Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.     

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.     

Onset of Buoyancy-Driven Convection in a Liquid-Saturated Cylindrical Anisotropic Porous Layer Supported by a Gas Phase

A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, and anisotropic cylindrical porous layer supported by a gas phase. Darcy’s law and Boussinesq approximation are used to explain the characteristics of fluid motion, and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the quasi-steady-state approximation, the stability equations are derived in a similar boundary layer coordinate and solved by the numerical shooting method. The critical $Ra_D$ is determined as a function of the anisotropy ratio. Also, the onset time and corresponding wavelength are obtained for the various anisotropic ratios. The onset time becomes smaller with increasing $Ra_D$ and follows the asymptotic relation derived in the infinite horizontal porous layer. Anisotropy effect makes the system more stable by suppressing the vertical velocity.     

Porous MHD Convection: Effect of Vadasz Inertia Term

The onset of porous convection in an electrically conducting fluid uniformly heated from below and embedded in an external transverse constant magnetic field is analysed. In particular the effect of Vadasz inertia term, measured through the Vadasz number \(V_a\), on the instability threshold, is investigated. For the three-dimensional perturbations and full nonlinear problem it is shown that sub-critical instabilities do not exist and the global nonlinear stability is guaranteed by the linear stability. The long-time behaviour is characterized via the existence of \(L^2\)-absorbing sets.     

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

The linear stability analysis of vertical throughflow of power law fluid for double-diffusive convection with Soret effect in a porous channel is investigated in this study. The upper and lower boundaries are assumed to be permeable, isothermal and isosolutal. The linear stability of vertical through flow is influenced by the interactions among the non-Newtonian Rayleigh number (Ra), Buoyancy ratio (N), Lewis number (Le), Péclet number (Pe), Soret parameter (Sr) and power law index (n). The results indicate that the Soret parameter has a significant influence on convective instability of power law fluid. It has also been noticed that buoyancy ratio has a dual effect on the instability of fluid flow. Further, it is noticed that the basic temperature and concentration profiles have singularities at \(Pe = 0\) and \(Le = 1\), the convective instability is looked into for the limiting case of \(Pe\rightarrow 0\) and \(Le \rightarrow 1\). For the case of pure thermal convection with no vertical throughflow, the present numerical results coincide with the solution of standard Horton–Rogers–Lapwood problem. The present results for critical Rayleigh number obtained using bvp4c and two-term Galerkin approximation are compared with those available in the literature and are tabulated.     

Three dimensional simulations of penetrative convection in a porous medium with internal heat sources

The problem of penetrative convection in a fluid saturated porous medium heated internally is analysed. The linear instability theory and nonlinear energy theory are derived and then tested using three dimensions simulation.Critical Rayleigh numbers are obtained numerically for the case of a uniform heat source in a layer with two fixed surfaces. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three dimensional simulation. Our results show that the linear threshold accurately predicts the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

The influence of geostrophic force on the stability of an heterogeneous conducting fluid with a radial gravitional force

The linear and nonlinear stability of a heterogeneous incompressible inviscid perfectly conducting fluid between two cylinders is investigated in the presence of a radial gravitational force and geostrophic force. The stability for linear disturbances is investigated using the normal mode method, while the nonlinear stability is investigated by applying the energy method. In the case of linear theory, it is found that a necessary condition for in stability is that the algebraic sum of hydrodynamic, hydromagnetic and rotation Richardson number is less than one quarter somewhere in the fluid. A semi-circle theorem similar to that of Howard is also obtained. In the case of nonlinear disturbances a universal stability estimate namely a stability limit for motions subject to arbitrary nonlinear disturbances is obtained in the form $$E \leqslant E_0 \exp ( - 2M\tau ).$$ The motion is asymptotically stable if $$\delta \leqslant 1 + J_m + J_H $$ somewhere in the fluid. This asymptotic stability limit is improved using the calculus of variation technique. We also find that whenδ=1/4, andJ _R=1, both the linear and nonlinear stability criteria coincide and in that particular case, we have a necessary and sufficient condition for stability.     

Stability of elastic bodies with nonmonotone multivalued boundary conditions of the quasidifferential type

Necessary conditions for the stability of elastic bodies subjected to nonmonotone multivalued boundary conditions are derived. These conditions are assumed to be derived from nonconvex and nonsmooth, quasidifferentiable energy functions. A difference convex approximation of the potential energy function is written based on an appropriate quasidifferential formulation. Under appropriate assumptions for the convex and the concave parts we prove the existence of at least one nontrivial solution to the nonlinear eigenvalue problem.     

Improvement for expansion of parabolized stability equation method in boundary layer stability analysis

An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.     

Onset of Buoyancy-Driven Convection in a Fluid-Saturated Porous Medium Bounded by a Long Cylinder

A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, cylindrical porous column. Darcy’s law and Boussinesq approximation are used to explain the characteristics of fluid motion and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the principle of exchange of stabilities, the stability equations are derived in self-similar boundary-layer coordinate. The present predictions suggest the critical $R_D$ , and the onset time and corresponding wavenumber for a given $R_D$ . The onset time becomes smaller with increasing $R_D$ and follows the asymptotic relation derived in the infinite horizontal porous layer.     

