Critical Wave Number Research Articles

Faraday instability of viscous liquid films on a heated substrate with Maxwell–Cattaneo heat flux

Faraday instability of viscous liquid films with Maxwell–Cattaneo (MC) heat flux on an infinite, heated horizontal substrate subject to vertical time-varying periodic vibration is investigated theoretically. The MC effect means that the response of the heat flux to a temperature gradient obeys a relaxation time law rather than a classical Fourier time law. Applying the classic Floquet theory to linear analysis, a recursive relation is obtained. When considering the MC effect, a new phenomenon appears at a large wave number k. The neutral stability curves branch new tongues that turn left rather than right as before, but the tongues still move up and right as the wave number increases. Furthermore, typical harmonic (H) and subharmonic (SH) alternation behavior continues to exist. Interestingly, the first tongue of a branch is H or SH, implying that there is a transition following the branches. However, near the critical wave number kc of a branch, the SH and H almost overlap. As Cattaneo number C increases, the tongue-like unstable zones of branches become wider, and the critical wave number kc of the appeared branch becomes small. As the driving frequency ω decreases, the branch tongues become elongated and the critical wave number kc of the appeared branch becomes small.

Nonlinear modulation of dispersive fast magnetosonic waves in an inhomogeneous rotating solar low-β magnetoplasma

Abstract We study the modulation of fast magnetosonic waves (MSWs) in rotating inhomogeneous low-β magnetoplasmas with the effects of gravitation and the Coriolis force. By employing the standard multiple-scale reductive perturbation technique (RPT), we derive a nonlinear Schrödinger (NLS) equation that governs the evolution of slowly varying MSW envelopes. The fast MSW becomes dispersive by the effects of the Coriolis force in the fluid motion, and the magnetic field and density inhomogeneity effects favor the Jeans instability in self-gravitating plasmas in a larger domain of the wave number (k, below the Jeans critical wave number, kJ) than homogeneous plasmas. The relative influence of the Jeans frequency (ωJ, associated with the gravitational force) and the angular frequency (Ω0, relating to the Coriolis force) on the Jeans carrier MSW mode and the modulational instability (MI) of the MSW envelope is studied. We show that the MSW envelope (corresponding to the unstable carrier Jeans mode with ωJ > 2Ω0 and k < kJ) is always unstable against the plane wave perturbation with no cut-offs for growth rates. In contrast, the stable Jeans mode with ωJ > 2Ω0 but k > kJ manifests either modulational stability or MI having a finite growth rate before being cut off. We find an enhancement of the MI growth rate by the influence of magnetic field or density inhomogeneity. The case with constant gravity force (other than the self-gravity) perpendicular to the magnetic field is also briefly discussed to show that the fast magnetosonic carrier mode is always unstable, giving MI of slowly varying envelopes with no cut-offs for the growth rates. Possible applications of MI in solar plasmas, such as those in the X-ray corona, are also briefly discussed.

Exploring stability of Jeffrey fluids in anisotropic porous media: incorporating Soret effects and microbial systems

Purpose This paper aims to present a study on stability analysis of Jeffrey fluids in the presence of emergent chemical gradients within microbial systems of anisotropic porous media. Design/methodology/approach This study uses an effective method that combines non-dimensionalization, normal mode analysis and linear stability analysis to examine the stability of Jeffrey fluids in the presence of emergent chemical gradients inside microbial systems in anisotropic porous media. The study focuses on determining critical values and understanding how temperature gradients, concentration gradients and chemical reactions influence the onset of bioconvection patterns. Mathematical transformations and analytical approaches are used to investigate the system’s complicated dynamics and the interaction of numerous characteristics that influence stability. Findings The analysis is performed using the Jeffrey-Darcy type and Boussinesq estimation. The process involves using non-dimensionalization, using the normal mode approach and conducting linear stability analysis to convert the field equations into ordinary differential equations. The conventional thermal Rayleigh Darcy number RDa,c is derived as a comprehensive function of various parameters, and it remains unaffected by the bio convection Lewis number Łe. Indeed, elevating the values of ζ and γ′ in the interval of 0 to 1 has been noted to expedite the formation of bioconvection patterns while concurrently expanding the dimensions of convective cells. The purpose of this investigation is to learn how the temperature gradient affects the concentration gradient and, in turn, the stability and initiation of bioconvection by taking the Soret effect into the equation. The results provide insightful understandings of the intricate dynamics of fluid systems affected by chemical and biological elements, providing possibilities for possible industrial and biological process applications. The findings illustrate that augmenting both microbe concentration and the bioconvection Péclet number results in an unstable system. In this study, the experimental Rayleigh number RDa,c was determined to be 4π2at the critical wave number ( δcˇ) of π. Originality/value The study’s novelty originated from its investigation of a novel and complicated system incorporating Jeffrey fluids, emergent chemical gradients and anisotropic porous media, as well as the use of mathematical and analytical approaches to explore the system’s stability and dynamics.

