Saturated Anisotropic Porous Medium Research Articles

Nonlinear system stabilization in an anisotropic porous medium with Oldroyd-B fluid based on an actuator and sensor array

In this research, the authors precisely focus on the analysis of the chaotic behavior in an Oldroyd-B fluid saturated anisotropic porous medium via a feedback control technique. A four-dimensional (4D) weakly nonlinear system emerging from a Galerkin method of the constitutive and preservation equations is developed to accord with a convective stabilization with various Darcy numbers (Da) and feedback control gain value [Formula: see text]. The chaotic dynamic convection is governed by the Darcy–Rayleigh number ([Formula: see text]) and feedback control, while the Da has a significant impact on system stabilization. Their results reveal the effects of the feedback gain parameter [Formula: see text], stress relaxation parameter ([Formula: see text]), strain retardation parameter ([Formula: see text]), Darcy number (Da), mechanical ([Formula: see text]) and thermal ([Formula: see text]) anisotropy parameter on the stability and destabilization of thermal convection. Stabilization of thermal convection are important in cooling, nuclear power, and a range of technical, biological and engineering processes. In particular, feedback control gain values are discovered to be the preferred mode for the controlled onset of oscillatory convection. Finally, a graphical representation is presented to demonstrate that the feedback control approach is more effective in regulating the entire system when aperiodic external disturbances occur.

Effects of variable heat source on convective motion in an anisotropic porous layer

The onset of convective motion in a fluid saturated anisotropic porous medium layer is examined analytically in the occurrence of the qualitative effect of the source of variable internal heat. Three different kinds of heat source functions: (i) N (z) = z (linear) (ii)N(z) = z2 (parabolic) and (iii) N (z) = z 3 (cubic) are considered. The resulting eigenvalue problem was analytically solved using regular perturbation technique. A parametric study is carried out by varying the following parameters: the heat source parameter (X), thermal anisotropy parameter(n), and mechanical anisotropy parameter(ξ). Results indicates that the effects of the increasing values of λ and ξ will enhance the convection of a anisotropic porous layer system while the increasing of n will help to stabilize the system. It is also noted that for case (iii) the system is more stable, while for case (i) the system is more unstable

Convective instability analysis of couple-stress dielectric fluid saturated anisotropic porous medium with radiation effect

PurposeThe effects of anisotropy and radiation cannot be considered negligible while investigating the stability of the fluid in convection. Hence, the purpose of this paper is to analyze how these effects could affect the system while considering a couple-stress dielectric fluid. Therefore, the study establishes the effect of thermal radiation in a couple-stress dielectric fluid with an anisotropic porous medium using Goody's approach (Goody, 1956).Design/methodology/approachTo analyze the effect of radiation on the onset of convection, the Milne–Eddington approximation is employed to convert radiative heat flux to thermal heat flux. The equations are further developed to approximate for transparent and opaque medium. Stability of the quiescent state within the framework of linear theory is performed. The principle of exchange of stabilities is shown to be valid by means of single-term Galerkin method. Large values of conduction–radiation and absorptivity parameters are avoided as fluid is considered as liquid rather than gas.FindingsThe radiative heat transfer effect on a couple-stress dielectric fluid saturated anisotropic porous medium is examined in terms of Milne–Eddington approximation. The effect of couple-stress, dielectric, anisotropy and radiation parameters are analyzed graphically for both transparent and opaque medium. It is observed that the conduction–radiation parameter stabilizes the system; in addition, the critical Darcy–Rayleigh number also shows a stabilizing effect in the absence of couple-stress, dielectric and anisotropy parameters, for both transparent and opaque medium. Furthermore, the absorptivity parameter stabilizes the system in the transparent medium, whereas it exhibits a dual effect in the case of an opaque medium. It was also found that an increase in thermal and mechanical anisotropy parameters shows an increase in the cell size, whereas the increase in Darcy–Roberts number and conduction–radiation parameter decreases the cell size. The validity of principle of exchange of stability is performed and concluded that marginal stability is the preferred mode than oscillatory.Originality/valueThe effects of anisotropy and radiation on Rayleigh–Bénard convection by considering a couple-stress dielectric fluid has been analyzed for the first time.

