This project studies the flow of a fluid inside an infinite pipe with periodically
forced extensional walls in the axial direction. The Navier-Stokes equations for an
incompressible fluid inside the pipe are solved to find self-similar solutions of the
problem. The system of PDEs is preconditioned using a spectral expansion in modulated
Legendre polynomials, and is solved using an IMEX method. Anumerical exploration of
time-periodic solutions in a wide parameter region has been conducted, and interesting
non-linear phenomena is found for specific parameter regions.
The system tends to converge to solutions with limit cycles of the same period as the
oscillation. The frequency response of the system is studied. For small frequencies, the
problem explores different solutions found in the ’non-forced’ problem [9], approaching
stable limit cycles of high frequencies when the pipe is shrinking very slowly, and
forming very fast axial jets. This behaviour is lost when the pipe oscillates at higher
frequencies. Moreover, no stable solutions with azimuthal velocity are found, but a
branch of unstable solutions with azimuthal velocity is born at a pitchfork bifurcation
of cycles. In addition, cascade of period doubling bifurcations is found, leading the
system to possible chaos.