Exact hydrodynamic manifolds for the linear Boltzmann BGK equation I: spectral theory

We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number kcrit\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{crit}}$$\\end{document} which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.

Effect of viscosity variation with temperature on convective instability in a hot dusty ferrofluid layer with permeable boundaries

As the subject under consideration has numerous industrial, biological, and environmental applications, so the prime objective of this article is to investigate the onset of thermal convection in a dusty ferrofluid layer under the effect of change in viscosity due to temperature in the presence of permeable boundaries. The linear stability analysis is employed to acquire the eigenvalue problem and then the eigenvalue problem is solved for the stationary convection mode using a single-term Galerkin method. The numerical values of critical wave number and critical Rayleigh number are calculated using MATLAB software, and the results are graphically displayed. The investigation shows that the permeability of the boundaries affects the size of the convection cells that form at the beginning of thermal convection. Further, it is found that the parameters related to temperature-dependent viscosity and permeable boundaries have a stabilizing effect on the system, while magnetic and dust particle parameters destabilize the system.

Amplitude equations for wave bifurcations in reaction–diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction–diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary n-component systems ( n⩾3 ) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction–diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

Gravitational instability in magnetized viscoelastic fluid with radiation pressure effect

This study examines the gravitational instability of a finitely conducting magnetized viscoelastic fluid, under the influence of radiation pressure within the framework of a generalized hydrodynamic fluid model. The general dispersion relation, valid for strongly and weakly coupled fluids under kinetic and hydrodynamic limits respectively, is derived using normal mode analysis for a transverse wave propagation mode. The obtained Jeans instability criteria, in terms of critical Jeans wavenumbers under both limits, are modified by the presence of radiative effects, viscoelastic compressional speed, and Alfvén wave velocity. The findings highlight that viscoelastic compressional speed and radiative pressure slow the Jeans instability's growth rate, exerting a stabilizing effect on the initiation of gravitational collapse. The present investigation provides valuable insights into the role of radiative mechanisms in the gravitational collapse within the strongly coupled region of molecular cloud clumps.

Transient internal wave excitation of resonant modes in a density staircase

The density of the ocean generally increases continuously with depth as a consequence of variations in salinity and temperature. In some regions, however, the density profile of the ocean adopts a (double diffusive) staircase structure in which successive layers of uniform density fluid are separated by rapid density jumps. Previous work has theoretically examined the transmission and reflection of periodic internal (gravity) waves incident upon a density staircase. This predicted the existence of transmission spikes (global modes) for certain combinations of frequency and horizontal wave number in which the incident waves transmit perfectly across a density staircase. It was hypothesized that the transmission spikes occur when the incident waves resonate with natural modes of disturbances in the staircase. Here we derive theory to investigate the interactions between incident internal waves and modes. We demonstrate a close correspondence between the frequency for incident waves at a transmission spike and the real part of the frequency of modes at the same horizontal wave number. However, the frequency of the corresponding modes has a negative imaginary part corresponding to exponential decay of the modes in time. We perform numerical simulations to examine the impact of this near-resonant coupling when a vertically localized, quasimonochromatic internal wave packet interacts with a density staircase. In a range of simulations with fixed incident wave frequency and varying horizontal wave number, the measured transmission coefficient does not exhibit transmission spikes, but decreases monotonically with increasing horizontal wave number about the critical wave number separating strong and weak transmission. We show this occurs because the incident wave excites modes that then slowly transmit energy above and below the staircase at a rate consistent with the predicted decay rate of the modes. This rate is slower for staircases with more steps with the decay time increasing as the cube of the number of steps. Published by the American Physical Society 2024

Surface wrinkling of a hyperelastic half-space coated by a liquid crystal elastomer film

We consider the stability of a hyperelastic substrate coated by a liquid crystal elastomer film and subjected to compressive forces. In this problem, the liquid crystal elastomer directors are free to evolve and this possible variation needs to be included in the stability analysis. We consider the case where the initial directors are aligned either in the horizontal or in the vertical direction and obtain an exact bifurcation condition for surface wrinkling. We show that director reorientation increases both the critical compressive strain and the critical wavenumber, hence stabilizing the material. In the small wavenumber limit we carry out an asymptotic analysis and obtain analytical solutions for the critical stretch and the critical wavenumber, which can be useful in applications.

