This thesis reports the methods to simulate flows withcomplex boundary such as orifice flow. The method is forgeneral purposes so that it has been tested on different flowsincluding orifice flow. Also it contains a chapter about theexperiment of orifice flow. Higher-order precision interpolation schemes are used inumerical simulation to improve prediction at acceptable gridrefinement. Because higher-order schemes cause instability inconvection-diffusion problems or involve a large computationalkernel, they are implemented with deferred correction method. Alower-order scheme such as upwind numerical scheme is used tomake preliminary guess. A deferred (defect) correction term isadded to maintain precision. This avoids the conflict betweenprecision order and implementation difficulty. The authorproposes a shifting between upwind scheme and centraldifference scheme for the preliminary guess. This has beenproven to improve convergence while higher order schemes havewider range of stability. Non-orthogonal grid is a necessity for complex flow. Usuallyone can map coordinate of such a grid to a transformed domainwhere the grid is regular. The cost is that differentialequations get much more complex form. If calculated directly innon-orthogonal grid, the equations keep simple forms. However,it is difficult to make interpolation in a non-orthogonal grid.Three methods can be used: local correction, shape function andcurvilinear interpolation. The local correction method cannotinsure second-order precision. The shape function method uses alarge computational molecule. The curvilinear interpolationthis author proposes imports the advantage of coordinatetransformation method: easy to do interpolation. A coordinatesystem staggered half control volume used in the coordinatetransformation method is used as accessory to deriveinterpolation schemes. The calculation in physical domain withnon-orthogonal grid becomes as easy as that in a Cartesianorthogonal grid. The author applies this method to calculate turbulentorifice flow. The usual under-prediction of eddy length isimproved with the ULTRA-QUICK scheme to reflect the highgradients in orifice flow. In the last chapter, the author quantifies hydraulicabruptness to describe orifice geometry. The abruptness canhelp engineers to interpolate existing data to a new orifice,which saves detailed experiments

<p>This thesis reports the methods to simulate flows withcomplex boundary such as orifice flow. The method is forgeneral purposes so that it has been tested on different flowsincluding orifice flow. Also it contains a chapter about theexperiment of orifice flow.</p><p>Higher-order precision interpolation schemes are used inumerical simulation to improve prediction at acceptable gridrefinement. Because higher-order schemes cause instability inconvection-diffusion problems or involve a large computationalkernel, they are implemented with deferred correction method. Alower-order scheme such as upwind numerical scheme is used tomake preliminary guess. A deferred (defect) correction term isadded to maintain precision. This avoids the conflict betweenprecision order and implementation difficulty. The authorproposes a shifting between upwind scheme and centraldifference scheme for the preliminary guess. This has beenproven to improve convergence while higher order schemes havewider range of stability.</p><p>Non-orthogonal grid is a necessity for complex flow. Usuallyone can map coordinate of such a grid to a transformed domainwhere the grid is regular. The cost is that differentialequations get much more complex form. If calculated directly innon-orthogonal grid, the equations keep simple forms. However,it is difficult to make interpolation in a non-orthogonal grid.Three methods can be used: local correction, shape function andcurvilinear interpolation. The local correction method cannotinsure second-order precision. The shape function method uses alarge computational molecule. The curvilinear interpolationthis author proposes imports the advantage of coordinatetransformation method: easy to do interpolation. A coordinatesystem staggered half control volume used in the coordinatetransformation method is used as accessory to deriveinterpolation schemes. The calculation in physical domain withnon-orthogonal grid becomes as easy as that in a Cartesianorthogonal grid.</p><p>The author applies this method to calculate turbulentorifice flow. The usual under-prediction of eddy length isimproved with the ULTRA-QUICK scheme to reflect the highgradients in orifice flow.</p><p>In the last chapter, the author quantifies hydraulicabruptness to describe orifice geometry. The abruptness canhelp engineers to interpolate existing data to a new orifice,which saves detailed experiments</p>

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This thesis deals with a one and two dimensional numerical modeling of a low-pressure DC glow discharge in argon gas. We develop two-component fluid model which uses the diffusion-drift theory for the gas discharge plasma and consists of continuity equations for electrons and ions, as well as Poisson equation for electric field. Numerical method is based on the control volume technique. Calculations are carried out in MATLAB environment. Computed results are compared with the classic theory of glow discharges and available experimental data.

In this thesis, a number of mathematical models of haptotactic cell migration are developed. The modelling of haptotaxis is presented in two distinct parts - the first comprises an investigation of haptotaxis in pre-necrotic avascular tumours, while the second consists of the modelling of adhesion-mediated haptotactic cell migration within tissue, with particular attention paid to the biological appropriateness of the description of cell-extracellular matrix adhesion. A model is developed that describes the effects of passive and haptotactic migration on the cellular dynamics and growth of pre-necrotic avascular tumours. The model includes a description of the extracellular matrix and its effect on cell migration. Questions are posed as to which cell types act as a source of the extracellular matrix, and the model is used to simulate the possible effects of different matrix sources. Simulations in one-dimensional and spherically symmetric geometry are presented, displaying familiar results such as three-phase tumour growth and tumours comprising a rim of proliferating cells surrounding a non-proliferating region. Novel effects are also described such as cell population splitting and tumour shrinkage due to haptotaxis and appropriate extracellular matrix construction. The avascular tumour model is then extended to describe the internalisation of labelled cells and inert microspheres within multicell tumour spheroids. A novel model of adhesion-receptor mediated haptotactic cell migration is presented and specific applications of the model to tumour invasion processes are discussed. This model includes a more biologically realistic description of cell adhesion than has been considered in previous models of cell population haptotaxis. Through assumptions of fast kinetics, the model is simplified with the identification of relationships between the simplified model and previous models of haptotaxis. Further simpli.cations to the model are made and travelling wave solutions of the original model are then investigated. It is noted that the generic numerical solution routine NAG D03PCF is not always appropriate for the solution of the model, and can produce oscillatory and inaccurate solutions. For this reason, a control volume numerical solver with .ux limiting is developed to provide a better method of solving the cell migration models.