Dynamics of semi-flexible fibres in viscous flow

The dynamics of semi-flexible fibres in shear flow and the effect of flexibility on the
swimming speed of helical flagella are investigated. High aspect ratio particles such as
carbon and glass fibres are often added as fillers to processed polymers. Although these
materials have high rigidity, the large aspect ratiomakes the fibres liable to bending during
flow. Other high aspect ratio fibres that behave as semi-flexible fibres include carbon
nano-tubes, paper fibres and semi-flexible polymers such as the muscle protein f-actin.
Most theoretical studies assume that fibres are either rigid or completely flexible, but in
this thesis fibres with a finite bending modulus are considered.

A semi-flexible fibre is modelled as a chain of shorter rods linked together. A bending
torque is included at the joints between the rods to account for the rigidity. In shear flow
the simulation reproduces the C and S turns observed in experiments on semi-flexible
fibres. The results for finite aspect ratio fibres predict changes to the period of rotation
and drift between Jeffery orbits. The direction of drift for a flexible fibre depends on both
the intial orientation and the fibre’s flexiblity.

We also present a linear analysis of how small distortions to a straight semi-flexible fibre
grow when the flow places the fibre under compression. These results are in agreement
with our full simulations and the growth rates of the distortions to a straight fibre allow us
to predict the most unstable mode at a particular flow rate.

To allow for intrinsically bent or helical equilibrium shapes a second simulation method
is developed that includes a twisting torque at the joints between the rods as well as a
bending torque. Using this simulation we measure the period of rotation and orbit drift of
permanently deformed fibres in shear flow and show that due to the asymmetry of a helix,
shear induced rotation results in translation and orbit drift for both rigid and semi-flexible
fibres.

Bacteria such as Vibrio alginolyticus and Escherichia coli swim by rotating one or more helical flagella. Vibrio alginolyticus has only one flagella and changes direction by
altering its sense of rotation. Experimental observations of Vibrio alginolyticus have
found that backwards swimming is 50% faster than forwards swimming speed however,
previous numerical simulation results have shown only a 4% difference for flagella of the
same dimensions. We use our simulation to consider how flexiblity affects the swimming
speed of helical flagella and show that for a constant angular velocity, difference between
forwards and backwards swimming speed ranges between 0-23%depending on the exact
stiffness chosen. We explain the differences in swimming speeds of semi-flexible fibres
by investigating the shape changes which occur and comparing them to the results for
swimming speeds of rigid flagella of varying dimensions.