Radiation pressure and galactic cosmic rays-driven gravitational instability in rotating and magnetized viscoelastic fluids

This paper studies the combined effects of radiation and galactic cosmic ray pressures on the gravitational instability of magnetized and rotating viscoelastic fluids. The dispersion relations are derived using the normal mode analysis and discussed in the hydrodynamic (weakly coupled fluid) and kinetic (strongly coupled fluid) limits. These dispersion relations are analyzed separately for transverse and longitudinal wave propagation modes. Jeans instability criteria are obtained for kinetic and hydrodynamic limits for both modes of wave propagation, and it is found that the critical Jeans wavenumbers in each case are modified due to the presence of viscoelastic effects, radiation and cosmic rays pressures, and Alfvên wave velocity. It is also observed that the radiation pressure, cosmic ray pressure and viscoelastic parameters suppress the growth rate and thus have stabilizing effects on the Jeans instability. However, cosmic ray diffusion has a destabilizing effect on the onset of gravitational instability. The effects of various parameters on the growth rate of instability are calculated numerically and the outcomes are depicted graphically. The results of the present analysis shall be helpful in understanding the impact of cosmic rays and radiative mechanisms on the gravitational collapse in the viscoelastic region of molecular cloud clumps.

Formation of convective roll patterns over a longitudinal throughflow in an active inhomogeneous porous medium with weak clogging

The convective stability of a plane-parallel longitudinal throughflow in a system consisting of two porous sublayers with distinct permeability is studied taking into account the weak pore clogging effect and given a finite difference in concentration between the upper and lower boundaries of the system. A system of differential equations for convection is derived within the framework of the continuum approach and the nonlinear MIM model. A linear stability analysis of the basic longitudinal flow is carried out. The boundary value problem is solved using the numerical technique for constructing a fundamental system of solutions. The neutral stability curves of the basic flow with respect to the disturbances in the form of rolls with different wavelength are obtained. For the absence of pore clogging, the convective stability maps of the critical solutal Rayleigh-Darcy number versus the permeability ratio for the initial uncontaminated sublayers, as well as the plots of the critical disturbance frequency and wave number versus the permeability ratio for some values of the Peclet number, are constructed. It is shown that these results have symmetry with respect to the solution for equal initial permeability of sublayers. The clogging effect associated with the solute adsorption and desorption in a porous medium is studied in the two-layered systems with the values 0.1 and 10 of the ratio of initial permeabilities of the sublayers; convection localization occurs in distinct sublayers. It was found that clogging increases the stability of the longitudinal basic flow relative to solutal inhomogeneity and breaks the symmetry of solutions with respect to the case when the two-layered system is reduced to a single layer with the same filtration properties of the porous medium throughout its volume. The influence of sorption effects on the critical parameters of oscillatory convection is investigated by plotting the critical solutal Rayleigh-Darcy number, disturbance frequency and wave number against the adsorption coefficient at fixed values of the desorption coefficient. The isolines of the stream function are constructed to analyze convective flow patterns. It was established that there is a weak relationship between the typical size of local convective roll patterns and the clogging parameters. It is shown that the sorption processes, which can lead to clogging, have a significant impact on the velocity of convective rolls moving along the two-layered system at a slight variation in their sizes.

Magnetic field influence on Casson fluid flow in rotating convection

The stability of buoyant flow in an infinite extended vertical fluid layer bounded by impermeable conducting isothermal rigid walls, known as magnetic field influence on Casson fluid flow in rotating convection, is investigated. A system of governing equations (Navier–Stokes, heat, and induction ones) is solved with isothermal rigid boundary conditions. When the majority of electrically conducting fluids are extremely small, the stability of governing equations can be simplified by taking the smallness of magnetic Prandtl number into account. In linear stability, the Chebyshev collocation method is used to solve numerically the system of eigenvalue problems. The Casson fluid parameter, Chandrasekhar number, magnetic Prandtl number, and Taylor number all have destabilizing effects on the system's basic velocity and basic magnetic field, resulting in instability. The critical Rayleigh number (Rc), critical wave number (ac), and critical wave speed (cc) are calculated using the influence of governing parameters. The Casson fluid parameter and magnetic Prandtl number were found to stabilize stationary disturbances in neutral stability curves.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Elastic instabilities of soft laminates with stiffening behavior

