Horizontal Porous Layer Research Articles

Linear and weakly nonlinear analyses of double‐diffusive convection in porous media with chemical reaction using LTNE model

AbstractThe onset of convection in a horizontal porous layer with chemical reaction and local thermal nonequilibrium is investigated. The nondimensional governing equations have been solved using the normal mode technique, which results in an eigenvalue problem. The analytical expressions for both stationary and oscillatory Rayleigh numbers are obtained. The effect of different parameters has been investigated and presented. The amplitude equation is derived using weakly nonlinear theory. Nusselt number is calculated using an amplitude equation to investigate heat transport. When modeling a fluid‐saturated porous medium, previous research on double‐diffusive convection has uniformly operated under the assumption of local thermal equilibrium (LTE) between the fluid and solid phases at all points within the medium. This standard practice assumes a minimal temperature gradient between the phases at any given location. However, in practical scenarios involving high‐speed flows or significant temperature differentials between the fluid and solid phases, the LTE assumption proves insufficient.

Numerical Modeling of Non-Boussinesq Natural Convection in a Square Cavity Containing a Porous Layer

Numerical simulations were conducted to study the fluid flow and heat transfer through a partially filled square enclosure with a horizontal porous layer at the bottom. The vertical walls of the enclosure were subjected to uniform heating and cooling at specified temperatures TH and TC, respectively. The problem formulation was based on the Darcy–Brinkman model, which took into account density variation using the Mach number law approximation and incorporated variable properties. Based on the developed code, a parametric study of fluid flow and heat transfer inside the cavity was carried out to examine the effects of temperature difference, medium permeability, and Darcy number. Additionally, the variation of the Nusselt number on the hot and cold walls was provided to demonstrate the overall nature of heat transfer within the enclosure. The results of this study highlight that fluid flow and heat transfer are influenced by the temperature difference between the vertical walls and the characteristic parameters of the porous layer.

Linear and non-linear stability analysis of double-diffusive Hadley-Prats flow through horizontal porous layer with non-uniform thermal and solutal gradients

A scientific model is devised to understand the onset of double-diffusion Hadley-Prats flow through an infinite parallel permeable channel with non-uniform thermal, solutal gradients, and concentration based internal heat generation under the influence of convection conditions is considered. Darcy’s model is implemented in the context of a porous medium, which is characterized as isotropic and homogeneous. A linear and nonlinear stability analysis is conducted with the help of energy functional method and longitudinal roll perturbations are examined. The emerged dimensionless eigenvalue problem in the linear and nonlinear stability analysis is solved numerically by utilizing fourth order Runge-Kutta method (RK-4) method and shooting scheme. The critical values of wave and vertical thermal Rayleigh numbers are examined. A comprehensive analysis of solutions concerning the onset of convection mechanism is conducted. The produced results are discussed for the various values of emerged physical parameters. It is evident that, the presence of non-uniform thermal and solutal gradients and the internal heat generation mechanism causes the significant variations in the critical values of vertical thermal Rayleigh number in the channel regime.

Effect of gravity settling on the onset of thermal convection in a nanofluid-saturated porous medium layer

The onset of convection in a horizontal porous medium layer saturated with a nanofluid and heated from below is investigated via linear stability analysis and numerical simulation. The Darcy–Buongiorno model is used to describe the convective transport behaviour of the nanofluid and the settling effect of nanoparticles due to gravity is considered in addition to thermophoresis and Brownian diffusion. The linear stability analysis shows that the gravity settling is a substantial stabilizing mechanism restraining the destabilizing factors such as thermal buoyancy and thermophoresis. The stability threshold is determined by the relative strength of thermophoresis to gravity settling. It is found that the system is destabilized when the thermophoretic mobility prevails. As the nanoparticle size increases, the gravity settling effect is promoted and makes the system more stable. In particular, the onset of instability is dominated by the oscillatory mode once the nanoparticle concentration is in a stably stratified profile across the porous layer. When the Rayleigh–Darcy number $Ra_D$ exceeds the critical value, the spectrum of the growth rates of the unstable modes rises with increasing $Ra_D$ and $Rn$ (i.e. the concentration Rayleigh number), and eventually the unstable modes in the high-wavenumber region exhibit the same instability. The evolution of the convection is further examined by numerical simulation. The results verify the stability characteristics predicted by linear stability analysis. Moreover, the pattern of fingering convection of the nanofluid concentration is observed once the nanofluid concentration is unstably stratified and the density difference across the porous layer is large enough.