Influence of Chemical Reaction on Stability of Thermo-Solutal Convection of Couple-Stress Fluid in a Horizontal Porous Layer

We consider the onset of thermo-solutal convection in a couple-stress fluid-saturated anisotropic porous medium, where the chemical equilibrium on the bounding surfaces and the solubility of the dissolved components depend on temperature. The entire study has been spilt into two parts: (i) linear stability analysis (ii) weakly non-linear stability analysis. Stationary case of linear stability analysis is discussed for two modes of bounding surfaces (a) realistic bounding surfaces i.e. Rigid-Rigid and Rigid-Free (R/R and R/F), (b) non-realistic bounding surfaces i.e. Free-Free (F/F). Howsoever, investigation of oscillatory state and weakly non-linear stability are restricted to F/F case. Galerkin method is used to solve the eigenvalue problem for R/R and R/F cases, whereas, exact solutions are obtained for F/F case.A comparative study among flow stability for above different cases is made as function of ratio of viscosities ( i.e., couple-stress viscosity to fluid viscosity which is defined as couple-stress parameter, $(C)$ ) and effective chemical reaction (i.e. chemical reaction parameter, $(\chi )$ ). It has been found that increasing viscosity of the couple-stress fluid, in terms of increasing $C$ , increases flow stability in all three cases, but among all cases its stabilization effect for R/R is maximum. However, in the absence of couple-stress parameter the maximum stability of flow is observed for F/F. Apart from this, the chemical reaction stabilizes the flow for all the three cases. Furthermore, stability analysis for F/F case indicates that couple-stress parameter stabilizes the system in all modes (stationary, oscillatory and finite amplitude) of convection.Damköhler number $(\chi )$ is found to delay the stationary convection, however, it speeds up the onset of oscillatory convection. The non-linear theory based on truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of $C$ and $\chi $ on heat and mass transfer is also examined.     

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

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Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λ$\lambda$-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.     

Linear Instability of a Horizontal Thermal Boundary Layer Formed by Vertical Throughflow in a Porous Medium: The Effect of Local Thermal Nonequilibrium

In this paper we investigate the onset of convection in a saturated porous medium where uniform suction into a horizontal and uniformly hot bounding surface induces a stationary thermal boundary layer. Particular attention is paid to how the well-known linear stability characteristics of this boundary layer are modified by the presence of local thermal nonequilibrium effects. The basic conduction state is determined and it is found that the boundary layer forms two distinct regions when the porosity is small or when the conductivity of the fluid is small compared with that of the solid. A linearised stability analysis is performed which results in an ordinary differential eigenvalue problem for the critical Darcy–Rayleigh number as a function of the wave number and the two nondimensional parameters, $H$ H and $\gamma $ γ , which are associated with local thermal nonequilibrium. This eigenvalue problem is solved numerically by first approximating the equations by fourth order compact finite differences, and then the critical Rayleigh number is computed iteratively using the inverse power method and minimised over the wavenumber. The variation of the critical Rayleigh number and wavenumber with $H$ H and $\gamma $ γ is presented. One of the unusual effects of local thermal nonequilibrium is that there exists a parameter regime within which the neutral curve is bimodal.     

Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics

This paper presents stability and convergence results on a novel approach for imposing holonomic constraints for a class of multibody system dynamics. As opposed to some recent techniques that employ a penalty functional to approximate the Lagrange multipliers, the method herein defines a penalized dynamical system using penalty-augmented kinetic and potential energies, as well as a penalty dependent constraint violation dissipation function. In as much as the governing equations are not typically cocreive, the usual convergence criteria for linear variational boundary value problems are not directly applicable. Still numerical simulations by various researchers suggest that the method is convergent and stable. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Likewise, stability and asymptotic stability results for the penalty formulation are derived from well-known stability results available from classical mechanics. Unfortunately, the convergence theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of a typical dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Lyapunov/Invariance Principle analysis. In short, the approach has many advantages as an alternative to other computational techniques:

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.     

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {}}}$$-symmetric potentials

A (2+1)-dimensional nonlinear Schrödinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.     

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincaré maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping ( \(p=0.5, 2, 3\) ) suspension as for the linear damping ( \(p=1\) ) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model ( \(p=0.5\) ) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping ( \(p=2\) ) shows similar results to linear damping with some special damping coefficients. The cubic damping ( \(p=3\) ) shows more stable response than the linear damping in general.     

Linear and weakly nonlinear variable viscosity convection in spherical shells

A theoretical study of linear and weakly nonlinear thermal convection in a spherical shell is performed. The Boussinesq fluid is of infinite Prandtl number and its viscosity is temperature dependent. The linear stability eigenvalue problem is derived and solved by a shooting method assuming isothermal, stress-free boundaries, a self-gravitating fluid, and corresponding to two heating models. The first is heating from below, and the second is a model of combined heating from below and within, such that convection is described by a self-adjoint linear stability formulation. In addition, nonlinear, hemispherical, axisymmetric convection is computed by a finite volume technique for a shell with 0.5 aspect ratio. It is shown that 2-cell convection occurs as transcritical bifurcation for a viscosity constrast across the shell up to about 150. Motions with four cells are also possible. As expected, the subcritical range is found to increase with increasing viscosity contrast, even when the linear operator is self-adjoint.This research was supported by the AT&T Foundation.     

A new lattice hydrodynamic model for bidirectional pedestrian flow considering the visual field effect

In this paper, a new lattice hydrodynamic model for bidirectional pedestrian flow is proposed by considering the pedestrian’s visual field effect. The stability condition of this model is obtained by the linear stability analysis. The mKdV equation near the critical point is derived to describe the density wave of pedestrian jam by applying the reductive perturbation method. The phase diagram indicates that the phase transition occurs among the freely moving phase, the coexisting phase, and the uniformly congested phase below the critical point \(a_c\) . Furthermore, the analytical results show that the visual field effect plays an important role in jamming transition. To take into account the visual information about the motion of more pedestrian in front can improve efficiently the stability of pedestrian system. In addition, the numerical simulations are in accordance with the theoretical analysis.