Dr. Neil Balmforth

Non-Newtonian Fluid Dynamics And Applications In Geophysics:

Aside from air and water, most of the fluids we encounter in physical and industrial processes are "non-Newtonian" (meaning that there is no simple relationship between the stress and the rate of strain). Often, the 'rheology' arises because the fluid in question builds up a microstructure at the molecular level which becomes sufficiently extensive to affect the macroscopic properties of the fluid. In lava, for example, the microstructure is provided by a network of interlocked silicate crystals, and endows the fluid with an internal  strength that allows lava to withstand a certain amount of imposed stress  before it flows. Unfortunately, fluid models that build in the rheology significantly complicate the governing equations. My work in this area is focused on simplifying these equations to gain insight into the flow dynamics. For example, for lava flow, the main goal is to develop a  "shallow-lava theory". That is, a general theoretical model that incorporates the essential physical ingredients and can be used to compute the flow of lava under terrestrial and extra-terrestrial conditions. Computations with the full three-dimensional equations is an inefficient route for such a model, partly because the rheology of cooling lava is so complicated. But by using asymptotic methods, simpler versions can be built for use by geologists. Related geological problems include mud flows and glacier mechanics. The rheology of mud and ice has several similarities with that of lava, and the same non-Newtonian fluid models can be used to describe how they flow. Similar models are also used for landslides, avalanches, and the dynamics of sand. These materials are mostly granular media, and require a different physical framework. However, the underlying idea (the reduction of governing equations to simpler, useful models) is a powerful tool that can be brought to bear in these applications also.

Much of my work is theoretical, focussing on mathematical modelling, but I also do experiments (some of which took place in the backyard and cellar; occasionally even in a lab).

I have many collaborators in these efforts, including Richard Craster and Richard Kerswell.

Click to enlarge
Experiment showing a clay slurry flowing over an inclined plane
A model of an expanding, inclined dome of viscoplastic fluid
Theoretical slump
Theoretical slump
Theoretical slump
Experiment showing a clay slurry flowing over an inclined plane

A model of an expanding, inclined dome of viscoplastic fluid
(theoretical calculations of a quasi-static dome extruded onto an inclined plane, computed analytically via the method of characteristics)
Theoretical slumps
(The shallow, slow flow approximation furnishes a simple model describing the evolution of the free surface (blue) and fake yield surface (red))
Isothermal viscoplastic domes
Cooling viscoplastic domes I
Cooling viscoplastic domes II
Viscoplastic convection
Viscoplastic lubrication
Viscoplastic Hele-Shaw
Asymptotic methods
Boundary layers
Viscoplastic flow on an inclined plane
More inclined viscoplastic flow
Shallow lava theory
Non-Newtonian GFD
Slump shapes
2D Slumps
Beams and threads
Dambreaks - Bostwick Consistometer
Fingering of a viscoplastic contact line
The viscoplastic Stokes layer
Viscoplastic drips I and II
Thixotropic gravity currents
Viscoplastic Bretherton problem
Swimming in mud

Department of Mathematics / Fluid Labs / Neil Balmforth / Research Interests