Limit cycles in the presence of convection, a first order analysis

We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads pattern outwards from the source. Convection adds instability to the reaction-diffusion system. We see the result of the instability in a readiness to create pattern. In the case of strong convection, we consider that the first-order approximation may be valid for some aspects of the solution behaviour. We employ the method of Riemann invariants and rescaling to transform the reduced system into one invariant under parameter change. We carry out numerical experiments to test our analysis. We find that most aspects of the solution do not comply with this, but we find one significant characteristic which is approximately first order. We consider the correspondence of the Partial Differential Equation with the Ordinary Differential Equation along rays from the initiation point in the transformed system. This yields an understanding of the behaviour

A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis

Convergence analysis of a vertex-centered finite volume scheme for a copper heap leaching model

In this paper a two-dimensional solute transport model is considered to simulate the leaching of copper ore tailing using sulfuric acid as the leaching agent. The mathematical model consists in a system of differential equations: two diffusion–convection-reaction equations with Neumann boundary conditions, and one ordinary differential equation. The numerical scheme consists in a combination of finite volume and finite element methods. A Godunov scheme is used for the convection term and an P1-FEM for the diffusion term. The convergence analysis is based on standard compactness results in L2. Some numerical examples illustrate the effectiveness of the scheme

Limit cycles in the presence of convection, a travelling wave analysis

We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We find a transformation which maps the irregular equations into model form. Next we transform the dependent variables into polar form. From here, a travelling wave analysis is performed on the radial variable. Results are complex, but we make some simple estimates. We carry out numerical experiments to test our analysis. An initial `knock' starts the propagation of pattern. The speed of the travelling wave is not quite as expected. We investigate further. The system demonstrates distinctly different behaviour to the left and the right. We explain how this phenomenon occurs by examining the underlying behaviour.Comment: 20 pages, 5 figure

More than Skew: Asymmetric Wave Propagation in a Reaction-Diffusion-Convection System

Convection-induced instability in reaction-diffusion systems produces complicated patterns of oscillations behind propagating wavefronts. We transform the system twice: into lambda-omega form, then into polar variables. We find analytical estimates for the wavefront speed which we confirm numerically.Our previous work examined a simpler system [E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 (2007)]; the onset of instability is qualitatively different  in numerical solutions of this system. We modify our estimates and connect the two different behaviours. Our estimate explains how the Turing instability fits with pattern found in reaction-diffusion-convection systems. Our results can have important applications to the pattern formation analysis of biological systems

Modeling convection-diffusion-reaction systems for microfluidic molecular communications with surface-based receivers in Internet of Bio-Nano Things.

We consider a microfluidic molecular communication (MC) system, where the concentration-encoded molecular messages are transported via fluid flow-induced convection and diffusion, and detected by a surface-based MC receiver with ligand receptors placed at the bottom of the microfluidic channel. The overall system is a convection-diffusion-reaction system that can only be solved by numerical methods, e.g., finite element analysis (FEA). However, analytical models are key for the information and communication technology (ICT), as they enable an optimisation framework to develop advanced communication techniques, such as optimum detection methods and reliable transmission schemes. In this direction, we develop an analytical model to approximate the expected time course of bound receptor concentration, i.e., the received signal used to decode the transmitted messages. The model obviates the need for computationally expensive numerical methods by capturing the nonlinearities caused by laminar flow resulting in parabolic velocity profile, and finite number of ligand receptors leading to receiver saturation. The model also captures the effects of reactive surface depletion layer resulting from the mass transport limitations and moving reaction boundary originated from the passage of finite-duration molecular concentration pulse over the receiver surface. Based on the proposed model, we derive closed form analytical expressions that approximate the received pulse width, pulse delay and pulse amplitude, which can be used to optimize the system from an ICT perspective. We evaluate the accuracy of the proposed model by comparing model-based analytical results to the numerical results obtained by solving the exact system model with COMSOL Multiphysics

The effects of natural convection on low temperature combustion

When a gas undergoes an exothermic reaction in a closed vessel, spatial temperature gradients can develop. If these gradients become sufficiently large, the resulting buoyancy forces will move the gas, i.e. there is natural convection. The nature of the resulting flow is determined by the Rayleigh number, Ra = (β g ΔT L^3) / (κ ν). The evolution of such a system will depend on the interactions of natural convection, diffusion of both heat and chemical species, and chemical reaction. This study is concerned with a gas-phase system undergoing Sal’nikov’s reaction: P → A → B, in the presence of natural convection. This kinetic scheme is used as a simplified representation of a cool flame, which is a feature of the low temperature combustion of a hydrocarbon vapour. Sal’nikov’s reaction is one of the simplest to display thermokinetic oscillations, such as those seen in cool flames. The behaviour of Sal’nikov’s reaction in the presence of natural convection was investigated using a combination of analytical and numerical techniques. First, a numerical model was developed to compute the temperature, velocity and concentrations when a simple exothermic reaction occurs in a spherical batch reactor, the results of which could be compared with previous experimental measurements. Subsequently, a scaling analysis of Sal’nikov’s reaction proceeding in a spherical reactor was performed. This yielded significant insight into the general behaviour of this and similar systems. The forms of the analytical scales were confirmed through comparison with the results from numerical simulations. These scales were used to predict how the system responds to changes in certain key process variables, such as the pressure and the size of the reactor. It was shown that the behaviour of this system is governed by the ratios of the characteristic timescales for diffusion, reaction and natural convection. These ratios were used to define a regime diagram describing the system. The behaviour in different parts of this regime diagram was characterised and regions in which oscillations occur were identified.Engineering and Physical Sciences Research Council and Society of Chemical Industr

Analytical and Numerical Study of Photocurrent Transients in Organic Polymer Solar Cells

This article is an attempt to provide a self consistent picture, including existence analysis and numerical solution algorithms, of the mathematical problems arising from modeling photocurrent transients in Organic-polymer Solar Cells (OSCs). The mathematical model for OSCs consists of a system of nonlinear diffusion-reaction partial differential equations (PDEs) with electrostatic convection, coupled to a kinetic ordinary differential equation (ODE). We propose a suitable reformulation of the model that allows us to prove the existence of a solution in both stationary and transient conditions and to better highlight the role of exciton dynamics in determining the device turn-on time. For the numerical treatment of the problem, we carry out a temporal semi-discretization using an implicit adaptive method, and the resulting sequence of differential subproblems is linearized using the Newton-Raphson method with inexact evaluation of the Jacobian. Then, we use exponentially fitted finite elements for the spatial discretization, and we carry out a thorough validation of the computational model by extensively investigating the impact of the model parameters on photocurrent transient times.Comment: 20 pages, 11 figure

The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting