Generalised theory and Darcy's law for shear-induced dispersion in fluids with variable density and transport properties

The proposed work is an original research project with two main objectives: (1) to provide a generalised mixing theory for shear-induced dispersion and (2) to test the theory by applying it to four iconic mixing problems. The theory is pertinent to many engineering and real-life situations involving transport phenomena with variable density, viscosity, and other transport properties. Such situations are encountered in a variety of contexts with vastly different length scales—from the oceanic to the microscopic. The project is an ambitious attempt to generalise theoretical concepts and tools recently developed in our combustion-focused investigations in order to develop a theoretical framework with broad interdisciplinary applications. The theory will be applied to the following four problems: Flame instabilities in a Hele-Shaw cell, focusing on the coupling between Taylor dispersion and classical instabilities associated with variable properties, including the Darrieus–Landau instability (associated with variable density), the Saffman–Taylor instability (associated with variable viscosity), and the diffusive-thermal instability (associated with unequal mass and heat diffusivities). In the project, specific emphasis will be dedicated to hydrogen combustion. Effect of Taylor dispersion on the Rayleigh–Benard instability (associated with buoyancy) in a Hele-Shaw cell. Effect of Taylor dispersion on the double-diffusive convection instability (associated with buoyancy and differential diffusion) in a Hele-Shaw cell. Effect of Taylor dispersion on the Marangoni convection (associated with variable interfacial surface tension) in a thin fluid layer. The objectives of the work will be accomplished through a two-phase approach, the first being based on theoretical tools such as scaling analyses and perturbation methods and the second relying both on theoretical tools such as modelling and linear stability analyses and numerical tools involving numerical computations applied to the four selected problems. The success of the project will pave the way for future proposals that will benefit various beneficiaries. Such beneficiaries include applied mathematicians, fluid dynamicists, combustion scientists, oceanographers, physiologists, and others. A concrete illustration of research areas on which the project is expected to have an impact is that of hydrogen combustion, to which one of the test problems in this proposal is dedicated. Specifically, the outcomes of this problem should contribute to improving our understanding of hydrogen-flame dynamics in micro-engines that can power devices such as micro air vehicles (MAVs) and micro-satellites, thereby assisting engineers to design these devices more efficiently. With hydrogen being regarded as one of the most promising fuels for the future in order to decarbonise current fuels, this research should contribute to placing the UK at the forefront of international research efforts to address global greenhouse gas concerns. This is also in line with the EPSRC's strategic plan towards achieving "engineering net zero" via "transformative low and zero carbon—hydrogen and alternative liquid fuels".