There are a number of possible further studies that may provide answers to some of the open questions raised by this work. To save computational resources, the Taylor-Couette flow simulations were performed with only the repulsive part of the Lennard-Jones interaction, which reduced the size of neighbour-lists consid- erably. It would be of interest to examine whether molecular dynamics simu- lations correctly predict the critical Reynolds at which Taylor vortices develop when attractive forces are also considered. This could further be extended by examining whether other Newtonian fluids with more complex interactions, such as water, transition in the same manner. The FENE-model of polymeric fluids in Taylor-Couette flow was not numerically stable, at the high temperatures that resulted from fast shear rates, without a thermostat in the bulk fluid. The study could be repeated with a hydrodynamics-conserving thermostat applied to the fluid, such as the DPD-thermostat [87], which could determine the extent to which energy transport affects the flow transitions.
Another clear opportunity for further work is in molecular dynamics simu- lations of Taylor-Couette flow where polymer chains are grafted to the cylinder surfaces. The simulation could even be performed with a hybrid model, where only the near-surface regions are treated with molecular dynamics and the bulk fluid is modelled with a continuum. Due to the complex three-dimensional
momentum flux-coupling algorithm would be required here with a compressible continuum solver. An understanding of how the presence of polymers may or may not act to dampen the flow instabilities could provide important details that may inform similar hybrid studies of flow stability in the planar-shear case. More studies could also be performed to examine the extent to which branching of the polymer chains might affect the configuration of the brush, or the dynam- ics of the solvent shear flow. These cases, however, would still be restricted to configurations in which fluid complexity is contained in a region far away from the continuum overlap.
To extend the domain-decomposition hybrid methodology to simulations where complex fluids are also present in the overlap region, the heterogeneous multiscale method could also be used to provide the continuum constitutive relation. However, the problem of performing constitutive modelling with MD in more than two dimensions still remains, due to the one-dimensional nature of the Lees-Edwards sliding boundary condition. The open boundary at the top of the MD domain would also require a spatial coarse-graining such as AdResS [183], or a large-molecule insertion algorithm such as Fade [227], in order to facilitate mass flux across the boundary.
It is known that very small additions of polymers to a solvent can reduce the drag on a surface in turbulent flow [228, 6]. If the aforementioned methodology were to be implemented and validated, hybrid simulations could perhaps be ex- ploited to assess the validity of any proposed mechanisms of the drag-reduction. When compared to continuum constitutive models that aim to capture complex fluid dynamics with very few parameters, multiscale simulations, if tractable, could provide many benefits to the modelling of increasingly complex molecular fluids.
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