Numerical solution of the onset of Buoyancy‐driven nanofluid convective motion in an anisotropic porous medium layer with variable gravity and internal heating

AbstractThe onset of buoyancy‐driven convective motion in a nanofluid saturated anisotropic porous medium layer is examined numerically in the occurrence of uniform internal heat source under the variable gravity field. Three kinds of gravity force variation functions: (a) G(z) = −z (linear), (b) G(z) = −z2 (parabolic), and (c) G(z) = −z3 (cubic) are considered. Wide‐range governing parameters impacts are inspected on the beginning of convective motion under the zero nanoparticle flux situation at the boundaries using the higher term Galerkin technique. It is established that the thermal anisotropy parameter η and the gravity variation parameter λ delay the arrival of convective motion, while the mechanical anisotropy parameter ξ, the internal heating parameter Hs, the nanoparticle Rayleigh‐Darcy number Rnp, the modified diffusivity ratio NAnf, and the modified nanofluid Lewis number Lenf rapid the start of convective motion. The size of the convective cells reduces on raising the internal heating parameter Hs, while the gravity variation parameter λ, the mechanical anisotropy parameter ξ, the thermal anisotropy parameter η, the nanoparticle Rayleigh‐Darcy number Rnp, the modified diffusivity ratio NAnf, and the modified nanofluid Lewis number Lenf amplify the dimension of the convective cells. It is also detected that the arrangement is more unstable for case (c), while it is more stable for case (a).

Magnetohydrodynamic stability of pressure-driven flow in an anisotropic porous channel: Accurate solution

The stability of fully developed pressure-driven flow of an electrically conducting fluid through a channel filled with a saturated anisotropic porous medium is studied under the influence of a uniform transverse magnetic field using a modified Brinkman equation. An analogue of Squire's transformation is used to show that two-dimensional motions are more unstable than three-dimensional ones. The modified Orr–Sommerfeld equation for the problem is solved numerically and a more accurate solution is obtained using the Chebyshev collocation method combined with Newton's and golden section search methods. The critical Reynolds number Rc and the corresponding critical wave number αc are computed for a wide range of porous parameter σp, the ratio of effective viscosity to the fluid viscosityΛ, the mechanical anisotropy parameter K1, the porosity ε and the Hartman number M. It is found that the system remains unconditionally stable to small-amplitude disturbances for the Darcy case and the energy stability analysis is also performed to corroborate this fact.

Anisotropic porous penetrative convection for a local thermal non-equilibrium model with Brinkman effects

The intent of the present problem is to observe the effect of internal heat source on the onset of convection in a fluid saturated anisotropic porous medium. Brinkman model is employed to formulate the momentum equation and fluid and solid phases are considered to be at local thermal non-equilibrium. The linear and non-linear theories are implemented to analyze the stability by computing the critical Rayleigh number and corresponding wave number. Normal mode technique is used for linear analysis and energy method is used for non-linear stability analysis. The presence of heat generation leads to the possibility of the existence of a subcritical instability. Effects of increase of Darcy-Brinkman number, inter-phase heat transfer parameter, anisotropy parameter, porosity-modified conductivity ratio and internal heat parameter on critical Rayleigh number was analyzed numerically using Chebyshev pseudospectral method.

