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1 
Mathematical modelling of avascular tumour growthWard, John P. January 1997 (has links)
No description available.

2 
Swimming Filaments in a Viscous Fluid with ResistanceHo, Nguyenho 28 April 2016 (has links)
In this dissertation, we study the behavior of microscopic organisms utilizing lateral and spiral bending waves to swim in a fluid. More specifically, spermatozoa encounter different fluid environments filled with mucus, cells, hormones, and other large proteins. These networks of proteins and cells are assumed to be stationary and of low volume fraction. They act as friction, possibly preventing or enhancing forward progression of the swimmers. The flow in the medium is described as a viscous fluid with a resistance term known as a Brinkman fluid. It depends on the Darcy permeability parameter affecting the swimming patterns of the flagella. To further understand these effects we study the asymptotic swimming speeds of an infinitelength swimmer propagating planar or spiral bending waves in a Brinkman fluid. We find that, up to the second order expansion, the swimming speeds are enhanced as the resistance increases. The work to maintain the planar bending and the torque exerted on the fluid are also examined. The Stokes limits of the swimming speeds, the work and the torque are recovered as resistance goes to zero. The analytical solutions are compared with numerical results of finitelength swimmers obtained from the method of Regularized Brinkmanlets (MRB). The study gives insight on the effects of the permeability, the length and the radius of the cylinder on the performance of the swimmers. In addition, we develop a gridfree numerical method to study the bend and twist of an elastic rod immersed in a Brinkman fluid. The rod is discretized using a Kirchhoff Rod (KR) model. The linear and angular velocity of the rod are derived using the MRB. The method is validated through a couple of benchmark examples including the dynamics of an elastic rod, and the planar bending of a flagellum in a Brinkman fluid. The studies show how the permeability and stiffness coefficients affect the waveforms, the energy, and the swimming speeds of the swimmers. Also, the beating pattern of the spermatozoa flagellum depends on the intracellular concentrations of calcium ([Ca2+]). An increase of [Ca2+] is linked to hyperactivated motility. This is characterized by highly asymmetrical beating, which allows spermatozoa to reach the oocyte (egg) or navigate along the female reproductive tract. Here, we couple the [Ca2+] to the bending model of a swimmer in a Brinkman fluid. This computational framework is used to understand how internal flagellar [Ca2+] and fluid resistance in a Brinkman fluid alter swimming trajectories and flagellar bending.

3 
Asymptotics of higherorder Painlevé equationsMorrison, Tegan Ann January 2009 (has links)
Doctor of Philosophy (PhD) / We undertake an asymptotic study of a second Painlevé hierarchy based on the JimboMiwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of nonlinear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higherorder second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourthorder analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leadingorder by two related genus2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leadingorder equations, and we investigate how these parameters change with respect to this variable. We also show that the fourthorder equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higherorder Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reactiondiffusion equations with quadratic and cubic source terms, with a spatiotemporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents.

4 
Mathematical Analysis of Film BlowingBennett, James Cameron, james.bennett@student.rmit.edu.au January 2008 (has links)
Film blowing is a highly complex industrial process used to manufacture thin plastic films for uses in a wide range of applications; for example, plastic bags. The mathematical modelling of this process involves the analysis of highly nonlinear differential equations describing the complex phenomena arising in the film blowing process, and requires a sophisticated mathematical approach. This dissertation applies an innovative combination of tools, namely analytic, numerical and heuristic mathematical techniques to the analysis of the film blowing process. The research undertaken examines, in particular, a twopoint boundary value problem arising from the modelling of the radial profile of the polymer film. For even the simplest modelling of this process, namely the isothermal Newtonian model, the resulting differential equation is a highly nonlinear, second order one, with an extra degree of difficulty due to the presence of a small parameter multiplying the highest derivative. Thus, the problem falls into the category of a nonlinear singular perturbation problem. Analytic techniques are applied to the isothermal Newtonian blown film model to obtain a closed form explicit approximation to the film bubble radius. This is then used as a base approximation for an iterative numerical scheme to obtain an improved numerical solution of the problem. The process is extended to include temperature variations, varying viscosity (Power law model) and viscoelastic effects (Maxwell model). As before, closed form approximations are constructed for these models which are used to launch numerical schemes, whose solutions display good accuracy. The results compare well with results obtained by purely numerical solutions in the literature.

5 
Instability Thresholds and Dynamics of Mesa Patterns in ReactionDiffusion SystemsMcKay, Rebecca Charlotte 19 August 2011 (has links)
We consider reactiondiffusion systems of two variables with Neumann boundary conditions on a finite interval with diffusion rates of different orders. Solutions of these systems can exhibit a variety of patterns and behaviours; one common type is called a mesa pattern; these are solutions that in the spatial domain exhibit highly localized interfaces connected by almost constant regions. The main focus of this thesis is to examine three different mechanisms by which the mesa patterns become unstable.
These patterns can become unstable due to the effect of the heterogeneity of the domain, through an oscillatory instability, or through a coarsening effect from the exponentially small interaction with the boundary.
We compute instability thresholds such that, as the larger diffusion coefficient is increased past this threshold, the mesa pattern transitions from stable to unstable. As well, the dynamics of the interfaces making up these mesa patterns are determined. This allows us to describe the mechanism leading up to the instabilities as well as what occurs past the instability threshold. For the oscillatory solutions, we determine the amplitude of the oscillations. For the coarsening behaviour, we determine the motion of the interfaces away from the steady state.
These calculations are accomplished by using the methods of formal asymptotics and are verified by comparison with numerical computations. Excellent agreement between the asymptotic and the numerical results is found.

