 Abstract | A weakly nonlinear stability analysis was conducted for the flow induced in an
incompressible, Newtonian, viscous fluid lying between two infinite parallel plates
which form a channel. The plates are oscillating synchronously in simple harmonic
motion. The disturbed velocity of the flow was written in the form of a series in
powers of a parameter which is a measure of the distance away from the linear
theory neutral conditions. The individual terms of this series were decomposed using
Floquet theory and Fourier series in time.
The equations at second order and third order in were derived, and solutions
for the Fourier coefficients were found using pseudospectral methods for the spatial
variables. Various alternative methods of computation were applied to check
the validity of the results obtained. The Landau equation for the amplitude of
the disturbance was obtained, and the existence of equilibrium amplitude solutions
inferred.
The values of the coefficients in the Landau equation were calculated for the
nondimensional channel half-widths h for the cases h = 5, 8, 10, 12, 14 and 16. It was
found that equilibrium amplitude solutions exist for points in wavenumber Reynolds
number space above the smooth portion of the previously determined linear stability
neutral curve in all the cases examined. Similarly, Landau coefficients were calculated
on a special feature of the neutral curve (called a “finger”) for the case h = 12.
Equilibrium amplitude solutions were found to exist at points inside the finger, and
in a particular region outside near the top of the finger.
Traces of the x-components of the disturbance velocities have been presented for
a range of positions across the channel, together with the size of the equilibrium
amplitude at these positions. As well, traces of the x-component of the velocity of
the disturbed flow and traces of the velocity of the basic flow have been given for
comparison at a particular position in the channel. |
