  Peritrichous bacteria swim in viscous fluids by rotating multiple helical
flagellar filaments. As the bacterium swims forward, all its flagella rotate in
synchrony behind the cell in a helical bundle. When the bacterium changes its
direction, the flagellar filaments unbundle and reorient the cell for a short
period of time before returning to their bundled state and resuming swimming.
This rapid bundling and unbundling is, at its heart, a mechanical process
whereby hydrodynamic interactions balance with elasticity to determine the
time-varying deformation of the filaments. Inspired by this biophysical
problem, we present in this paper what is perhaps the simplest model of
bundling whereby two, or more, straight elastic filaments immersed in a viscous
fluid rotate about their centreline, inducing rotational flows which tend to
bend the filaments around each other. We derive an integro-differential
equation governing the shape of the filaments resulting from mechanical balance
in a viscous fluid at low Reynolds number. We show that such equation may be
evaluated asymptotically analytically in the long-wavelength limit, leading to
a local partial differential equation governed by a single dimensionless
Bundling number. A numerical study of the dynamics predicted by the model
reveals the presence of two configuration instabilities with increasing
Bundling numbers: first to a crossing state where filaments touch at one point
and then to a bundled state where filaments wrap along each other in a helical
fashion. We next provide an intuitive physical model for the crossing
instability and show that it may be used to predict analytically its threshold,
and adapted to address the transition to a bundling state. We then use a
macro-scale experimental implementation of the two-filament configuration in
order to validate our theoretical predictions and obtain excellent agreement.
