Non-Newtonian effects and Taylor-Aris despersion in rivulet flow
In this thesis we use a combination of analytical and numerical methods to analyse two aspects of the steady flow of rivulets of fluid, namely the effects of non-Newtonian rheology, and the transport of a passive solute in a rivulet of Newtonian fluid. In Chapters 2-4 we consider rivulet flow of non-Newtonian fluids. Firstly, we obtain the solution for unidirectional gravity-driven flow of a uniform thin rivulet of a power-law fluid down a planar substrate, and then we use this solution to describe the flow of a rivulet with prescribed constant contact angle but slowly varying semi-width down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder. Secondly, we use the solution for unidirectional flow to describe the flow of arivulet with prescribed constant semi-width but slowly varying contact angledown a slowly varying substrate. Thirdly, we consider rivulet flow of generalised Newtonian fluids down a vertical planar substrate. In particular, we obtain the solutions for rivulet flow of a Carreau fluid and of an Ellis fluid, highlighting their similarities and differences. In Chapters 5 and 6 we investigate both the short-time advection and the long-time Taylor-Aris dispersion of a passive solute in uniform non-thin and thin rivulets, respectively, of a Newtonian fluid undergoing steady unidirectional flow driven by gravity and/or a prescribed uniform surface shear stress on a verticalplanar substrate. In particular, we obtain an explicit expression for the effective diffusivity of the solute in a thin rivulet as a function of the surface shear stress, the volume flux along the rivulet, and either the semi-width or the contact angle of the rivulet.
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