Slender
Body Theory

Dr O.G. Harlen

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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**

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**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.

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