Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Solitons and modulation theory (4)
- Combustion theory (2)
- EQUATION (2)
- Microwave heating (2)
- KdV equation; modulation theory; initial boundary-value problems (1)
-
- MODEL; RADIATION (1)
- Microwave heating; Limit-cycles; Stability analysis; Hopf bifurcations; Control process; Thermal runaway (1)
- Microwave heating; thermal runaway; Arrhenius law; TM waveguide mode; three-dimensional bodies (1)
- Reaction-diffusion equations; Gray-Scott model; singularity theory; Hopf bifurcations; semi-analytical solutions (1)
Articles 1 - 8 of 8
Full-Text Articles in Physics
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 …