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Articles 91 - 116 of 116
Full-Text Articles in Physics
Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant
Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant
Tim Marchant
The diffractive resolution on a collisionless shock formed along the spatial profile of a beam in a nematic liquid crystal is considered, this material being an example of a self-focusing, nonlocal medium. It is found that the shock is resolved through the formation of an undular bore structure which persists for experimentally relevant propagation distances due to nonlocality delaying the onset of modulational instability. Both 1+1 and 2+1 dimensional bores with circular symmetry are considered (termed line and circular bores, respectively). A semianalytical solution is developed for the line undular bore, approximating it as a train of uniform solitary waves. …
Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant
Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant
Tim Marchant
An ionic version of the Gray–Scott chemical reaction scheme is considered in a reaction–diffusion cell, with an applied electric field, which causes migration of the reactant and autocatalyst in a preferred direction. The Galerkin method is used to reduce the governing partial differential equations to an approximate model consisting of ordinary differential equations. This is accomplished by approximating the spatial structure of the reactant and autocatalyst concentrations. Bifurcation analysis of the semi-analytical model is performed by using singularity theory to analyse the static multiplicity and a stability analysis to determine the dynamic multiplicity. The application of the electric field causes …
Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant
Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant
Tim Marchant
The propagation of coherent, polarized light in a nematic liquid crystal, governed by the nematicon equations, is considered. It is found that in the special case of 1 + 1 dimensions and the highly nonlocal limit, the nematicon equations have an asymptotic bulk solitary wave solution, termed a nematicon, which is given in terms of Bessel functions. This asymptotic solution gives both the ground state and the symmetric and antisymmetric excited states, which have multiple peaks. Numerical simulations of nematicon evolution, for parameters corresponding to experimental scenarios, are presented. It is found, for experimentally reasonable parameter choices, that the validity …
Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant
Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant
Tim Marchant
The interaction of two symmetric solitary waves, termed nematicons, in a liquid crystal is considered in the limit of nonlocal response of the liquid crystal. This nonlocal limit is the applicable limit for most experimentally available liquid crystals. In this nonlocal limit, two separate cases for the initial separation of the nematicons are considered, these being large and small separation. Both spinning and nonspinning nematicons are considered. It is found that in the case of large initial separation, the nematicons can form a spinning or nonspinning bound state with a finite steady separation, this being called a nematicon dipole, when …
Simulation And Analysis Of Ultrafast Laser Pulse Induced Plasma Generation In Dielectric Materials, Jeremy Gulley, Sebastian Winkler, William Dennis
Simulation And Analysis Of Ultrafast Laser Pulse Induced Plasma Generation In Dielectric Materials, Jeremy Gulley, Sebastian Winkler, William Dennis
Jeremy R. Gulley
Recent experiments on optical damage by ultrashort laser pulses have demonstrated that the temporal pulseshape can dramatically influence plasma generation in fused silica and sapphire. In this work a modified 3+1D nonlinear Schroedinger equation for the pulse propagation coupled to a rate equation for the plasma density in the dielectric material is used to simulate pulse propagation and plasma formation in a range of dielectric materials. We use these simulations to analyze the influence of pulse-width, pulse-shape and beam geometry on the formation of the electron plasma and hence damage in the bulk material. In particular, when possible, we simulate …
Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant
Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant
Tim Marchant
The increase in temperature in compost piles/landfill sites due to micro-organisms undergoing exothermic reactions is modelled. A simplified model is considered in which only biological self-heating is present. The heat release rate due to biological activity is modelled by a function which is a monotonic increasing function of temperature over the range 0⩽T⩽a, whilst for T⩾a it is a monotone decreasing function of temperature. This functional dependence represents the fact that micro-organisms die or become dormant at high temperatures. The bifurcation behaviour is investigated for 1-d slab and 2-d rectangular slab geometries. In both cases there are two generic steady-state …
Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant
Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant
Tim Marchant
Solitary wave interaction for a higher-order version of the nonlinear Schrödinger (NLS) equation is examined. An asymptotic transformation is used to transform a higher-order NLS equation to a higher-order member of the NLS integrable hierarchy, if an algebraic relationship between the higher-order coefficients is satisfied. The transformation is used to derive the higher-order one- and two-soliton solutions; in general, the N-soliton solution can be derived. It is shown that the higher-order collision is asymptotically elastic and analytical expressions are found for the higher-order phase and coordinate shifts. Numerical simulations of the interaction of two higher-order solitary waves are also performed. …
Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant
Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant
Tim Marchant
The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts. Numerical simulations of both elastic and inelastic collisions are performed. For the example covered …
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant
Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant
Tim Marchant
Wool scouring is the process of washing dirty wool after shearing. Our model numerically simulates contaminant movement in a wool scour bowl using the advection–dispersion equation. This is the first wool scour model to give time-dependent results and to model the transport of contaminants within a single scour bowl. Our aim is to gain a better understanding of the operating parameters that will produce efficient scouring. Investigating the effects of varying the parameters reveals simple, interesting relationships that give insight into the dynamics of a scour bowl.
