Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 132–142, February 2008.     

An analytical study of nonlinear double-diffusive convection in a porous medium under temperature/gravity modulation

The article deals with nonlinear thermal instability problem of double-diffusive convection in a porous medium subjected to temperature/gravity modulation. Three types of imposed time-periodic boundary temperature (ITBT) are considered. The effect of imposed time-periodic gravity modulation (ITGM) is also studied in this problem. In the case of ITBT, the temperature gradient between the walls of the fluid layer consists of a steady part and a time-dependent periodic part. The temperature of both walls is modulated in this case. In the problem involving ITGM, the gravity field has two parts: a constant part and an externally imposed time-periodic part. Using power series expansion in terms of the amplitude of modulation, which is assumed to be small, the problem has been studied using the Ginzburg–Landau amplitude equation. The individual effects of temperature and gravity modulation on heat and mass transports have been investigated in terms of Nusselt number and Sherwood number, respectively. Further the effects of various parameters on heat and mass transports have been analyzed and depicted graphically.     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a linear instability analysis is performed. To establish conditional and unconditional nonlinear stability results, both the Darcy and Forchheimer models are employed to describe fluid flow. Due to the presence of significant regions of potential subcritical instabilities, the results indicate that linear theory may only be accurate enough to predict the onset of convective motion when the model for the internal heat source is predominantly linear.Received: 6 May 2003, Accepted: 9 August 2003, Published online: 12 December 2003     

Magnetic effect on instability and nonlinear stability of double-diffusive convection in a reacting fluid

We study the problem of double-diffusive convection in a reacting fluid with a concentration and magnetic field effect–based internal heat source. A linear instability analysis and nonlinear stability analysis are performed, and using the finite element method of p order, we get the numerical results of each case. The numerical results are presented for fixed–fixed and free–free boundary conditions.     

Global nonlinear stability for a triply diffusive convection in a porous layer

A triply convective-diffusive fluid mixture saturating a porous horizontal layer in the Darcy–Oberbeck–Boussinesq scheme is studied. The nonlinear stability analysis of the conduction solution is performed when the layer is heated from below and salted from above by one salt and below by another salt. Denoting by P _i , (i = 1, 2), the salts Prandtl numbers, it is shown that in the cases {P ₁ = 1; P ₂ = 1; P ₁ = P ₂} do not exist subcritical instabilities and the thermal Rayleigh critical number of global stability in a simple closed form is given. The methodology used and the results obtained appear to be new in the existing literature and useful for the applications.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Küppers-Lortz instability in rotating Rayleigh-Bénard convection in a porous medium

The Darcy-Lapwood-Brinkman model with the Boussinesq approximation is used to study Küppers-Lortz (KL) instability in the nonlinear regime of rotating Rayleigh-Bénard convection in a sparsely packed porous medium near the onset of stationary convection. The threshold Taylor numbers and critical angles for the onset of KL instability are obtained for different values of Λ, M for finite Prandtl numbers (1.5≤Pr≤100). Heat transfer is studied from Nusselt number at the onset of stationary convection.     

Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems

At the state of statistical stationarity, the response of a nonlinear system under multiplicative random excitations can be either trivial or non-trivial, depending on the spectral levels of the excitations and the values of certain system parameters. Assuming that the random excitations are Gaussian white noises, the two types of response may be investigated by way of their stationary densities, which are obtainable for first order dynamical systems and for higher order dynamical systems belonging to the class of generalized stationary potential. Alternatively, the Lyapunov exponents can be computed for perturbation from either the trivial or non-trivial solution, since a negative sign for the greatest Lyapunov exponent provides both the necessary and sufficient conditions for the stability of sample functions with probability one. It is shown in two specific examples, that the boundary at which the greatest Lyapunov exponent changes its sign coincides with the boundary for regularity (or being normalizable) for the probability density in both the trivial and non-trivial solutions. Thus, the stability conditions in the strong sense of probability one and the weak sense in distribution are identical in these cases.     

Study of nonlinear convection in a sparsely packed porous medium using spectral analysis

Nonlinear study cellular convection in a sparsely packed fluid saturated porous medium is investigated, considering the Brinkman model, using the technique of spectral analysis. It is established how the Brinkman model with free-free boundaries generalizes the study of convection in a porous medium in the sense that it yields the results tending to those of viscous and Darcy flows respectively for very small and very large values of the permeability parameter σ². It also provides results for the transition zone (i.e. 10²<σ²<10³). The cross-interaction of the linear modes caused by non-linear effects are considered through the modal Rayleigh number R_γ. The possibility of the existence of steady solution with two self-excited modes in certain regions is predicted. The similarities of present analysis with and advantages over that of the power integral technique are brought out. A detailed discussion of the heat transport, with the effect of permeability thereon, is made. The theoretical values of the Nusselt number are found to be in good agreement with experimental results.     

Necessary and sufficient conditions for the absolute stability of discrete type lurie control system

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Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Onset of double-diffusive convection in a saturated porous layer with time-periodic surface heating

The stability of a fluid saturated, horizontal porous layer in the presence of a solute concentration gradient and time-periodic thermal gradient is examined. The modulated gradient is the result of a sinusoidal upper surface temperature which models the effect of variable solar radiation heating of the layer. Darcy's law and the Boussinesq approximation are employed, and we assume an equation of state linear in temperature and concentration. A linear stability analysis is carried out to obtain predictions for the onset of convection and critical wavenumbers for the system. The critical conditions are obtained via the Galerkin method and Floquet theory. The effects of variable concentration gradient, temperature modulation amplitude and frequency are examined, and compared with the results obtained analytically from the corresponding unmodulated problem. It is shown that instabilities can occur as convective motions which are synchronous or subharmonic with the surface heating, or can be identified via complex conjugate Floquet exponents. The neutral stability curves at the transitions between instabilities are found to be bimodal when the temperature is time-periodic, and are characterized by jumps in the critical wavenumbers. Received February 5, 1998     

Horizontal thermal convection in a porous medium

Linearised instability and nonlinear stability bounds for thermal convection in a fluid-filled porous finite box are derived. A nonuniform temperature field in the steady state is generated by maintaining the vertical walls at different temperatures. The linearised instability threshold is shown to be well above the global stability boundary, which is strongly suggested by the lack of symmetry in the perturbed system. The numerical results are evaluated utilising a newly developed Legendre-polynomial-based spectral method.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

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