A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.      

Entropy generation of double-diffusive convection in the presence of rotation

In the present study, the entropy generation of double-diffusive convection in the presence of rotation is investigated for the first time. In order to reveal the effects of rotating of the enclosure (described by the ratio of buoyancy frequency to Coriolis frequency St) and the ratio of buoyancy forces R_p on entropy generation systematically, we divide the cases investigated in the present study into two groups: (1) R_p is fixed but St varies from 0.001 to 10; (2) St is fixed but R_p varies from 0.1 to 2. We find that only fast rotation has significant influence on entropy generation distribution. Moreover, the share of irreversibility due to concentration diffusion increases quickly with R_p and it becomes the main contributor to entropy generation since R_p>0.6.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, T_A1T_{A_1}, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T_A1T_{A_1} , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.     

Effects of roughness on nonlinear stationary vortices in rotating disk flows

Primary instability of rotating disk boundary layer flow over a rough surface for stationary modes was investigated by using the weakly nonlinear theory where the Reynolds number R is close to its critical value R_c as determined by linear theory. Both the single mode case, where the wave vector K equals its critical K_c at the onset of stationary primary instability, and the bimodal case, where the wave vectors K_n (n = 1, 2) are close to K_c for the primary instability of the flow, are considered. The analysis leads to stable solutions for particular roughness forms and magnitude, and particular wave vectors ˜K_n (n = 1, 2) of the surface roughness.     

On stability concepts in nonlinear programming

Kojima's strong stability of stationary solutions can be characterized by means of first and second order terms. We treat the problem whether there is a characterization of the stability concept allowing perturbations of the objective function only, keeping the feasible set unchanged. If the feasible set is a convex polyhedron, then there exists a characterization which is in fact weaker than that one of strong stability. However, in general it appears that data of first and second order do not characterize that kind of stability. As an interpretation we have that the strong stability is the only concept of stability which both admits a characterization and works for large problem classes.Supported by the Deutsche Forschungsgemeinschaft, Graduiertenkolleg Analyse und Konstruktion in der Mathematik.Partial support under Support Center for Advanced Telecommunications Technology Research.     

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value ReE depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, ReE is shown to be at least 174.     

Global stability for nonlinear difference equations with variable coefficients

Consider the following nonlinear difference equation with variable coefficients:

     

Long time stability and convergence for fully discrete nonlinear galerkin methods

The aim of this paper is to analyze the fully discrete nonlinear Galerkin methods, which are well suited to the long time integration of dissipative partial differential equations.

With the help of several time discrete Gronwall lemmas, we are able to prove L ^∞(IR⁺,H ^α ) (α=0,1) stabilities of the fully discrete nonlinear Galerkin methods under a less restrictive time step constraint than that of the classical Galerkin methods.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

A general fuzzy control scheme for nonlinear processes with stability analysis

In this paper we develop a general fuzzy control scheme for nonlinear processes. Assuming little knowledge about the dynamics of the controlled process, the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation. A simple local controller is then designed at each fuzzy locality. A fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm. By employing a newly developed structure-adaptive fuzzy modelling scheme, the appropriate fuzzy rule-base for the inference mechanism can be extracted stably and efficiently. The conditions for the stability of the global controller are rigourously established. Simulation results are presented to illustrate the effectiveness of the scheme.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations

This paper is concerned with the problem of exponential stability for uncertain neutral switched systems with interval time-varying mixed delays and nonlinear perturbations. By using the average dwell time approach and the piecewise Lyapunov functional technique, some sufficient conditions are first proposed in terms of a set of linear matrix inequalities (LMIs), to guarantee the robustly exponential stability for the uncertain neutral switched systems, where the decay estimate is explicitly given to quantify the convergence rate. Three numerical examples are finally illustrated to show the effectiveness of the proposed method.     

Robust stability and stabilization methods for a class of nonlinear discrete-time delay systems

This paper establishes new robust delay-dependent stability and stabilization methods for a class of nonlinear discrete-time systems with time-varying delays. The parameter uncertainties are convex-bounded and the unknown nonlinearities are time-varying perturbations satisfying Lipschitz conditions in the state and delayed-state. An appropriate Lyapunov functional is constructed to exhibit the delay-dependent dynamics and compensate for the enlarged time-span. The developed methods for stability and stabilization eliminate the need for over bounding and utilize smaller number of LMI decision variables. New and less conservative solutions to the stability and stabilization problems of nonlinear discrete-time system are provided in terms of feasibility-testing of new parametrized linear matrix inequalities (LMIs). Robust feedback stabilization methods are provided based on state-measurements and by using observer-based output feedback so as to guarantee that the corresponding closed-loop system enjoys the delay-dependent robust stability with an L₂ gain smaller that a prescribed constant level. All the developed results are expressed in terms of convex optimization over LMIs and tested on representative examples.     

A note on the stability criterion for a class of nonlinear fractional differential systems

This paper is devoted to dealing with a flaw that existed in a recent paper (Zhou et al. 2014). We give a new proof of Th. 3.1 in Zhou et al. (2014), which is a correction of the original proof.     

Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate

A numerical study of magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate is presented. The onset position characterized by the local Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The Coriolis force and the Lonertz force have significant effects on the formation of longitudinal vortices and the associated instability. Positive rotation stabilizes the flow on the rotating flat surface. On the contrary, a negative rotation destabilizes the flow. The flow is found more stable as the value of Hartmann number increases. The numerical data show reasonable agreement with the experimental results with the case of thermal instability in natural convection over a flat plate heated from below.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Exponential stability criterion for time-delay systems with nonlinear uncertainties

Exponential stability of time-delay systems with nonlinear uncertainties is studied in this paper. Based on the Lyapunov method and the approaches of decomposing the matrix, a new exponential stability criterion is derived in terms of a matrix inequality, which allows to compute simultaneously the two bounds that characterize the exponential nature of the solution. Some numerical examples are also given to show the superiority of our result to those in the literature.     

General theorems for stability and boundedness for nonlinear functional discrete systems

We consider the nonlinear functional discrete system

     

A general stability result for a nonlinear wave equation with infinite memory

In this work we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, among which those of exponential decay type, usually considered in the literature, are only special cases.     

A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems

Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when α_k are determined by Hanke criterion.