An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant
An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant
Tim Marchant
Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. The undular bore solution for the higher-order Korteweg-de Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. The higher-order KdV equation considered includes all possible third-order correction terms (where the KdV equation retains second-order terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higher-order undular bore. Examples of higher-order undular bores, describing both surface and internal waves, are presented. …
Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant
Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant
Tim Marchant
Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and …
Modelling A Wool Scour Bowl, Tim Marchant
Modelling A Wool Scour Bowl, Tim Marchant
Tim Marchant
Wool scouring is the process of washing dirty wool after shearing. Our model simulates, using the advection-diffusion equation, the movement of contaminants within a scour bowl. The effects of varying the important parameters are investigated. Interesting, but simple, relationships are found which give insight into the dynamics of a scour bowl.
Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant
Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant
Tim Marchant
A 1D partial differential equation (pde) describing the flow of magma in the Earth's mantle is considered, this equation allowing for compaction and distension of the surrounding matrix due to the magma. The equation has periodic travelling wave solutions, one limit of which is a solitary wave, called a magmon. Modulation equations for the magma equation are derived and are found to be either hyperbolic or of mixed hyperbolic/elliptic type, depending on the specific values of the wave number, mean height and amplitude of the underlying modulated wave. The periodic wave train is stable in the hyperbolic case and unstable …
Microwave Thawing Of Cylinders., Tim Marchant
Microwave Thawing Of Cylinders., Tim Marchant
Tim Marchant
Microwave thawing of a cylinder is examined. The electromagnetic field is governed by Maxwell's equations, where the electrical conductivity and the thermal absorptivity are both assumed to depend on temperature. The forced heat equation governs the absorption and diffusion of heat where convective heating occurs at the surface of the cylinder, while the Stefan condition governs the position of the moving phase boundary. A semi-analytical model, which consists of ordinary differential equations, is developed using the Galerkin method. Semi-analytical solutions are found for the temperature, the electric-field amplitude in the cylinder and the position of the moving boundary. Two examples, …
Asymptotic Solitons For A Third-Order Kortweg-De Vries Equation, Tim Marchant
Asymptotic Solitons For A Third-Order Kortweg-De Vries Equation, Tim Marchant
Tim Marchant
Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic and …
Semi-Analytical Solutions For One - And Two-Dimensional Pellet Problems., Tim Marchant
Semi-Analytical Solutions For One - And Two-Dimensional Pellet Problems., Tim Marchant
Tim Marchant
The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law …
Cubic Autocatalysis With Michaelis - Menten Kinetics: Semi-Analytical Solutions For The Reaction - Diffusion Cell, Tim Marchant
Cubic Autocatalysis With Michaelis - Menten Kinetics: Semi-Analytical Solutions For The Reaction - Diffusion Cell, Tim Marchant
Tim Marchant
Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of …
A Comparison Of Semi-Analytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A Three-Dimensional Waveguide, Prof. Tim Marchant
A Comparison Of Semi-Analytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A Three-Dimensional Waveguide, Prof. Tim Marchant
Tim Marchant
The microwave heating of a three-dimensional block in an infinitely long rectangular waveguide propagating the TE10 mode is considered. The electrical conductivity (the dielectric loss) is assumed to be a function of temperature, and modelled by the Arrhenius law. A coupled set of equations is obtained that describes the electromagnetic fields and the temperature distribution in the block. The numerical solutions of this problem are obtained by two methods, the well known FD-TD scheme and a frequency domain method which makes the further assumption that a single TE10 mode exists in the waveguide and material. The results show that an …
Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant
Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant
Tim Marchant
Solitary wave interaction for a higher-order modified Korteweg–de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the example …
High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant
High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant
Tim Marchant
The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right- and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right- and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of …
The Occurrence Of Limit-Cycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant
The Occurrence Of Limit-Cycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant
Tim Marchant
The microwave heating of one- and two-dimensional slabs, subject to linear feedback control, is examined. A semianalytical model of the microwave heating is developed using the Galerkin method. A local stability analysis of the model indicates that Hopf bifurcations occur; the regions of parameter space in which limit-cycles exist are identified. An efficient numerical scheme for the solution of the governing equations, which consist of the forced heat equation and a Helmholtz equation describing the electric-field amplitude, is also developed. An excellent comparison between numerical solutions of the semianalytical model and the governing equations is obtained for the temporal evolution …
Cubic Autocatalytic Reaction-Diffusion Equations: Semi-Analytical Solutions, Prof. Tim Marchant
Cubic Autocatalytic Reaction-Diffusion Equations: Semi-Analytical Solutions, Prof. Tim Marchant
Tim Marchant
The Gray-Scott model of cubic-autocatalysis with linear decay is coupled with diffusion and considered in a one-dimensional reactor (a reaction-diffusion cell). The boundaries of the reactor are permeable, so diffusion occurs from external reservoirs that contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations. The ordinary differential equations are then analysed to obtain semi-analytical results for the reaction-diffusion cell. Steady-state concentration profiles and bifurcation diagrams are obtained …
Semi-Analytical Solutions For Continuous-Flow Microwave Reactors, Prof. Tim Marchant
Semi-Analytical Solutions For Continuous-Flow Microwave Reactors, Prof. Tim Marchant
Tim Marchant
A prototype chemical reaction is examined in both one and two-dimensional continuous-flow microwave reactors, which are unstirred so the effects of diffusion are important. The reaction rate obeys the Arrhenius law and the thermal absorptivity of the reactor contents is assumed to be both temperature- and concentration-dependent. The governing equations consist of coupled reaction-diffusion equations for the temperature and reactant concentration, plus a Helmholtz equation describing the electric-field amplitude in the reactor. The Galerkin method is used to develop a semi-analytical microwave reactor model, which consists of ordinary differential equations. A stability analysis is performed on the semi-analytical model. This …
The Initial Boundary Problem For The Korteweg-De Vries Equation On The Negative Quarter-Plane, Prof. Tim Marchant
The Initial Boundary Problem For The Korteweg-De Vries Equation On The Negative Quarter-Plane, Prof. Tim Marchant
Tim Marchant
The initial boundary-value problem for the Korteweg-de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter-plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate …
The Microwave Heating Of Three-Dimensional Blocks: Semi-Analytical Solutions, Prof. Tim Marchant
The Microwave Heating Of Three-Dimensional Blocks: Semi-Analytical Solutions, Prof. Tim Marchant
Tim Marchant
The microwave heating of three-dimensional blocks, by the transverse magnetic waveguide mode TM11, is considered in a long rectangular waveguide. The governing equations are the forced heat equation and a steady-state version of Maxwell's equations, while the boundary conditions take into account both convective and radiative heat loss. Semi-analytical solutions, valid for small thermal absorptivity, are found using the Galerkin method. The electrical conductivity and the thermal absorptivity are assumed to be temperature dependent, while both the electrical permittivity and magnetic permeability are taken to be constant. Both a quadratic relation and an Arrhenius-type law are used for the temperature …
Numerical Solitary Wave Interaction: The Order Of The Inelastic Effect, Prof. Tim Marchant
Numerical Solitary Wave Interaction: The Order Of The Inelastic Effect, Prof. Tim Marchant
Tim Marchant
Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to …