Stability analysis of dissolution-driven convection in porous media

We study the stability of dissolution-driven convection in the presence of a capillary transition zone and hydrodynamic dispersion in a saturated anisotropic porous medium, where the solute concentration is assumed to decay via a first-order chemical reaction. While the reaction enhances stability by consuming the solute, porous media anisotropy, hydrodynamic dispersion, and capillary transition zone destabilize the diffusive boundary layer that is unstably formed in a gravitational field. We perform linear stability analysis, based on the quasi-steady-state approximation, to assess critical times, critical wavenumbers, and neutral stability curves as a function of anisotropy ratio, dispersivity ratio, dispersion strength, material parameter, Bond number, Damköhler number, and Rayleigh number. The results show that the diffusive boundary layer becomes unstable in anisotropic porous media where both the capillary transition zone and dispersion are considered, even if the geochemical reaction is significantly large. Using direct numerical simulations, based on the finite difference method, we study the nonlinear dynamics of the system by examining dissolution flux, interaction of convective fingers, and flow topology. The results of nonlinear simulations confirm the predictions from the linear stability analysis and reveal that the fingering pattern is significantly influenced by combined effects of reaction, anisotropy, dispersion, and capillarity. Finally, we draw conclusions on implications of our results on carbon dioxide sequestration in deep saline aquifers.

Modeling Isothermal Flow in Saturated Porous Media Compressing between Two Parallel Plates

Many of manufacturing processes in composite industry consist of injecting/squeezing a metallic, ceramic, or polymeric melt in-to/out-of a deformable porous material, which can be made of solid constituents in the form of mats, fibers, whiskers, particulates, flakes, or wires. Examples of practical application of such processes are compression molding, injection/compression type liquid composite molding and squeeze casting, with additional applications in the fields of lubrication and bio-medicine. This paper presents a unified approach of developing governing equations for pressure during the squeezing of saturated porous media between two parallel plates. It starts with a combination of Darcy's law and continuity equation that yields a partial differential equation for pressure inside saturated porous media. The suggested governing equation was also solved analytically for two simple cases: 1) squeezing a saturated anisotropic porous medium between two rectangular plates, 2) squeezing a saturated isotropic porous medium between two circular plates. The results of the corresponding pressure and velocity fields were discussed.

Thermovibrational Instability in a Fluid Saturated Anisotropic Porous Medium

A comprehensive investigation is made to understand the effect of harmonic vibration on the onset of convection in a horizontal anisotropic porous layer heated either from below or from above. The layer is subject to vertical mechanical vibrations of arbitrary amplitude and frequency. The porous medium is assumed to be both mechanically and thermally anisotropic, and Brinkman’s law is invoked to model the momentum balance. Both continued fraction and Hill’s infinite determinant methods are used to determine the convective instability threshold with the aid of Floquet theory. The synchronous and subharmonic resonant regions of dynamic instability are determined and their critical boundaries are found. The results show that anisotropy in permeability favors convection whereas that in thermal conductivity suppresses it with a wider cellular pattern at the instability threshold. The influence of vibration parameters and heating condition on the anisotropy effects and the competition between the synchronous and subharmonic modes are discussed. This study also reveals the existence of a closed disconnected instability region in certain areas of the parameter space for the first time in literature.

Double diffusive convection in a vertical enclosure filled with anisotropic porous media

This paper summarises a numerical study of double-diffusive natural convection in a rectangular cavity filled with a saturated anisotropic porous medium. The side walls of the cavity are maintained at constant temperatures and concentrations, while the horizontal walls are adiabatic and impermeable. The buoyancy forces that induce the fluid motion are assumed to be cooperative. Numerical results are presented for 10 2≤ R T≤10 4, 0≤ N≤30, 10 −5≤ K≤10 3, 1≤ Le≤10, and A=1, where R T, N, K, Le and A denote the thermal Rayleigh number, buoyancy ratio, permeability ratio, Lewis number and the aspect ratio. In the two extreme cases of heat-driven ( N≪1) and solute-driven ( N≫1) natural convection, scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime. Also, based on the numerical results, correlations for the average Nusselt and Sherwood numbers are proposed for the range of parameters considered in this study. It is demonstrated that the anisotropic properties of the porous medium considerably modify the heat and mass transfer rates from that expected under isotropic conditions. The Brinkman's extension of Darcy's law is also used in this study to investigate double-diffusive convection in anisotropic porous media with high porosity.