This paper investigates the elastic instability behavior in soft periodic laminates subjected to finite strains, with a focus on both macroscopic and microscopic instabilities. Considering the deformation-induced phase stiffening, the Gent model with a high bulk-to-shear modulus ratio describes the behavior of incompressible phases. This non-Gaussian statistics-based model captures the non-linear constitutive results from the limited extensibility of polymeric molecular chains. This paper derives an analytical prediction for the onset of macroscopic (or longwave) instability and microscopic instability as functions of material parameters. Moreover, a numerical Bloch-Floquet analysis is imposed on identifying the instability behavior under compression. We consider a wide range of phase combinations and find that the relatively rapid stiffening of the matrix compared to the stiff layer increases the stability of laminates by decreasing the critical stretch ratio. Essentially, properly manipulating the stiffening parameters can produce an absolutely stable region without observed instability. This paper also systematically illustrates the changes in instability and the transition between macro and micro instability in fully Gent laminates, which show higher stability than fully neo-Hookean laminates with larger critical stretch ratios. The critical characteristics of instabilities, such as critical stretch ratios and critical wavenumbers, can be controlled by the choice of stiffening parameters and other material properties, enlarging the tuning of soft laminates for desired buckling patterns in practical applications.

Interface coupling effect and multi-mode Faraday instabilities in a three-layer fluid system

We investigate the Faraday instabilities of a three-layer fluid system in a cylindrical container containing low-viscosity liquid metal, sodium hydroxide solution and air by establishing the Mathieu equations with considering the viscous model derived by Labrador et al. (J. Phys.: Conf. Ser., vol. 2090, 2021, 012088). The Floquet analysis, asymptotic analysis, direct numerical simulation and experimental method are adopted in the present study. We obtain the dispersion relations and critical oscillation amplitudes of zigzag and varicose modes from the analysis of the Mathieu equations, which agree well with the experimental result. Furthermore, considering the coupling strength of two interfaces, besides zigzag and varicose modes, we find a beating instability mode that contains two primary frequencies, with its average frequency equalling half of the external excitation frequency in the strongly coupled system. In the weakly coupled system, the $A$ -interface instability, $B$ -interface instability and $A$ & $B$ -interface instability are defined. Finally, we obtain a critical wavenumber $k_c$ that can determine the transition from zigzag or varicose modes to the corresponding $A$ -interface or $B$ -interface instability.

Thermo-magnetic convection and convective magneto-gravitational parametric resonance in a horizontal ferrofluid layer under time quasi-periodic vibrations: A balance harmonic analysis

Convective instability in a horizontal ferromagnetic liquid layer with rigid boundaries is investigated in the presence of both an uniform magnetic field and a vertical quasi-periodic forcing having two incommensurate frequencies ω1 and ω2. A linearized convective instability analysis predicting the onset of convection is performed and by means of a Galerkin projection truncated to the first order, the linear dynamic of the system is reduced to a damped quasi-periodic Mathieu oscillator. Using a balance harmonic analysis, it turns out that at sufficiently low vibration frequencies, we recover the critical instability parameters related to the unmodulated case where the convection is mainly due to the magnetic mechanism. This feature means that the buoyancy mechanism does not contribute into the convective cells formation and consequently heating from below or from above does not alter the onset of convection. However, at sufficiently high vibration frequencies, both the magnetic and buoyancy mechanisms are operative and here we are dealing with a magneto-gravitational parametric resonance convection. The transition between these convective modes is occurred via bicritical states giving rise to cusp points in the magnetic Rayleigh number besides discontinuities in its corresponding critical wave number. In addition, it is shown that a proper tuning of the irrational frequency ratio ω=ω2/ω1 provides a control of the convection onset as well as the nature of its mechanism. Moreover, special attention is devoted to highlight the effect of the Prandtl number on the dynamic of the system at different values of the frequency ratio ω.

