Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio

Effect of Interfacial Tension on Propagating Polymerization Fronts

This paper is devoted to the investigation of polymerization fronts converting a liquid monomer into a liquid polymer. We assume that the monomer and the polymer are immiscible and study the influence of the interfacial tension on the front stability. The mathematical model consists of the reaction-diffusion equations coupled with the Navier-Stokes equations through the convection terms. The jump conditions at the interface take into account the interfacial tension. Simple physical arguments show that the same temperature distribution could not lead to Marangoni instability for a nonreacting system. We fulfill a linear stability analysis and show that interaction of the chemical reaction and of the interfacial tension can lead to an instability that has another mechanism: the heat produced by the reaction decreases the interfacial tension and initiates the liquid motion. It brings more monomer to the reaction zone and increases even more the heat production. This feedback mechanism can lead to the instability if the frontal Marangoni number exceeds a critical value. (C) 2000 American Institute of Physics. [S1054-1500(00)01701-8]

The Effect of Convection on a Propagating Front with a Liquid Product: Comparison of Theory and Experiments

This work is devoted to the investigation of propagating polymerization fronts converting a liquid monomer into a liquid polymer. We consider a simplified mathematical model which consists of the heat equation and equation for the depth of conversion for one-step chemical reaction and of the Navier-Stokes equations under the Boussinesq approximation. We fulfill the linear stability analysis of the stationary propagating front and find conditions of convective and thermal instabilities. We show that convection can occur not only for ascending fronts but also for descending fronts. Though in the latter case the exothermic chemical reaction heats the cold monomer from above, the instability appears and can be explained by the interaction of chemical reaction with hydrodynamics. Hydrodynamics changes also conditions of the thermal instability. The front propagating upwards becomes less stable than without convection, the front propagating downwards more stable. The theoretical results are compared with experiments. The experimentally measured stability boundary for polymerization of benzyl acrylate in dimethyl formamide is well approximated by the theoretical stability boundary. (C) 1998 American Institute of Physics

Coronavirus: Scientific insights and societal aspects

In December 2019, the first case of infection with a new virus COVID-19 (SARS-CoV-2), named coronavirus, was reported in the city of Wuhan, China. At that time, almost nobody paid any attention to it. The new pathogen, however, fast proved to be extremely infectious and dangerous, resulting in about 3–5% mortality. Over the few months that followed, coronavirus has spread over entire world. At the end of March, the total number of infections is fast approaching the psychological threshold of one million, resulting so far in tens of thousands of deaths. Due to the high number of lives already lost and the virus high potential for further spread, and due to its huge overall impact on the economies and societies, it is widely admitted that coronavirus poses the biggest challenge to the humanity after the second World war. The COVID-19 epidemic is provoking numerous questions at all levels. It also shows that modern society is extremely vulnerable and unprepared to such events. A wide scientific and public discussion becomes urgent. Some possible directions of this discussion are suggested in this article

The role of platelets in blood coagulation during thrombus formation in flow

Hemostatic plug covering the injury site (or a thrombus in the pathological case) is formed due to the complex interaction of aggregating platelets with biochemical reactions in plasma that participate in blood coagulation. The mechanisms that control clot growth and which lead to growth arrest are not yet completely understood. We model them with numerical simulations based on a hybrid DPD-PDE model. Dissipative particle dynamics (DPD) is used to model plasma flow with platelets while fibrin concentration is described by a simplified reaction-diffusion-convection equation. The model takes into account consecutive stages of clot growth. First, a platelet is weakly connected to the clot and after some time this connection becomes stronger due to other surface receptors involved in platelet adhesion. At the same time, the fibrin network is formed inside the clot. This becomes possible because flow does not penetrate the clot and cannot wash out the reactants participating in blood coagulation. Platelets covered by the fibrin network cannot attach new platelets. Modelling shows that the growth of a hemostatic plug can stop as a result of its exterior part being removed by the flow thus exposing its non-adhesive core to the flow

On a Conjecture of Goriely for the Speed of Fronts of the Reaction--Diffusion Equation

In a recent paper Goriely considers the one--dimensional scalar reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction term $f(u)$ and conjectures the existence of a relation between a global resonance of the hamiltonian system $u_{xx} + f(u) = 0$ and the asymptotic speed of propagation of fronts of the reaction diffusion equation. Based on this conjecture an explicit expression for the speed of the front is given. We give a counterexample to this conjecture and conclude that additional restrictions should be placed on the reaction terms for which it may hold.Comment: 9 pages Revtex plus 4 postcript figure

Preface. Bifurcations and Pattern Formation in Biological Applications

In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena

Propagation of a Solitary Fission Wave

Reaction-diffusion phenomena are encountered in an astonishing array of natural systems. Under the right conditions, self stabilizing reaction waves can arise that will propagate at constant velocity. Numerical studies have shown that fission waves of this type are also possible and that they exhibit soliton like properties. Here, we derive the conditions required for a solitary fission wave to propagate at constant velocity. The results place strict conditions on the shapes of the flux, diffusive, and reactive profiles that would be required for such a phenomenon to persist, and this condition would apply to other reaction diffusion phenomena as well. Numerical simulations are used to confirm the results and show that solitary fission waves fall into a bistable class of reaction diffusion phenomena. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4729927]United States Nuclear Regulatory Commission NRC-38-08-946Mechanical Engineerin

Flame Enhancement and Quenching in Fluid Flows

We perform direct numerical simulations (DNS) of an advected scalar field which diffuses and reacts according to a nonlinear reaction law. The objective is to study how the bulk burning rate of the reaction is affected by an imposed flow. In particular, we are interested in comparing the numerical results with recently predicted analytical upper and lower bounds. We focus on reaction enhancement and quenching phenomena for two classes of imposed model flows with different geometries: periodic shear flow and cellular flow. We are primarily interested in the fast advection regime. We find that the bulk burning rate v in a shear flow satisfies v ~ a*U+b where U is the typical flow velocity and a is a constant depending on the relationship between the oscillation length scale of the flow and laminar front thickness. For cellular flow, we obtain v ~ U^{1/4}. We also study flame extinction (quenching) for an ignition-type reaction law and compactly supported initial data for the scalar field. We find that in a shear flow the flame of the size W can be typically quenched by a flow with amplitude U ~ alpha*W. The constant alpha depends on the geometry of the flow and tends to infinity if the flow profile has a plateau larger than a critical size. In a cellular flow, we find that the advection strength required for quenching is U ~ W^4 if the cell size is smaller than a critical value.Comment: 14 pages, 20 figures, revtex4, submitted to Combustion Theory and Modellin