This repository contains the works of Ivar Svalheim Haugerud for his master thesis at the University of Oslo. The thesis will soon be found at it's own github site and will also be published at University of Oslo's archives. In the thesis, we are analytically and numerically investigating three different systems to understand the geometry’s effect on the flow and resulting solute dispersion. Numeric solutions of the Navier-Stokes equations and Brenner theory are found using the finite element method and the lattice-Boltzmann method. The three different systems have their own directory containing all the code. Though the data files generated are not included, the code used to visualize the results are.
As a solute spreads in a flowing liquid, the interplay between advection and diffusion can make the macroscopic dispersion order of magnitude larger than what can be predicted from the molecular diffusivity alone. This has lead to the concept of effective diffusion, which remains essential for describing the transport of pollutants and chemical agents in a wide range of natural systems and industrial applications. The aim of this thesis is to better understand how the macroscopic dispersion is influenced by spatial and temporal oscillations; that is, by spatial periodic modulations of the confining boundary and tem- poral oscillations in the driving force of the flow. To do so, we build on Brenner’s theory of effective diffusion for steady incompressible flow in periodic channels. A discontinuous boundary roughness leads to recirculation zones at all boundary amplitudes, resulting in significantly enhanced dispersion for creeping flow. Contrary to previous research, increas- ing fluid inertia with rough boundaries is found to be able to both increase or decrease the dispersion depending on the Peclet number; a possible explanation of this phenomenon is proposed. Brenner’s theory is generalised to time-dependent velocity fields to investigate single-frequency flow in channels with sinusoidal boundary profiles. A novel resonance phenomenon between the wavelength of the boundary roughness and the frequency of the oscillating driving force is found to maximise the dispersion; a physical explanation is given. Lastly, a reciprocity relation for symmetric transport properties under the reversal of creeping flow was generalised for the purpose of possible medical applications.
The first area of investigation is to find and understand the effective dispersion coefficient’s dependency on the boundary amplitude, Peclet, and Reynolds number, in a channel with a periodic rough square boundary. The investigation is completely numerical, using the FEM to solve Brenner’s equation, as the discontinuous geometry makes the perturbative approach prohibited. This thesis aims to better understand the effects of inertial flow in rough channels, with our geometry as a prototype for the more general investigation. The code is found in the directory rough_channel.
The second area of investigation is the interactions between an oscillating flow and a smoothly varying sinusoidal boundary on the effective dispersion. To understand the system, we investigate it both analytically and numerically. For the analytical investigation, a perturbative approach is made, where the equations of interest are expanded in powers of the boundary amplitude. Brenner's equation has hitherto only been valid for time-independent flows, and the theory is generalized for the purpose of our investigation. Therefore, a part of the result is deriving the generalized theory and verifying it in our geometry. The perturbative result is compared with numerical solutions of Brenner's equations using the FEM. The main goal of the investigation is to analyze the behaviour of the effective diffusion coefficient with the addition of a varying boundary. The code is found in the directory osc_wavy.
Lastly, a general symmetric transport property on the form of a reciprocity relation for dispersion is investigated. For reversible creeping flow, the reciprocal relation allows for the prediction of a concentration at a specific point in the fluid. A generalization of previous work on this relationship is investigated analytically and compared to lattice-Boltzmann simulations. The relation is relevant for medical applications, as it can be used to access and optimize the predicted concentration profile of for example chemotherapeutic agents in the context of cancer treatment, at an otherwise inaccessible region. Therefore, the research questions are closely related to extending the applicability of the reciprocal relation with such motivations in mind. The code is found in the directory reciproc_relation