6 
Mathematical modelling of modulatedtemperature differential scanning calorimetryNikolopoulos, Christos January 1997 (has links)
No description available.

7 
Asymptotics of higherorder Painlevé equationsMorrison, Tegan Ann January 2009 (has links)
Doctor of Philosophy (PhD) / We undertake an asymptotic study of a second Painlevé hierarchy based on the JimboMiwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of nonlinear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higherorder second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourthorder analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leadingorder by two related genus2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leadingorder equations, and we investigate how these parameters change with respect to this variable. We also show that the fourthorder equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higherorder Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reactiondiffusion equations with quadratic and cubic source terms, with a spatiotemporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents.

8 
Ασυμπτωτικά αναπτύγματα ολοκληρωμάτων / Asymptotic expansions of integralsΔρούλια, Σοφία 15 October 2012 (has links)
Ενώ η πραγματική ανάλυση φαίνεται να έχει προβάδισμα όσο αφορά στον τρόπο
επίλυσης των περισσότερων προβλημάτων λογισμού που διδάσκονται τόσο σε
σχολικό όσο και σε πανεπιστημιακό επίπεδο, η πραγματικότητα είναι διαφορετική.
Ουσιαστικά, ελάχιστα προβλήματα της εφαρμοσμένης ανάλυσης λύνονται αναλυτικά,
καθώς οι λύσεις που προκύπτουν είναι συχνά υπό μορφή ολοκληρωμάτων που δεν
υπολογίζονται στοιχειωδώς. Στην παρούσα διπλωματική εργασία γίνεται προσπάθεια
αντιμετώπισης κάποιων ολοκληρωμάτων με τεχνικές της ασυμπτωτικής ανάλυσης.
Αφότου αποσαφηνιστούν κάποιες βασικές έννοιες της ασυμπτωτικής ανάλυσης,
παρουσιάζονται πέντε μέθοδοι υπολογισμού ολοκληρωμάτων μέσω ασυμπτωτικών
αναπτυγμάτων. Το σύνολο τους, καλύπτει ένα αρκετά ευρύ φάσμα ανάλυσης και
υπολογισμού τέτοιου τύπου ολοκληρωμάτων και η κάθε μια από αυτές, εξιδεικεύεται
σε συγκεκριμένες περιπτώσεις, ανάλογα με το χώρο στον οποίο ανήκουν οι υπό
ολοκλήρωση συναρτήσεις καθώς και το πεδίο ολοκλήρωσης. / 

9 
Mathematical modelling of asymmetrical metal rolling processesMinton, Jeremy John January 2017 (has links)
This thesis explores opportunities in the mathematical modelling of metal rolling processes, specifically asymmetrical sheet rolling. With the application of control systems in mind, desired mathematical models must make adequate predictions with short computational times. This renders generic numerical approaches inappropriate. Previous analytical models of symmetrical sheet rolling have relied on ad hoc assumptions about the form of the solution. The work within this thesis begins by generalising symmetric asymptotic rolling models: models that make systematic assumptions about the rolling configuration. Using assumptions that apply to cold rolling, these models are generalised to include asymmetries in roll size, roll speed and rollworkpiece friction conditions. The systematic procedure of asymptotic analysis makes this approach flexible to incorporating alternative friction and material models. A further generalisation of a cladsheet workpiece is presented to illustrate this. Whilst this model was formulated and solved successfully, deterioration of the results for any workpiece inhomogeneity demonstrates the limitations of some of the assumptions used in these two models. Attention is then turned to curvature prediction. A review of workpiece curvature studies shows that contradictions exist in the literature; and complex nonlinear relationships are seen to exist between asymmetries, roll geometry and induced curvature. The collated data from the studies reviewed were insufficient to determine these relationships empirically; and neither analytical models, including those developed thus far, nor linear regressions are able to predict these data. Another asymmetric rolling model is developed with alternative asymptotic assumptions, which shows nonlinear behaviour over ranges of asymmetries and geometric parameters. While quantitative curvature predictions are not achieved, metrics of mechanisms hypothesised to drive curvature indicate these nonlinear curvature trends may be captured with further refinement. Finally, coupling a curved beam model with a curvature predicting rolling model is proposed to model the ring rolling process. Both of these parts are implemented but convergence between them is not yet achieved. By analogy this could be extended with shell theory and a threedimensional rolling model to model the wheeling process.

10 
An investigation into compliance and the rotating discJohn, JoAnne Louise January 2000 (has links)
No description available.

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