The molecular dynamics simulations of Taylor-Couette flow in chapter 3 were extensions of a similar study by Hirshfeld and Rapaport [17, 18], who employed no-slip boundary conditions on the cylinder walls, specular reflecting walls at the axial boundaries, and a special quarter-cylinder azimuthal periodicity. The critical Reynolds number was found to be Rc = 86.5, which was 13% greater than the value predicted by theory, Rc = 76.6. The boundary conditions in this work differed in all directions. The radial confinement of the fluid was pro- vided by particles ‘tethered’ to rotating sites, the axial boundaries were periodic and the domain spanned the full azimuth θ ∈ [0, 2π). In a similar geometry to those in references [17, 18], the critical Reynolds number was found to be in the range 76.3 < Rc < 77.3, which is within 2-3% of the theoretical predic- tion, Rc = 74.7. In addition, simulating the full azimuth did not restrict any azimuthal modes that could appear to those with integer wavenumbers that are multiples of kθ = 4. A wavy vortex mode with kθ = 1 was identified at higher rotation speeds, and was shown to be non-unique to the geometry by its persistence in domains with longer axial periodicity. Replacement of peri- odic boundary conditions with specular walls in the axial direction suppressed the wavy vortex mode. In a second, larger geometry, similar simulations were performed at a greater Taylor number. The domain size was too large for a full parameter study of rotation speed to be tractable with reasonable compu-
about the transient growth of a spiral wavy vortex mode with three azimuthal oscillations. Many flow configurations were observed as the wavy vortices devel- oped, and the growth of the primary Taylor vortex mode (kz= 2) was consistent with the theoretical prediction of Landau [7]. Fourier analysis showed that all spectral harmonics of the waves had the same phase velocity, confirming that the waves were coherently convected with the mean flow in the channel.
To ensure that flow instabilities were neither excited nor dampened in the bulk fluid, the temperature of the system was controlled only by applying a ther- mostat to the molecules that formed the cylinder walls. However, this method was insufficient for maintaining a constant fluid temperature at all rotation speeds. This could have made estimating the viscosity, which is strongly de- pendent on temperature, difficult. However, this problem was addressed with a direct measurement of the viscosity via the stress tensor in cylindrical coordi- nates.
The calculation of Rc to within 3% of the theoretical value implies that molecular dynamics simulations may be used as one part of a hybrid computa- tion to accurately predict flow instabilities. However, the tethered molecules in the cylinder walls did not provide a perfect no-slip condition, and the tangen- tial velocity of the first layer of fluid molecules at the inner cylinder was up to 10% lower than the wall molecules. If the Reynolds number were to be defined with the velocity in this layer, the estimate of the critical value for the primary instability would be significantly less consistent with theory.
Periodic boundaries in the axial direction are not possible in experiment, but allow the simulations here to provide conditions that are a close approximation to very long cylinders with minimal computational expense. The simulations in this work have shown that periodic boundaries provide conditions that are favourable to the appearance of the wavy vortex mode, which is suppressed in cases with end-walls. However, periodic boundaries do not actively add energy to the system, so they are not expected to actively excite or dampen any az- imuthal modes unless there is a significant self-interaction across the domain. The persistence of the wavy mode in the long cylinder cases suggests that this interaction is unlikely to be present.
To allow the parameter study of inner-cylinder rotation speed to be per- formed in a reasonable computational time, each rotational speed was realised in independent simulations that were performed in parallel. The time for the inner cylinder to be brought from stationary to full speed - the ‘ramp’ time - was approximately four times longer than the period of one rotation at the
fastest speed. The simulations could more closely replicate the conditions of experiments if they were to be performed in serial, where the final microstate of a simulation at the slowest angular speed could be used as the initial state for ‘ramping’ to the next speed in the parameter study (and so on). With this method the overall rate of change of angular speed would be much slower, even with the same ramp time. The question of whether the critical Reynolds number Rcremains the same in a regime such as this is unresolved. A similar parameter study of inner cylinder speed, performed in serial, could also be undertaken in the larger cylinder geometry. A study of this kind could determine if the appear- ance of the spiral vortex with three azimuthal waves is unique to the particular flow history modelled here. It would also be of interest to continue the original simulation for a long time, to establish whether the spiral-wave configuration is temporally stable.
An attempt to use the same methodology for studying Taylor-Couette flow in polymer solutions was numerically unstable at high rotation speeds. In these cases the temperature of the system became too high to ensure that, in simu- lations with a reasonable time-step, thermal fluctuations were not large enough to stretch the FENE-bonds beyond the maximum extension. The study of hy- drodynamic instabilities with fully molecular simulations was therefore limited to the inertial case. Studies with a more numerically stable polymer model, for example harmonic springs that have a defined potential at all extensions, is a clear opportunity to extend this work to the elastic instability case. Alterna- tively, should a suitable thermostat algorithm be proven to not affect the point at which flow transition occurs, the temperature could be controlled in the bulk fluid to a value that is stable with a reasonable timestep. This would have the additional benefit of extending the relaxation time of the polymers, which would increase the Weissenberg number and make configurations in which an elastic instability is expected more accessible.