Non-Darcy natural convection in a hydrodynamically and thermally anisotropic porous medium

The natural convective flow and heat transfer in a fluid saturated anisotropic porous medium have been investigated using the generalised non-Darcy model. A semi-implicit, Galerkin, velocity correction procedure has been employed to solve the governing partial differential equations. Numerical results are presented for different inclinations of the principal permeability directions, Rayleigh and Darcy numbers and permeability and thermal conductivity ratios. The comparison of present results with existing analytical and numerical results shows that present formulation is accurate. A parametric study has been carried out and the difference between the Brinkman and the generalised porous medium models in the context of anisotropic flow is discussed. Significant difference between these models is noticed at higher Rayleigh and Darcy numbers.

Notes on convection in a porous medium (i): an effective Rayleigh number for an anisotropic layer, (ii) the Malkus hypothesis and wavenumber selection

Studies of convection in a layer of a saturated anisotropic porous medium uniformly heated from below are reviewed. Emphasis is placed on the usefulness of an effective Rayleigh number, defined (for certain boundary conditions) as the square harmonic-mean square root of horizontally-based and vertically-based Rayleigh numbers. The status of the Malkus hypothesis, together with its relationship with observed cell size, is also discussed. A possible explanation is provided for the observed phenomenon that in a porous medium the cell size decreases as the amplitude of convection increases, whereas in a clear fluid the cell size increases.

NATURAL CONVECTION IN A VERTICAL SLOT FILLED WITH AN ANISOTROPIC POROUS MEDIUM WITH OBLIQUE PRINCIPAL AXES

Natural convection heat transfer in a rectangular cavity filled with a saturated anisotropic porous medium is investigated analytically and numerically. The side walls of the cavity are heated and cooled isothernutlly, while the horizontal walls are adiabatic. The porous medium is anisotropic in permeability with its principal axes oriented in a direction that is oblique to the gravity vector. In the large-Rayleigh-number limit a boundary layer solution, based on the integral relations approach developed by Simpkins and Blythe [11] is proposed. Scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime. Comparisons between the numerical solution of the full equations and analytical solutions are presented for a wide range of the governing parameters. The numerical experiments confirm the flow features and scales anticipated by the approximate boundary layer solution. It is demonstrated that the anisotropic properties and different orientation of the porous medium co...

Natural convection in a vertical cylinder filled with anisotropic porous media

A SIMPLE numerical algorithm is used to analyze the natural convection in a vertical cylindrical enclosure filled with a saturated anisotropic porous medium. The boundary conditions of the enclosure are assumed to be maintained at a uniform high temperature except that the temperature of the bottom is low. The effects of anisotropic permeability ratio, anisotropic thermal conductivity ratio, geometrical aspect ratio and Rayleigh number on the flow field and heat transfer are investigated. Numerical results show that the heat transfer rates of the side and bottom walls increase, either decreasing the values of anisotropic permeability ratio or increasing the values of anisotropic thermal conductivity ratio, and the side average Nusselt number increases as the aspect ratio decreases.

Natural Convection in an Anisotropic Porous Medium Heated from the Side. Effects of Amisotropic Properties of Porous Matrix.

Natural convection heat transfer in a square cavity filled with a saturated anisotropic porous medium has been investigated analytically. One vertical wall is kept at a constant temperature which is higher than that of the other. Top and bottom walls are insulated. All boundaries are assumed to be nonpenetrable for the fluid. The temperature and stream function are expanded about Ra, the Rayleigh number of the system. The series solutions are then substituted in the governing equations. Equating terms of equal power of Ra, a number of sets of the homogeneous or nonhomogeneous linear systems are generated. The linear systems are solved by Fourier series. Anisotropy of both permeability and thermal conductivity is considered in the process of solving these linear systems. The degrees of anisotropy of fluid permeability Kx/Ky and of thermal conductivity kx/ky are varied from 0.1 to 100 and from 0.01 to 100, respectively. The effect of both anisotropy of permeability and thermal conductivity on the overall heat transfer is found to be equally significant. It is also found that the anisotropy of a porous medium produces similar effects on flow and temperature patterns as does the geometric aspect ratio of the rectangular cavity.