Models of flow and diffusion: I : Selective withdrawal from a stratified fluid II: Dispersal of hydrogen in the retina

Wafaa Faisal Mansoor

This dissertation is divided into two parts. In the ﬁrst part we investigate the withdrawal of ﬂuid from a region of ﬂuid with the outlet situated at an arbitrary location. A model for the dispersal of hydrogen released into the retina in an attempt to estimate blood ﬂow in the eye is considered in the second part. In the ﬁrst part, the ﬂow induced by a line sink at an arbitrary location in a ﬂuid of ﬁnite depth with a free surface, relevant to ﬂows in reservoirs, lakes and cooling ponds, is examined. A rigid-lid solution for small ﬂow rates is obtained and a numerical method based on fundamental singularities techniques is applied to the full nonlinear problem. Both linear and numerical steady solutions are obtained for the shape of the free surface and show good agreement. The results suggest that steady non-linear solutions are limited to ﬂow rates below some critical value that depends on the sink location, the surface tension and the strength of the ﬂow. A theorem has been proven regarding the behaviour of the ﬂuid surface and some interesting surface shapes are obtained. The two-dimensional steady ﬂow of an inviscid ﬂuid induced by a line sink located at an arbitrary location in a region of inﬁnite depth is computed. The solution in the limit of small Froude number is obtained analytically, and numerically for the nonlinear problem. The asymptotic solution is found to have a property that if the horizontal location of the sink, xs < 1, there is only one stagnation point on the surface, at the wall. However, if the horizontal location xs > 1 a second stagnation point forms on the free surface. This has implications for the design of outlets in dams and reservoirs. Numerical solution for the nonlinear problem conﬁrms these properties. The effect of moving the sink horizontally has also been considered. The maximum Froude numbers at which steady solutions exist are computed and compared with previous work and the effects of surface tension are investigated. Finally in this part, we examine the unsteady ﬂow due to a line sink in a ﬂuid of ﬁnite depth with surface tension where the sink is situated at an arbitrary depth and location. Here we focus on critical values of ﬂow rate that lead to steady or drawn down surfaces and the transitions between the different cases. A solution to the un-steady, linear problem is derived using an integral equation technique. The unsteady, nonlinear equation is then solved numerically using a novel fundamental singularities method. The shape of the free surface is computed for a range of parameter values where the effects of surface tension are taken into account. The linear solution is shown to be in a good agreement with the full nonlinear solution until the depth becomes signiﬁcant, at which point the nonlinearities become apparent and the two solutions begin to differ slightly. We will examine the behaviour of the ﬂow when the ﬂuid is stagnant and the sink is turned on suddenly. Initially, the free surface is pulled down everywhere regardless of Froude number, F , and surface tension, β, ﬁrstly, either near the wall, if the sink is close to the wall, or above the sink, in cases where the sink is situated further from the origin. If the Froude number, F , is large enough the initial dip will keep going until drawdown occurs. However, if the Froude number, F , is smaller, the central region rebounds upward and that leads to a more complicated situation where there are several possible outcomes. There are three general states at the breakdown point of the simulations. These are, evolution to a steady state, drawdown of the surface and something in between which may include breaking waves or splashes. In particular, we obtained critical drawdown values of F for a range of values of β, sink location, xs, and different layer depths, H. In the second major part of the thesis, two mathematical models of advection and diffusion of hydrogen within the retina are discussed to assist in interpretation of the “Hydrogen clearance technique” that is used to estimate blood ﬂow. The ﬁrst model assumes the retina consists of three, well-mixed layers with different thickness, and the second is a two-dimensional model consisting of three regions that represent the layers in the retina. Diffusion between the layers and leakage through the outer edges are considered. Solutions to the governing equations are obtained by employing Fourier series and ﬁnite difference methods for the two models, respectively. The effect of important parameters on the hydrogen concentration is examined and discussed, and a formula is derived for the speed of travel of the bolus of hydrogen. The results contribute to understanding the dispersal of hydrogen in the retina and in particular the effect of ﬂow in the vascular retina. It is shown that the predominant features of the process are captured by the simpler model, meaning that the predictions in experiments can be interpreted without detailed simulation.

Models of flow and diffusion: I : Selective withdrawal from a stratified fluid II: Dispersal of hydrogen in the retina

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