ON THERMAL CONVECTION IN ROTATING CASSON NANOFLUID PERMEATED WITH SUSPENDED PARTICLES IN A DARCY-BRINKMAN POROUS MEDIUM

The present paper investigates the effect of suspended particles on thermal convection in rotating Casson nanofluids saturating a Darcy-Brinkman porous medium which has various applications in different sectors, including those that process food, paint, water generators, electricity generators, hydrology, and geophysics, heavily rely on rotation in thermal convection. With the aid of the Galerkin 1st approximation technique, the numerical examination is carried out. The Darcy-Brinkman porous media and particles suspension are taken into consideration throughout the conduct of this study. The non-Newtonian Casson nanofluid, Darcy-Brinkman porous medium, particle suspension and rotation parameter, and their impact on thermal convection have been analyzed and presented graphically for free-free, rigid-rigid, and rigid-free boundaries. It is found that for all boundary conditions the Casson nanofluid and suspended particle parameters have destabilizing impact on the stationary convection, whereas the parameter which occurred due to presence of rotation, i.e., Taylor number and Brinkman porous medium parameters, both delayed the stationary convection. In addition, we have discovered that for realistic rigid-rigid boundary condition, the system is determined to be more stable than for the other two boundary conditions. Also, on the basis of several approximations on the Taylor number and other parameters, the critical wave number and value for stationary convection are determined.

THE INFLUENCE OF THE MARANGONI EFFECT ON THE ONSET OF GRAVITACTIC BIOCONVECTION

Self-movement of such microorganisms as Tetrahymena against gravity leads to the accumulation of cells at the free surface and, as a consequence, to an unstable density distribution. The study considers the onset of gravitactic bioconvection in a horizontal layer of a nonisothermal liquid with a free nondeformable boundary, taking into account the dependence of surface tension on the concentration of microorganisms. The dependence of the critical concentration Rayleigh number characterizing the onset of instability on the Marangoni number is analytically obtained in the long-wave approximation. It is shown that the long-wave concentration Rayleigh number does not depend on the thermal Rayleigh number, as well as on the Schmidt and Lewis numbers. The parameter ranges (the Marangoni and thermal Rayleigh numbers) in which the long-wave instability exists are found. Critical wave numbers and critical concentration Rayleigh numbers are determined from the numerical solution of the linear stability problem.

Floquet-based analysis on three-dimensional non-axisymmetric instabilities in oscillatory-driven Taylor–Couette flows and their low-frequency asymptotic behavior using Wentzel–Kramers–Brillouin method

This paper revisits the linear stability analysis of oscillatory-driven flows between two oscillating cylinders against non-axisymmetric disturbances. This study is motivated by the lack of a sufficiently reliable theoretical analysis giving insight into the experimentally observed spiral-like non-axisymmetric patterns when the cylinders are counter-oscillating. A new generalized time-dependent algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional Navier–Stokes equations around the purely azimuthal basic state. Numerical evaluation of the critical eigenvalues combining both Floquet theory and spectral method reveals the existence of frequency ranges where this basic state becomes unstable against three-dimensional non-axisymmetric disturbances before it does so for two-dimensional axisymmetric ones. Indeed, as the oscillation frequency of the cylinders increases, the azimuthal wave number of the critical eigensolution is found to change from 0 to 2 to 1 and then back to 0. The primary bifurcation exchange between two instability modes with different azimuthal wave numbers occurs via different types of codimension-2 bifurcation points giving rise to discontinuities in the critical axial wave number where reversing and non-reversing non-axisymmetric Taylor vortex flows are identified. In addition, by extending our numerical calculations to the co-oscillating case, we show that the axisymmetric disturbances are the most unstable confirming thus existing experimental findings. Furthermore, a Wentzel–Kramers–Brillouin (WKB) analysis is performed to shed light on the asymptotic behavior of these time-dependent flows in the low-frequency limit when the cylinders are slowly oscillating.

Dissipation induced instability in porous medium using Brinkman model: Thermal non-equilibrium effect

The convective instability that arises in a porous layer solely because of dissipative energy in the medium is studied. The thermal non-equilibrium model is considered for energy equation. The boundary at top is kept at constant temperature, and an adiabatic lower boundary is considered. The momentum equation is written according to the Brinkman model. A basic flow in the horizontal plane is considered, and the basic velocity and temperature profile have been derived. A comparison between the basic temperature profiles for solid and fluid of the porous medium is presented for Darcy and Brinkman models. Infinitesimal disturbance is introduced to the basic flow. A linear stability analysis has been carried out to study the stability of the basic flow. The parameters influencing the stability of the system are critical Rayleigh number (RaC), Gebhart number (Ge), ξ associated with the Darcy number (Da), and interphase heat transfer coefficient (H). The values of the critical Rayleigh number and wavenumber are compared by varying other variables. The flow is more stable when the Brinkman medium is considered. With the increase in interphase heat transfer coefficient, the critical Rayleigh number increases in both Darcy and Brinkman media. The longitudinal rolls appear as the most unstable rolls for all cases. The variation of convective rolls with respect to ξ and heat transfer coefficient is presented.