Saturated Porous Ferroconvection in a Ferrofluid Layer with Viscosity as a Function of Magnetic Field: Focus on Convective Boundary Condition

The present work aims to examine the influence of magnetic field dependent (MFD) viscosity on the onset of ferroconvection (FC) in a horizontal porous layer saturated with a quiescent ferrofluid (FF) and subjected to a uniform vertical magnetic field. It is assumed that the porous boundaries at the bottom and top are rigid-paramagnetic. The thermal conditions consist of a constant heat flux at the lower surface and a convective boundary condition at the upper surface, encompassing fixed temperature and uniform heat flux cases. The application of the Galerkin technique to the resulting eigenvalue problem reveals that the stability region expands as the porous parameter, Biot number, MFD viscosity parameter and magnetic susceptibility increase in magnitude. Conversely, the stability region contracts as the magnetic number and non-linearity of magnetization increase. Furthermore, it is noted that under uniform heat flux boundary conditions, the criterion for the initiation of ferroconvection remains unaffected by the non-linearity of fluid magnetization.

Linear and nonlinear stability analyses of micropolar fluid flow in horizontal porous layers

The linear and nonlinear stability analyses of micropolar fluid flow in a horizontal porous layer heated from below in the presence of throughflow is numerically investigated. The Brinkman model is considered to govern the micropolar fluid flow within the porous region. The main purpose of the present study is to investigate the behavior of the subcritical region for micropolar fluid parameters in the presence of throughflow. The energy approach is used to analyze nonlinear stability, whereas the normal mode scheme is used to investigate linear stability. The obtained eigenvalue problems related to linear and nonlinear stability analyses are solved numerically using the bvp4c routine in MATLAB. Finally, the critical thermal Rayleigh number is determined for the given values of the governing parameters. It is observed that the subcritical area decreases as the Darcy number (Da), micropolar parameter (m), and absolute value of throughflow parameter (|Pe|) decrease. Furthermore, there is no subcritical gap in the absence of the throughflow effect for micropolar fluid flow, which is a good agreement for the linear and nonlinear thresholds.

The effect of Péclet number on the onset of Casson fluid convective motion in a porous layer: Analytical and numerical investigations

In this article, we present analytical and numerical investigations of the influence of Péclet number on the arrival of Casson fluid convective motion in a horizontal porous layer utilizing the Galerkin technique. The flow in the porous matrix is modeled by an amended Casson-altered Darcy equation that considers the rheological behavior of Casson fluid. The outcomes indicate that the stability of the arrangement drops with growing the Casson parameter, while a reverse result is detected with Péclet number. We demonstrated that the oscillatory instability is not promising for the considered problem. It is also important to note that the extent of the convective cell declines with increasing the Péclet number on using the higher order Galerkin approximation while, on using the single term Galerkin approximation, the Péclet number has no impact on the magnitude of the convection cell. Further, the present results are equated with the available literature under the limiting situation of this study.

Numerical study of linear and nonlinear stability in double-diffusive Hadley-Prats flow through a horizontal porous layer with Soret and internal heat source effects

Numerical analysis is constructed to study the onset of the double-diffusion convection mechanism through an infinite parallel permeable channel with heat generation, mass flow and Soret impacts. Homogenous isotropic Darcy’s flow model is deployed to elucidate the porous features. The roll instabilities pertaining to longitudinal and transverse cases are examined through a linear and nonlinear stability analysis. The unit-less eigenvalue problem is constructed through linear and nonlinear stability assumptions and which is solved numerically using a fourth order Runge-Kutta scheme. The critical values of wave and thermal Rayleigh parameters are evaluated. Extended graphical and tabular visualization is presented to describe the onset of the convection mechanism. The results of this semi-numerical investigation may be useful in environmental, geothermal and chemical engineering processes. In addition, the critical R z value is also determined for a range of thermo-physical numbers. Significant modifications in the flow patterns are computed with mass flow parameter and vertical thermal and solutal Rayleigh number. With an increment in Soret (thermo-diffusive) parameter Sr the regime become more unstable. Based on the nonlinear stability analysis, an elevation in solutal Rayleigh number also induces earlier instability. The collective influence of the Lewis number and horizontal concentration parameter is observed to render the system more stable.

