Cosymmetric families of steady states in 3D convection of incompressible fluid in a porous medium

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial decay in a double diffusive convection problem in Darcy flow

This paper deals with a model of time-dependent double diffusive convection in Darcy flow. In particular it is concerned with the spatial decay of solutions when the flow is confined to a semi-infinite cylinder. Decay bounds for an energy expression are derived.     

Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow

In this paper, the Darcy model is used to describe the double diffusive flow of a fluid containing a solute. Continuous dependence of the solution on the Soret coefficient is established.     

On steady states in a collisionless plasma

Motivated by the collisionless shock problem in plasma, we study the stationary Vlasov-Maxwell system in one space dimension via a dynamical system approach. We construct flat-tail and oscillatory-tail solutions, as well as solitons in the presence of a nontrivial magnetic field. Flat-tail solutions connect different densities, pressures, and strengths of constant fields at x = ±∞, while oscillatory-tail solutions oscillate as x → −∞. We also construct topological horseshoes, which imply very complicated structures of some steady states. © 1996 John Wiley & Sons, Inc.. Inc.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Shooting methods for some steady diffusion and convection problems

For diffusion-dominated steady flows, classical second-order methods are usually used. A large number of iterations, and hence a long computing time, is required to solve the set of discretized equations using an iterative method. On the other hand, a direct solver is degraded because of the accumulation of round- off errors. For convection-dominated flows, first-order upwinding has been used over the past few decades but suffered from severe inaccuracy. In this paper we first discuss the accuracy improvement of solving a diffusion equation by shooting methods. We manage to achieve the theoretical order of accuracy as the mesh size decreases as far as single-precision arithmetic is concerned. We then discuss an application to the interface coupling of subproblems in the context of domain decomposition methods. Finally, we discuss high-order nonoscillatory solutions of a convection-diffusion equation based on shooting methods.     

Non-constant positive steady states of the Sel'kov model

This paper deals with the reaction-diffusion system known as the Sel'kov model with the homogeneous Neumann boundary condition. This model has been applied to various problems in chemistry and biology. We first give a priori estimates (positive upper and lower bounds) of positive steady states, and then study the non-existence, bifurcation and global existence of non-constant positive steady states as the parameters λ and θ are varied.     

Natural convection in porous media ‐ a meshless solution of the Brinkman extended Darcy model

A numerical study is performed on steady natural convection inside a differentially heated square cavity. The cavity is filled with porous media which exhibits the Brinkman extended Darcy behavior. The solution procedure for coupled mass, momentum, and energy equations is based on primitive variables and RBF collocation method with r⁷ function. Numerical examples include calculations at filtration with Rayleigh number 100, and Darcy numbers 10^–3 and 10^–5. The solution is compared with reference results of the fine‐grid finite volume method.     

On the stability of steady states in a granuloma model

We consider a free boundary problem for a system of two semilinear parabolic equations. The system represents a simple model of granuloma, a collection of immune cells and bacteria filling a 3-dimensional domain Ω(t)$\Omega (t)$ which varies in time. We prove the existence of stationary spherical solutions and study their linear asymptotic stability as time increases to infinity.     

An unified numerical approach of steady convection between two parallel plates

This paper considers the problem of laminar forced convection between two parallel plates. We present an unified numerical approach for some problems related to this case: the problem of viscous dissipation with Dirichlet and Neumann boundary conditions and the Graetz problem. The solutions of these problems are obtained by a series expansion of the complete eigenfunctions system of some Sturm-Liouville problems. The eigenfunctions and eigenvalues of this Sturm-Liouville problem are obtained by using Galerkin’s method. Numerical examples are given for viscous fluids with various Brinkman numbers.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Tracking and controlling unstable steady states of dynamical systems

An adaptive controller for stabilization of unknown unstable steady states (spirals, nodes and saddles) of nonlinear dynamical systems is considered and its robustness under the changes of the location of the fixed point in the phase space is demonstrated. An analog electronic controller, based on a low-pass filter technique, is described. It can be easily switched between a stable and an unstable mode of operation for stabilizing either spirals/nodes or saddles, respectively. Numerical and experimental results for two autonomous systems, the damped Duffing–Holmes oscillator and the chaotic Lorenz system, are presented.     

Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics

In this paper we study a nonlocal reaction–diffusion–advection system modeling the growth of multiple competitive phytoplankton species in a vertical water column with incomplete mixing. We find that when the diffusion of the system is large, there is no positive steady states, and when the diffusion is not large, there exists at least one positive steady states under certain conditions. The main tools we use are the fixed point index theory, a refined comparison theorem and fine properties of the principal eigenvalues.