Slender Body Theory

Dr O.G. Harlen

There are many important examples of dispersions of elongated particles such as rods and fibres in fluids and solids. For example short glass and carbon fibres are commonly added to moulded plastics to improve their mechanical, thermal and electrical properties.   Analytical solutions to the equations of fluid flow (Stokes equations) and heat conduction (Laplace’s equation) only exist for flow past a very limited range of particle shapes. However, if the particle in question is “slender”, i.e. its length L is much larger than its width, d then there is an elegant mathematical theory first developed by Burgers  call ed slender body theory that produces an approximate asymptotic solution.

The solution is formed by considering two limits

1.      Large scale in which the particle is infinitessimally thin and of finite length, and so can be treated as a line.

2.      Small scale in which the particle is of finite thickness but infinitely long.

The trick is to match these two limits together to form a consistent solution.

The aim of the project is to understand the technique of slender body theory and to use it calculate the flow of heat and of a fluid around a fibre. These results will be used to calculate the sedimentation of a rod through a viscous fluid and the viscosity of a dilute suspension of fibres. If time permits we may also consider other problems such as a flow around a curved fibre or a ring.

Prerequisites

There are no essential prerequisites for this project, however both the Perturbation Methods course MATH 3552 and Viscous Flow course MATH 3512 will provide useful background.

References

Batchelor Journal of Fluid Mechanics volume 44, 419-440.

Clift Bubbles, Drops and Particles.

Cox Journal of Fluid Mechanics volume 44, 791-810 and volume 45, 625-657.

Hinch Perturbation Methods.