The appearance of oscillatory flows within a horizontal packed bed partly heated from below

We report on a numerical investigation of the unsteady mixed convective flow in a channel through a horizontal porous layer under localised isothermal heating from below to probe the oscillatory flow and thermal characteristics. The study was carried out for water flow within a plane channel containing stainless steel spherical particles. For this study, the channel height/particle diameter ratio was H/d=100 and the porosity was ɛ=0.35. The effects of the imposed pressure-driven flow characterised by the Péclet number (0.1≤Pe≤1000), and heating represented by the Rayleigh number (105≤Ra≤2×107), are investigated. The results reveal that for low Péclet numbers, steady free convective flows are dominant, while for high Péclet numbers, steady forced convective flows dominate. Importantly, no oscillatory flows are found to develop in the free and forced convection regimes. However, for moderate Péclet numbers, oscillatory mixed convective flows occur with different periodic, quasi-periodic, and chaotic flow behaviours. In addition, for low Rayleigh numbers, Ra≤106, the Nusselt number is not influenced by the Péclet number for Pe≤1. Nevertheless, for high Rayleigh numbers, Ra≥2×106, it is observed that there always exists a critical Péclet number for which the Nusselt number is a minimum. The effects of the Péclet and Rayleigh numbers on the local and average temperature disparity between the fluid and solid phases are then investigated.

Variable gravity effects on penetrative porous convection

The effect of variable gravity on the onset of penetrative convection in a horizontal fluid-saturated porous layer is studied, performing the linear instability analysis and the nonlinear stability analysis (via the weighted energy method) of the thermal conduction solution. The linear and nonlinear critical Rayleigh numbers for the onset of penetrative porous convection are determined and their behaviour with respect to the fundamental physical parameters of the model is studied. In particular, we found that there exists a region of subcritical instabilities and that the increase of the upper bounding plane temperature and the decrease of gravity have a stabilizing effect on the onset of convection.

Temporal to spatial instability in a flow system: a comparison

AbstractThe definitions of temporal instability and of spatial instability in a flow system are comparatively surveyed. The simple model of one-dimensional Burgers’ flow is taken as the scenario where such different conceptions of instability are described. The temporal analysis of instability stems from Lyapunov’s theory, while the spatial analysis of instability interchanges time and space in defining the evolution variable. Thus, the growth rate parameter for temporally unstable perturbations of a basic flow state is to be replaced by a spatial growth rate when a coordinate assumes the role of evolution variable. Finally, the idea of spatial instability is applied to a Rayleigh-Bénard system given by a fluid-saturated horizontal porous layer with an anisotropic permeability and impermeable boundaries kept at different uniform temperatures.

A Revised Work on the Rayleigh-Bénard Instability of Nanofluid in a Porous Medium Layer

To reveal the mechanism of the enhanced heat transfer in nanofluids, Buongiorno (ASME J. Heat Transfer, vol. 128, 2006, pp. 240–250) developed a convective transport model by considering the slip mechanisms of nanoparticles migration. By now, many extended researches are based on his model. Among them, the study on porous medium flow pioneered by Nield & Kuznetsov (Int. J. Heat & Mass Transfer, vol. 52, 2009, pp.5796–5801) has received much attention. Their work employed the Darcy model and Buongiorno’s model to investigate the thermal instability in a horizontal porous medium layer saturated by a nanofluid. Through a sophisticated analysis, they obtained an approximate formula capable of predicting the stability threshold. However, a potential contradiction exists in their analysis owing to an improper assumption about the thermophoretic coefficient, which may lead to an unphysical result. To date, much of current works still adopted this improper assumption in various extended problems. To resolve this contradiction, the present study revises their work by considering the dependence of thermophoretic coefficient on the volume fraction of nanoparticles. A nonlinear basic-state solution of concentration is obtained and then used to implement the linear stability analysis. In comparison with Nield’s formula, the present result shows that the threshold of instability shifts to a lower concentration by more than one order of magnitude. The mechanism causing the shift is discussed and the novelty of the present study is stressed.

