Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity

Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u$, which causes the density to disperse. However, RDEs with negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a non-classical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are initially multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u=1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce an new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position.Comment: 35 pages, 10 figure

Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations

The behaviour of many systems in chemistry, combustion and biology can be described using nonlinear reaction diffusion equations. Here, we use nonclassical symmetry techniques to analyse a class of nonlinear reaction diffusion equations, where both the diffusion coefficient and the coefficient of the reaction term are spatially dependent. We construct new exact group invariant solutions for several forms of the spatial dependence, and the relevance of some of the solutions to population dynamics modelling is discussed

Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis

We consider a family of exact solutions to a nonlinear reaction–diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher–KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher–KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.</p

Ferric Ion Diffusion for MOF-Polymer Composite with Internal Boundary Sinks

Simple and economical ferric ion detection is necessary in many industries. An europium-based metal organic framework has selective sensing properties for solutions containing ferric ions and shows promise as a key component in a new sensor. We study an idealised sensor that consists of metal organic framework (MOF) crystals placed on a polymer surface. A two-dimensional diffusion model is used to predict the movement of ferric ions through the solution and polymer, and the ferric ion association to a MOF crystal at the boundary between the different media. A simplified one-dimensional model identifies the choice of appropriate values for the dimensionless parameters required to optimise the time for a MOF crystal to reach steady state. The model predicts that a large non-dimensional diffusion coefficient and an effective association with a small effective flux will reduce the time to steady-state. The effective dissociation is the most significant parameter to aid the estimation of the ferric ion concentration. This paper provides some theoretical insight for material scientists to optimise the design of a new ferric ion sensor

Interfacial Tension Sensor for Low Dosage Surfactant Detection

Currently there are no available methods for in-line measurement of gas-liquid interfacial tension during the flotation process. Microfluidic devices have the potential to be deployed in such settings to allow for a rapid in-line determination of the interfacial tension, and hence provide information on frother concentration. This paper presents the development of a simple method for interfacial tension determination based on a microfluidic device with a flow-focusing geometry. The bubble generation frequency in such a microfluidic device is correlated with the concentration of two flotation frothers (characterized by very different adsorption kinetic behavior). The results are compared with the equilibrium interfacial tension values determined using classical profile analysis tensiometry

Convective and diffusive effects on particle transport in asymmetric periodic capillaries - Fig 1

<p>(a) Schematic of an infinitely long, periodic tube of longitudinally asymmetric profile exhibiting particles introduced () in the central wave-section and migrating out under diffusion and convection. (b) Detailed schematic of an axi-symmetric, smoothed, saw-tooth tube. Non-dimensional geometric construction particulars are indicated in the figure. The leading edge of the saw-tooth profile is that part in the positive <i>z</i> half of the wave-section having the steeper sloped boundary. The trailing edge is the shallower sloped boundary.</p

(a) The first moment, and (b) the ratio between the first and zeroth moment.

<p>Two different tube profiles are shown, both with the same throat radius, <i>B</i> = 0.2. Solid lines are for <i>A</i> = 0.24 (no recirculation case), dashed lines are for <i>A</i> = 0.48 (recirculation). Results are for various <i>β</i> values, <i>β</i> = 0, 100, 200, 300 as indicated by arrows; <i>α</i> = 0.1 in all cases. In all cases, simulations were performed with Δ<i>P</i>_<i>a</i>(<i>t</i>).</p

Comparison of results for different expansion amplitudes, <i>A</i>, with fixed <i>B</i> = 0.2.

<p>All results are for <i>α</i> = 0.1, <i>β</i> = 200. The solid lines are cases with no-recirculation (<i>A</i> = 0.12, 0.24), while the dashed lines are for cases with recirculation (<i>A</i> = 0.36, 0.48). (a) The first moment and (b) the ratio between the first and zeroth moments. The inset to (a) depicts schematically the physical significance of negative values of the first moment.</p

The time at which particles first reach <i>k</i> = ±10 (i.e., when the total mass in wave-section <i>W</i>_±10 is greater than 10⁻⁴) for different tube profiles.

<p>The solid lines are for different <i>A</i> with fixed <i>B</i> = 0.2, while the dashed lines are for different <i>B</i> with fixed <i>A</i> = 0.48. In all cases <i>α</i> = 0.1.</p