On the Stress State of a Heterogeneous Plate Made of a Material with a Step Change in Elastic Characteristics in the Presence of an Inclusion System and Temperature Influence

In the formulation of thermoelasticity and in the framework of the conventional theory of thermal stresses, the problem on the stress state of an elastic piecewise-homogeneous plane or an infinite plate at non-uniform steady-state heating is considered. On the interface of dissimilar materials, the compound plane is reinforced by a collinear system of absolutely rigid thin inclusions and is subjected to mechanical and thermal influences. First, to determine the temperature distribution in a piecewise-homogeneous plane the corresponding boundary value problem of the theory of steady-state heat conduction is solved using the integral Fourier transform. Solving this problem is reduced to solving a singular integral equation (SIE) that allows an exact solution. Further, the elastic displacements of points of the compound plane, caused by mechanical and temperature influence, are determined by the known methods of thermoselasticity. Based on these results, solving the problem of the contact interaction between the system of inclusions and a compound plane is again reduced to solving SIE, which also allows an exact solution. A special case is considered.

Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.

A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

CROSS-DIFFUSION INDUCED INSTABILITY IN LVLEV–TANNER MODEL

A Lvlev–Tanner model with cross-diffusion is considered. We analyze the positive uniform steady state and obtain conditions on the parameter values such that the homogeneous steady state is locally asymptotically stable both in the related ODE system and in the PDE system with self-diffusion. Once also cross-diffusion is considered in the model, the uniform steady state is shown to be unstable under some conditions. Numerical simulations are also presented.

Dynamics and stability of an annular electrolyte film

We investigate the evolution of an electrolyte film surrounding a second electrolyte core fluid inside a uniform cylindrical tube and in a core-annular arrangement, when electrostatic and electrokinetic effects are present. The limiting case when the core fluid electrolyte is a perfect conductor is examined. We analyse asymptotically the thin annulus limit to derive a nonlinear evolution equation for the interfacial position, which accounts for electrostatic and electrokinetic effects and is valid for small Debye lengths that scale with the film thickness, that is, charge separation takes place over a distance that scales with the annular layer thickness. The equation is derived and studied in the Debye-Hückel limit (valid for small potentials) as well as the fully nonlinear Poisson–Boltzmann equation. These equations are characterized by an electric capillary number, a dimensionless scaled inverse Debye length and a ratio of interface to wall electrostatic potentials. We explore the effect of electrokinetics on the interfacial dynamics using a linear stability analysis and perform extensive numerical simulations of the initial value problem under periodic boundary conditions. An allied nonlinear analysis is carried out to investigate fully singular finite-time rupture events that can take place. Depending upon the parameter regime, the electrokinetics either stabilize or destabilize the film and, in the latter case, cause the film to rupture in finite time. In this case, the final film shape can have a ring- or line-like rupture; the rupture dynamics are found to be self-similar. In contrast, in the absence of electrostatic effects, the film does not rupture in finite time but instead evolves to very long-lived quasi-static structures that are interrupted by an abrupt re-distribution of these very slowly evolving drops and lobes. The present study shows that electrokinetic effects can be tuned to rupture the film in finite time and the time to rupture can be controlled by varying the system parameters. Some intriguing and novel behaviour is also discovered in the limit of large scaled inverse Debye lengths, namely stable and smooth non-uniform steady state film shapes emerge as a result of a balance between destabilizing capillary forces and stabilizing electrokinetic forces.

Development of particle migration in pressure-driven flow of a Brownian suspension

An experimental investigation into the influence of Brownian motion on shear-induced particle migration of monodisperse suspensions of micrometre-sized colloidal particles is presented. The suspension is pumped through a 50 μm × 500 μm rectangular cross-section glass channel. The experiments are characterized chiefly by the sample volume fraction (φ = 0.1 − 0.4), and the flow rate expressed as the Péclet number (Pe = 10 − 400). For each experiment we measure the entrance length, which is the distance from the inlet of the channel required for the concentration profile to develop to its non-uniform steady state. The entrance length increases strongly with increasing Pe for Pe ≪ 100, in marked contrast to non-Brownian flows for which the entrance length is flow-rate independent. For larger Pe, the entrance length reaches a constant value which depends on the other experimental parameters. Additionally, the entrance length decreases with increasing φ; this effect is strongest for low φ. Modelling of the migration based on spatial variation of the normal stresses due to the particles captures the primary features observed in the axial evolution over a range of Pe and φ.