Global stability of Bénard–Marangoni convection in an anisotropic porous medium

Surface tension is essential in many industrial applications, especially where the liquid surface is in contact with the environment, such as crystal growth, semiconductor manufacturing, and welding. The present article reports a numerical analysis of convection induced by the combined effects of buoyancy force and surface tension in an infinitely extended horizontal fluid-saturated anisotropic porous layer with high permeability. We assume that the bottom boundary is rigid and the top is exposed to the air. Biot numbers establish general thermal conditions at both ends instead of commonly used adiabatic and isothermal boundary conditions. The anisotropy of the porous structure results in thermal and mechanical anisotropy parameters while examining layer's stability. The Chebyshev Tau technique yields the critical Marangoni number, MaLc and MaEc, representing linear and energy stability boundaries. We compare the constraints obtained from linear and energy analyses and conclude that the energy bounds for the current problem are less than linear bounds, indicating subcritical instabilities may exist. It is also observed that thermal anisotropy and Biot numbers stabilize the system. In contrast, mechanical anisotropy and the Darcy number advance the onset of convection. The existing results of limiting cases of the present problem are recovered with remarkable accuracy.

Rayleigh–Bénard instability in a horizontal porous layer with anomalous diffusion

The analysis of the Rayleigh–Bénard instability due to the mass diffusion in a fluid-saturated horizontal porous layer is reconsidered. The standard diffusion theory based on the variance of the molecular position growing linearly in time is generalized to anomalous diffusion, where the variance is modeled as a power-law function of time. A model of anomalous diffusion based on a time-dependent mass diffusion coefficient is adopted, together with Darcy's law, for momentum transfer, and the Boussinesq approximation, for the description of the buoyant flow. A linear stability analysis is carried out for a basic state where the solute has a potentially unstable concentration distribution varying linearly in the vertical direction, and the fluid is at rest. It is shown that any, even slight, departure from the standard diffusion process has a dramatic effect on the onset conditions of the instability. This circumstance reveals a strong sensitivity to the anomalous diffusion index. It is shown that subdiffusion yields instability for every positive mass diffusion Rayleigh number, while superdiffusion brings stabilization no matter how large is the Rayleigh number. A discussion of the linear stability analysis based on the Galilei-variant fractional-derivative model of subdiffusion is eventually carried out.

Soret-induced convection of ternary fluid in a horizontal porous layer heated from below

The present paper deals with the investigation of the Soret-induced convection of a three-component mixture of hydrocarbons in a horizontal porous layer. This problem is important for geological applications. The first part of the paper is devoted to the linear stability of the conductive state. The longwave instability is studied analytically by the expansion into the power series with respect to the wave number. A new long-wave oscillatory instability mode existing at negative separation ratios is found out. It is shown that this mode is more dangerous than the long-wave monotonic instability mode in the entire range of its existence. The instability to the perturbations with nonzero wave numbers is studied numerically by the shooting method. Stability map is obtained. The results confirm the predictions of the longwave analysis. In the second part of the paper, the nonlinear convection regimes are studied by the finite difference method. The calculations give the results consistent with the linear stability analysis. It is found that the primary bifurcation as a result of which the conductive state losses its stability is supercritical. The transformations of the convective flow structure with the change in the Rayleigh–Darcy number are accompanied by the hysteresis phenomena.

Exploration of rheological behavior of an Ellis fluid on the onset of thermosolutal porous convection

The initiation of thermosolutal convective instability is investigated in a horizontal porous layer saturated by a shear-thinning fluid following Ellis’ rheology. The porous layer is heated as well as salted from below and a basic horizontal throughflow prompted by the prescribed pressure gradient is considered. The linear stability analysis is performed using normal mode analysis and the threshold conditions for the onset of convection are obtained in a closed form. The imposed horizontal pressure gradient and the solute concentration gradient reinforce together in resulting the onset of convection through oscillatory motions. The transverse rolls are the most unstable which are found to be both travelling and non-travelling in the reference frame comoving with the basic throughflow. The effect of increasing the Ellis power-law index and the solute Darcy-Rayleigh number is to stabilise, while an increase in the Darcy-Ellis number is to destabilise the base flow. The pressure and temperature/solute concentration lines are presented at the critical state. The results delineated under the limiting cases are shown to be in agreement with those published previously.

