits. Physical Review Fluids. Full text. C. Moreau, K. Ishimoto, E. A. Gaffney, B. J. Walker (2021). Control and controllability of microswimmers by a shearing flow. Royal Society Open Science.
E. A. Gaffney, K. Ishimoto, B. J. Walker (2021). Modelling motility: the mathematics of spermatozoa. Frontiers in Cell and Developmental Biology. Full text. 
B. J. Walker, E. A. Gaffney (2021). Regularised nonuniform segments and efficient noslip elastohydrodynamics. Journal of Fluid Mechanics. Full text. 
L. S. Kimpton, B. J. Walker, C. L. Hall, B. Bintu, D. Crosby, H. M. Byrne, A. Goriely (2021). A morphoelastic shell model of the eye. Journal of Elasticity. B. J. Walker, K. Ishimoto, E. A. Gaffney (2020). Efficient simulation of filament elastohydrodynamics in three dimensions. Physical Review Fluids. Full text.
K. Ishimoto, E. A. Gaffney, B. J. Walker (2020). Regularized representation of bacterial hydrodynamics. Physical Review Fluids. Full text. 
B. J. Walker, M. P. Curtis, K. Ishimoto, E. A. Gaffney (2020). A regularised slenderbody theory of nonuniform filaments. Journal of Fluid Mechanics. Full text. 
B. J. Walker, S. Phuyal, K. Ishimoto, C. Tung, E. A. Gaffney (2020). Computerassisted beatpattern analysis and the flagellar waveforms of bovine spermatozoa. Royal Society Open Science. Full text. B. J. Walker, K. Ishimoto, H. Gadêlha, E. A. Gaffney (2019). Filament mechanics in a halfspace via regularised Stokeslet segments. Journal of Fluid Mechanics. Full text.B. J. Walker, K. Ishimoto, E. A. Gaffney (2019). The pairwise hydrodynamic interactions of synchronized spermatozoa. Physical Review Fluids. Full text.B. J. Walker, R. J. Wheeler (2019). Highspeed multifocal plane fluorescence microscopy for threedimensional visualisation of beating flagella. Journal of Cell Science. Full text.B. J. Walker, K. Ishimoto, R. J. Wheeler (2019). Automated identification of flagella from videomicroscopy via the medial axis transform. Scientific Reports. Full text.B. J. Walker, R. J. Wheeler, K. Ishimoto, E. A. Gaffney (2019). Boundary behaviours of Leishmania mexicana: A hydrodynamic simulation study. Journal of Theoretical Biology. Full text.B. J. Walker, R. J. Wheeler, K. Ishimoto, E. A. Gaffney (2018). Response of monoflagellate pullers to a shearing flow: A simulation study of microswimmer guidance. Physical Review E. Full text.