A mathematical model of meat cooking based on polymer-solvent analogy

Mathematical modelling of transport phenomena in food processes is vital to understand the process dynamics. In this work, we study the process of double sided cooking of meat by developing a mathematical model for the simultaneous heat and mass transfer. The constitutive equations for the heat and mass transport are based on Fourier conduction, and the Flory–Huggins theory respectively, formulated for a two-phase transport inside a porous medium. We investigate a reduced one-dimensional case to verify the model, by applying appropriate boundary conditions. The results of the simulation agree well with experimental findings reported in literature. Finally, we comment upon the sensitivity of the model to the porosity of meat.Science Foundation Ireland Grant SFI/12/IA/1683, and the South African DST/NRF SARChI Chair on Mathematical Methods in Bioengineering and Biosciences (M3B2).http://www.elsevier.com/locate/apm2016-07-31hb201

An explicit nonstandard finite difference scheme for the Allen-Cahn equation

We design explicit nonstandard finite difference schemes for the nonlinear Allen-Cahn reaction diffusion equation in the limit of very small interaction length ". In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimis- ing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space independent and the time independent different models. An exact nonstandard scheme is proposed for the space independent model and energy conservative schemes are proposed for the time independent model. We show the power of the derived scheme over the existing schemes through several numerical examples.South African DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M3B2).http://www.tandfonline.com/loi/gdea202016-07-31hb201

An instability theory for the formation of ribbed moraine, drumlins and mega-scale glacial lineations

We present a theory for the coupled ow of ice, subglacial water, and sub- glacial sediment, which is designed to represent the processes which occur at the bed of an ice sheet. The ice is assumed to ow as a Newtonian viscous uid, the water can ow between the till and the ice as a thin lm, which may thicken to form streams or cavities, and the till is assumed to be transported, either through shearing by the ice, squeezing by pressure gradients in the till, or by uvial sediment transport processes in streams or cavities. In previous studies, it was shown that the dependence of ice sliding veloc- ity on e ective pressure provided a mechanism for the generation of bedforms resembling ribbed moraine, while the dependence of uvial sediment transport on water lm depth provides a mechanism for the generation of bedforms re- sembling mega-scale glacial lineations (MSGL). Here we combine these two processes in a single model, and show that, depending largely on the granulom- etry of the till, instability can occur in a range of types which range from ribbed moraine through three-dimensional drumlins to mega-scale glacial lineations.Science Foundation Ireland PI grant 12/1A/1683.http://rspa.royalsocietypublishing.org2015-11-30hb201

Some optimised schemes for 1D Korteweg-de-Vries equation

Two new explicit finite difference schemes for the solution of the one-dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (1965) and Wang et al. (2008). An optimisation technique based on minimisation of the dispersion error is implemented to compute the optimal value of the spatial step size at a given value of the temporal step size and this is validated by some numerical experiments. The performance of the four methods are compared in regard to dispersive and dissipative errors and their ability to conserve mass, momentum and energy by using two numerical experiments which involve solitons.South African DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M3B2) of the University of Pretoria and the National Research Foundation of South Africa [95864, 93476].http://www.inderscience.com/jhome.php?jcode=PCFD2017-12-21hj2017Mathematics and Applied Mathematic

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R0, that is the average number of new infective individuals produced by one infective individual intro- duced into a completely susceptible population. If R0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptot- ically stable endemic equilibrium when R0 > 1. A nonstandard nite di erence scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.DST/NRF SARChI Chair in Mathematics Models and Methods in Bioengineering and Biosciences.http://link.springer.com/journal/133982017-09-30hb2016Mathematics and Applied Mathematic

Traveling wave solution of the Kuramoto-Sivashinsky equation : a computational study

This work considers the numerical solution of the Kuramoto-Sivashinsky equation using the fractional time splitting method. We will investigate the numerical behavior of two categories of the traveling wave solutions documented in the literature (Hooper & Grimshaw (1998)), namely: the regular shocks and the oscillatory shocks. We will also illustrate the ability of the scheme to produce convergent chaotic solutions.http://proceedings.aip.org/hb201

Nonstandard finite difference schemes for Michaelis-Menten type reaction-diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis-Menten (M-M) ordinary differential equation with several new non-standard finite difference (NSFD) schemes that we construct by using Mickens’ rules. Furthermore, the exact scheme of the M-M equation is used to design several dynamically consistent NSFD schemes for related reactiondiffusion equations, advection-reaction equations and advection-reaction-diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.South African National Research Foundationhttp://www3.interscience.wiley.com/journal/35979/homehb201

From enzyme kinetics to epidermiological models with Michealis-Menten contact rate : design of nonstandard finite difference schemes

We consider the basic SIR epidemiological model with the Michaelis-Menten formulation of the contact rate. From the study of the Michaelis-Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert W function and schemes obtained by using Mickens’s rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms. We compare and investigate the performance of the two types of schemes by showing that they are dynamically consistent with the continuous model. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.The South African National Research Foundationhttp://www.elsevier.com/locate/camw

On a fractional step-splitting scheme for the Cahn-Hilliard equation

PURPOSE – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. DESIGN / METHODOLOGY / APPROACH – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. FINDINGS – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. ORIGINALITY / VALUE – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.http://www.emeraldinsight.com/loi/echb201