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The removal of milk fouling deposits often requires the diffusion of electrolyte solutions such as sodium hydroxide through a gel. Very often more than one single anion and one single cation are involved and thus the modelling of such diffusion requires a multicomponent description. Diffusion of electrolyte solutions through gels can be modelled using the Maxwell-Stefan equation. The driving forces for diffusion are the chemical potential gradients of ionic species and the diffusion potential, i.e., the electrostatic potential induced by diffusion of the ions. A model based on the Maxwell-Stefan equation was applied to electrolyte solutions and electrolyte solutions with a gel. When modelling the diffusion of electrolyte solutions, the resulting equations were found to be a partial differential algebraic equation system with a differentiation index of two. The identification of this characteristic of the system enabled a solution method using the method of lines to be developed. When modelling the diffusion of electrolyte solutions through a gel an explicit expression for diffusion potential was developed and hence the diffusion equations were solved. Numerical solutions were presented for a number of case studies and comparisons were made with solutions from literature and between different electrolyte systems. It was found that the results of diffusion of electrolytes were in good agreement with those of experiments and literature. In the case of diffusion of electrolytes through a gel, swelling of the gel was predicted. The model can be improved by adding thermodynamic factors and can be easily extended to multiple ion systems.

The Stefan-Maxwell equations are a system of nonlinear partial differential equations that describe the diffusion of multiple chemical species in a container. These equations are of particular interest for their applications to biology and chemical engineering. The nonlinearity and coupled nature of the equations involving many variables make finding solutions difficult, so numerical methods are often used. In the engineering literature the system is inverted to write fluxes as functions of the species gradient before any numerical method is applied. In this thesis it is shown that employing a mixed finite element method makes the inversion unnecessary, allowing the numerical solution of Stefan-Maxwell equations in their primitive form. The plan of the thesis is as follows, first a mixed variational formulation will be derived for the Stefan-Maxwell equations. The nonlinearity will be dealt with through a linearization. Conditions for well-posedness of the linearized formulation are then determined. Next, the linearized variational formulation is approximated using mixed finite element methods. The finite element methods will then be shown to converge to an approximate solution. A priori error estimates are obtained between the solution to the approximate problem and the exact solution. The convergence order is then verified through an analytic test case and compared to standard methods. Finally, the solution is computed for another test case involving the diffusion of three species and compared to other methods.

Optical precursor is the sharp optical pulse front that does not show delay in absorptive media. In this thesis, optical precursor behavior in rubidium (Rb) vapor was investigated in the picoseconds regime. An amplified femtosecond laser was shaped to a 7-ps square pulse with sharp rising and trailing edges. This pulse was then sent into a hot rubidium vapor, and the center frequency of the laser pulse was absorbed. The output pulses were measured by a fast streak camera with 2-picosecond resolution. By varying the temperature of the Rb vapor, the measured pulse shapes showed the progression of formation of optical precursors. The measured pulses shapes showed good agreement with theory. On the other hand, a connection between optical precursors and femtosecond laser pumped 3-photon superradiance was investigated in this thesis. Maxwell-Bloch equations were numerically solved in two steps with commercial software Mathematica 8. A good agreement was found between simulation and experiment. It was confirmed that, at low excitation regime, superradiance generated from hot rubidium vapor, which were pumped by a femtosecond laser, can be understood as the formation of optical precursors.

The Vlasov equation describes the evolution of the distribution function of particles in phase space (x,v), where the particles interact with long-range forces, but where shortrange "collisional" forces are neglected. A space plasma consists of low-mass electrically charged particles, and therefore the most important long-range forces acting in the plasma are the Lorentz forces created by electromagnetic fields. What makes the numerical solution of the Vlasov equation a challenging task is that the fully three-dimensional problem leads to a partial differential equation in the six-dimensional phase space, plus time, making it hard even to store a discretised solution in a computer’s memory. Solutions to the Vlasov equation have also a tendency of becoming oscillatory in velocity space, due to free streaming terms (ballistic particles), in which steep gradients are created and problems of calculating the v (velocity) derivative of the function accurately increase with time. In the present thesis, the numerical treatment is limited to one- and two-dimensional systems, leading to solutions in two- and four-dimensional phase space, respectively, plus time. The numerical method developed is based on the technique of Fourier transforming the Vlasov equation in velocity space and then solving the resulting equation, in which the small-scale information in velocity space is removed through outgoing wave boundary conditions in the Fourier transformed velocity space. The Maxwell equations are rewritten in a form which conserves the divergences of the electric and magnetic fields, by means of the Lorentz potentials. The resulting equations are solved numerically by high order methods, reducing the need for numerical over-sampling of the problem. The algorithm has been implemented in Fortran 90, and the code for solving the one-dimensional Vlasov equation has been parallelised by the method of domain decomposition, and has been implemented using the Message Passing Interface (MPI) method. The code has been used to investigate linear and non-linear interaction between electromagnetic fields, plasma waves, and particles.

Thesis(M.S.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references (p.30-31).

Thèse (Ph. D.)--Université Laval, 2007. / Bibliogr.: f. 140-144. Publié aussi en version électronique dans la Collection Mémoires et thèses électroniques.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (p. [81]-83).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. [87-90]).

Thesis--Uppsala university, 1999. / Bibliogr. p. 209-219.