By Joseph Johnson, The University of Melbourne
My research this summer was into numerical solutions to the Boltzmann equation for rarefied gas flows. This is relevant because it is necessary in modelling the interaction between nano and micro scale electromechanical devices and the surrounding air. Accurate and efficient modelling of this interaction is essential to building devices like smartphone cameras.
The first step to understanding how this interaction is modelled is to see air as small devices do. On the nano and micro scale air doesn’t seem like a continuum as it does to us, in fact it seems like a sea of discrete molecules. When the gas seems discretised in this way it is called “rarefied” on the scale of the system. The degree of rarefaction is given by a parameter known as Knudsen Number and as Knudsen number changes, the stresses on the device change. Engineers must account for these changes in the devices construction. The equation that describes a fluids distribution with varying Knudsen number is called the Boltzmann equation, solving this equation is difficult and must often be done numerically.
The next step in modelling this interaction is to model the motion of the device. Couette flow is fluid flow between two parallel, shearing plates. Hence solving the Boltzmann equation for Couette flow models the fluid-device interaction with the boundary conditions of the shearing plates representing the motion of the device.
The right hand side of the Boltzmann equation is a collision term that represents how collisions impact the fluid distribution. The collision term scales as one over Knudsen number so for very rarefied flows, Knudsen number is high and collisions make a much smaller impact. With fewer collisions due to the fluids degree of rarefaction (particle separation) the impact of the plate’s motion isn’t smeared out by fluid particle collisions and the perturbation to the fluids mass distribution is steep in velocity space. That is the fluid distributions perturbation rapidly achieves its maximum (as defined by the plates shearing) with a small variation in velocity from zero. To better capture the variation of rarefied solutions to the Boltzmann equation a rescaling of velocity space by Knudsen is done before numerical solutions are calculated. This rescaling makes the computation of rarefied solutions to the Boltzmann equation more efficient.
Joseph Johnson was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.