Transport, reaction and mixing in fluid flows

An important area of interest in microfluidics research is the transport of suspended particles in micro-devices. Particle transport in low Reynolds number flows is also important in sedimentation, blood flow, polymer processing and other chemical and biological processes. In this dissertation, we present a study of transport, mixing and reaction of orientable rods and present backwards probabilistic techniques for solving transport equations.; The problem of enhancing reaction between orientable particles in a microdevice is considered. Achieving this objective is more complex than in the case of particulate matter because of the apparently conflicting requirements of alignment in orientation space while at the same time enhancing the mixing in physical space. A model framework is developed within which one may study reaction dynamics under various velocity fields.; We study the evolution of individual particle densities in position-orientation space. Trends in the amount and distribution of reaction product are obtained with variations in Peclet, rotational Peclet and Damkohler numbers in laminar microchannel. We demonstrate that a shear superposition micromixer can be used to achieve the dual objective of mixing and alignment. Simulations show that it is possible to achieve a significant enhancement of the reaction rate by using a shear superposition micromixer.; Traditional methods of solving transport PDEs become very expensive with high Peclet numbers and when the problem is high dimensional. For these reasons, we consider Monte Carlo methods that do not suffer from these difficulties. The commonly used forward Monte Carlo methods, however, suffer from sampling difficulties and can be very expensive when we wish to obtain statistically meaningful results. We develop backwards methods to study the evolution of particle densities which have several advantages when compared with the more traditional methods. These methods are based on the link between PDEs and Brownian motion, known as the Feynman-Kac relation.; These backward Monte Carlo methods to compute the effect of diffusion are ideally suited for application to microfluidic devices where velocity profiles are obtainable relatively easily and Peclet numbers are often very high. We demonstrate the efficiency and flexibility of this approach via applications to studying density evolution in a laminar microchannel flow as well as in a chaotic flow in the shear superposition micromixer. We present an extension of this method that takes into account anisotropic diffusion and lack of volume preservation by the velocity field, which are present in the case of orientable particles. Numerical experiments demonstrate the further advantages of this technique when the density distribution in the domain is highly non-uniform and when we only wish to solve the density in a small sub-domain.; Mixing is also studied in the context of macroscopic flows as well. One important mixing mechanism is through the free shear layer, which is found in atmospheric flow, oceans and in industrial applications such as combustion chambers and diffusers. A vortex sheet is an idealization of a free shear layer. Vortex sheet evolution is, however, an ill-posed problem. We employ the recently discovered Euler-alpha equations to regularize vortex sheet evolution. We perform a linear stability analysis and determine the dispersion relation for the problem. We find that the smoothed transport velocity from the Euler-alpha model helps stabilize the core of the sheet during roll-up and prevents self-intersection of the curve. In the non-linear regime, we discover a scaling of size of the rolled-up sheet with alpha, the regularization parameter.