I am an applied mathematician, interested in mathematical modelling and applying computational, analytical and occasionally experimental tools to answer questions in the natural sciences, especially the biological world. My work is focussed primarily on the continuum scale, in particular I am interested in: mechanical biology and physiology, growth and pattern formation, morphoelasticity, and elastic mechanisms in nature. Earlier work involved ferrohydrodynamics, minimal surfaces, and electrostatics.
Growth is the process by which a material gains mass. It is ubiquitous in nature, common in some form to all living entities, and found in many industrial applications and physical phenomena as well. It is the process that links the graceful neck of the giraffe to the powerful horns of the ram to the formation of a planet as a dust cloud collapses under gravity. In biological tissues, such as arteries, skin tissues, airways, and plants, growth is a very complex process, which can be analysed from multiple perspectives (biological, chemical, evolutionary, ...) as well as on multiple length and time scales.
My interest is on the mechanics of growth processes, and understanding the form of biological structures through analysis of the mechanical forces underlying their morphogenesis.
Surface growth refers to the deposition of new material on the surface of a body. A striking example of surface growth found in nature is the shape of seashells. Seashells have intrigued scientists for centuries, often forming a paradigm for evolutionary theories. Yet the developmental mechanisms underlying shell formation are largely not understood. We have developed a general model for shell growth based entirely on the local geometry and mechanics of the aperture, where shell growth occurs, and the mollusc, the creature that lives in and builds the shell.
Ammonites (pictured above) are an iconic group of Cephalopods that have been extinct for over 100 million years yet are still of great interest to paleontologists. Our modelling framework has provided the first quanitative explanation for the formation of the ribbing pattern on these beautiful shells.
Everything done in the biological world requires energy, it is the fundamental unit of currency. And one of the great wonders of evolution is the multitude of solutions that have been devised for the simply stated but crucial task of motion: moving something from point A to point B. It is a trick that has been accomplished in myriad remarkable ways.
I am interested in understanding the mechanisms underlying motion in plants and animals, both from developmental, evolutionary, and ``engineering'' points of views. In particular I am studying innovative uses of elastic energy, exemplified for instance by the ballistic projection of the chameleon tongue or explosive seed dispersal in certain plants such as Cardamine hirsuta (left).
Physiological function depends on the delicate balance between growth, stress, geometry, and external forces. These relationships are critical in understanding function and targeting the root cause underlying disfunction.
Biological structures are often highlighted by a complex make-up, including anisotropy, differential growth, and multiple layers. The theory of finite elasticity provides the most sophisticated tools for analysing such complex mechanical behaviour. My modelling approach is to work within simplified geometries -- such as cylindrical and spherical -- where analytical progress may be made. Recently we have used this framework to study such diverse topics as asthma, wound healing and the growth of tumour spheroids.
Elastic filaments, structures characterised by a length much greater than the cross-sectional width, are very common - for instance bacterial fibers, roots, and spine - and can have complicated 3D geometry. Understanding the mechanical behavior of these structures in the context of growth poses a significant challenge. We have developed a general framework to study such structures, which we term morphoelastic rods. Our approach combines the typical Kirchhoff equations for elastic rods with fundamental ideas from 3D morphoelasticity.
Recently, we have extended this work to the situation of 2 connected elastic rods with potentially different mechanical properties and growing at different rates. Our framework enables complex shapes and behaviour, such as perversion of helically wound filaments, to be studied efficiently and analytically.
I am exploring the stability of knotted elastic rods. In particular, we have found theoretically for the first time a stable elastic trefoil knot with no points of self-contact. Aside from increasing the understanding of the rich solution structure of the Kirchhoff equations, in recent years the study of elastic knots has found relevance in other areas as well, particularly in the micro biological world, where knots can be found in long polymers, proteins, and DNA plasmids.
A vertical soap film draining under gravity is a classic experiment, explored since the time of Newton. In our lab at the University of Delaware, we created magnetic soap-films by adding magnetic nano-particles to a standard soap-film recipe.
We have developed a 2D model for a draining magnetic film using lubrication theory. We have captured well the basic phenomenon of reverse draining; we have also explored the effect of a disjoining pressure due to molecular forces. Pictured on the left is a regular draining soap film, on the right a magnetic film placed under a strong magnet drains upward, against gravity (the black signals the thinnest, drained region).