Darcy–Benard–Oldroyd convection in anisotropic porous layer subject to internal heat generation

An anisotropic horizontal porous layer saturated with viscoelastic liquids of the Oldroyd-B type is explored to determine how the internal heat source affects thermal convection. As a momentum equation, a modified Darcy–Oldroyd model is used that takes into account the anisotropy of the porous layer. The energy equation is formulated in such a way that the influence of internal heat sources and anisotropy in thermal diffusivity on the stability criterion may be easily identified. The effects of anisotropy, viscoelasticity, and internal heat generation on the onset of thermal convection are investigated using linear stability analysis. It is understood that convection begins via an oscillatory mode instead of a stationary mode because viscous relaxation, thermal diffusions, and internal heat generation mechanisms compete with one another. Both steady and unsteady finite-amplitude convections are studied using nonlinear stability analysis with the truncated Fourier series method. The effect of different governing parameters on the system’s stability and on convective heat transfer is studied. The present investigation has been significantly validated by the recovery of several prior results as special situations. The findings presented in this work are anticipated to have significant implications for a number of real-world applications, including modeling of oil reservoirs, crude oil extraction, crystal growth, the pharmaceutical and medical industries, and the use of geothermal energy, among others.

Effect of variable gravity field on the onset of convection in a Brinkman porous medium under convective boundary conditions

The effect of variable gravity on the onset of convection in a horizontal porous layer has been analyzed due to external heating. The third kind of thermal boundary conditions is considered on both the boundaries which are connected by Biot numbers B0 and B1. Also, the Brinkman model has been employed in the momentum equation. Further, linear and quadratic varying downward gravity fields depending on height have been taken to investigate the above observation. The bvp4c scheme is used in MATLAB to solve the eigenvalue problem for neutral stability mode. It is observed that the system's stability increases monotonically due to the rising effect of gravity variation parameter, Darcy number, and Biot numbers. Further, the onset of convection has been discussed for the two limiting cases of external heating on the boundaries where one boundary is associated with constant heat flux while another one is associated with an isothermal state. Finally, it is observed that the convection comes earlier for quadratic varying gravity field compared to linear varying gravity field when the external heat supplies from one boundary or both boundaries.

Predicting the effect of inertia, rotation, and magnetic field on the onset of convection in a bidispersive porous medium using machine learning techniques

Effects of the magnetic field and inertia on the onset of thermal convection in a horizontal bidispersive porous layer, rotating about a vertical axis, are analyzed. The Darcy equation with same temperature in the micro- and macrophases is used to characterize the fluid motion. The Vadasz number is taken into account in a generalized Darcy equation for the macrophase. The eigenvalue problem obtained from the linear stability analysis is solved analytically for free–free boundaries. Moving one step further from the traditional linear stability analysis, machine learning tools are introduced in this paper to include the effect of multiple parameters on the marginal state of the system. Machine learning techniques have been implemented to identify the mode of instability with respect to different parameters. In particular, classification algorithms, namely, Artificial Neural Networks (ANN) and Support vector machine, are used to examine the onset of oscillatory convection and stationary convection. The required data for training of the algorithms are generated from the results of linear stability analysis. It is found that ANN with the sufficient number of hidden layers along with good choice of training dataset can predict the mode of instability even on the small variation in a given parameter. The combined effect of rotation, magnetic field, and inertia is to reduce the oscillatory mode of instability; hence, the system exhibits the steady mode of instability for a significant region in the three dimensional space comprising the Taylor number, the Hartman number, and the Vadasz number.

Numerical simulation on pore-scale pool boiling mechanisms of horizontal gradient porous metals

Horizontal gradient porous metals are proposed to solve the problem of nonuniform temperature distribution on the heating surface. The model of horizontal gradient porous metal is established based on the thermal response of metal skeletons. Pool boiling heat transfer performances of horizontal gradient porous metals are studied by phase-change lattice Boltzmann model. Horizontal porous metal layer thickness, wettability and thermal conductivity gradient effects on bubble behavior and heating surface temperature distribution uniformity are statistically analyzed. The results show that porous metals with horizontal thickness gradient can realize liquid-vapor channel separation, reduce the resistance of bubble rising, and improve the fresh liquid supplement. Compared with the uniform porous metal, the horizontal gradient porous metal with lateral hydrophobic surface or higher thermal conductivity has better boiling heat transfer performance and promotes heating surface uniform temperature distribution at low heat fluxes. The results provide a novel solution to improve the chip wall temperature uniformity for the stable operation of electronic equipment.