Critical Darcy Rayleigh Number Research Articles

Unstable buoyant viscoelastic fluid flow in a vertical porous layer with temperature-dependent viscosity

The stability of thermally driven buoyant flow of a viscoelastic fluid saturating a vertical porous layer with viscosity depending linearly on temperature is investigated numerically. The rheological behavior of the fluid is described through the Oldroyd-B model, leading to a modified Darcy's law of momentum transfer in the porous medium. The study explores the linear stability of the base flow by analyzing the behavior of normal modes of perturbation. Neutral stability curves and the critical Darcy–Rayleigh number are determined for a wide range of viscoelastic and viscosity parameters. Transition curves from stability to instability in the viscoelastic parameters space are also provided for both constant and variable viscosity cases. Additionally, the results for Newtonian, Boger, and Maxwell fluids are delineated as particular cases from this study.

The onset of anisotropic Darcy–Brinkman penetrative convection with local thermal nonequilibrium due to heat generation and extraction

AbstractAn anisotropic, fluid‐saturated porous medium with local thermal nonequilibrium is the subject of this paper's analysis of the stability of the commencement of convection caused by internal heat generation. The classical Darcy's momentum equation is applied with the Brinkman model. To observe the thermal instabilities in the fluid layer, it is assumed that there is steady heat extraction through the bottom boundary and constant heat generation inside the fluid layer. The critical internal Darcy Rayleigh and associated wave numbers are computed using linear instability and energy approaches using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. The fluid layer becomes unstratified as a result of heat generation inside the fluid, which encourages the occurrence of any subcritical instability. The critical Darcy–Rayleigh number has been found to be significantly influenced by a number of parameters, including the anisotropy parameter, internal heat parameter, porosity‐modified conductivity ratio, inter‐phase heat transfer parameter, and Darcy–Brinkman number.

Further thoughts on buoyancy-induced instability of a variable viscosity fluid saturating a porous slab

The stability of buoyant flow in a vertical porous layer bounded by impermeable-isothermal boundaries is studied insofar as the variability of fluid viscosity with temperature is concerned. The flow is governed by modified Darcy's law with two different forms of viscosity–temperature relationships namely, quadratic and exponential. The modal analysis is carried out with a velocity–temperature formulation of the governing equations for the perturbations. The temperature-dependence of the viscosity forbids the energy analysis of Gill [“A proof that convection in a porous vertical slab is stable,” J. Fluid Mech. 35, 545–547 (1969)] in embarking upon any definite conclusion on the stability of fluid flow even under the limit of an infinite Prandtl–Darcy number, and consequently, the stability eigenvalue problem is solved numerically. Types of temperature-dependent viscosity laws are found to demonstrate conflicting behavior on the stability characteristics of the base flow. The results show that the base flow is linearly stable if the viscosity varies with temperature quadratically. On the contrary, instability emerges for an exponential type of variation in viscosity beyond a certain range of values of the corresponding viscosity parameter depending on the Prandtl–Darcy number. It is established that an increase in the value of the viscosity parameter is to decrease the critical Darcy–Rayleigh number markedly and thereby destabilizes the fluid flow.

Temperature Dependent Viscosity Effect on Oscillatory Mode of Darcy-Rayleigh Convection in a Double Diffusive Binary Viscoelastic Fluid Saturated Anisotropic Porous Layer

An anisotropic porous layer saturated with a viscoelastic double diffusive binary fluid is examined numerically for the onset of Darcy-Rayleigh convection. From below, the system is heated, while from above, it is cooled. The temperature-dependent viscosity was added to the double diffusive binary fluid, and the Galerkin expansion method was used to determine the critical Darcy-Rayleigh number. When their values are raised, the impacts of strain retardation, thermal anisotropy parameter and Dufour number slow the production of heat transfer and stabilise the system. When the values are increased, the stress relaxation, Darcy-Prandtl, mechanical anisotropy parameters, temperature dependent viscosity and Soret number accelerate the heat transfer process in convection, which destabilises the system

A weakly nonlinear analysis of the effect of vertical throughflow on Darcy–Bénard convection

The aim of this paper is to analyze the effect of a downward vertical net mass flow on the type of instability that occurs in a horizontal fluid-saturated porous layer that is heated from below. The strength of the downflow is modelled by Pe, the Péclet number. First, we prove the validity of the principle of exchange of stabilities; consequently, we perform a linear instability analysis of the basic steady flow to determine the critical Darcy–Rayleigh number for the onset of steady convective instability as a function of Pe. Then, a weakly nonlinear stability analysis is performed to determine the smallest value of the Péclet number for which the onset of convection corresponds to a subcritical instability.

COUPLED EFFECT OF SOLID FRACTION AND INTERNAL ENERGY SOURCE ON CONVECTIVE HEAT TRANSPORT IN THREE-LAYERED AIR-POROUS-AIR DOMAIN: COMPARISON WITH DIFFERENT TWO-LAYERED DOMAINS

This paper studies convective heat transport driven by a uniform internal energy source coupled with the solid fraction in the partial air-saturated porous domains bounded by the top and bottom impermeable thermally conductive surfaces. The consideration of the coupled effect distinguishes our work from the papers of previous authors. We provide a comprehensive numerical study of internal convection and compare results of the linear stability analysis for three distinct domains. The first domain is air-porous-air (APA), where the heat-generating porous matrix is between the upper and lower air layers. In the second air-porous (AP) domain, the air layer overlays the porous medium. In the third porous-air (PA) domain, the air layer underlays the porous medium. The bimodal marginal stability curves and regime map with a demarcation line between the local and large-scale convective flows are obtained only for the APA and AP domains due to the division of each domain into the upper unstably stratified and lower stably stratified parts. The local convection cannot originate in the PA domain because the air layer belongs to the lower stably stratified part. A remarkable destabilizing effect of additional air layers has been revealed. For example, at the fixed solid fraction of 0.1, we achieve a 40-fold reduction of the critical Darcy-Rayleigh number in the APA domain by increasing the depth ratio from 0 to 0.5, and by 14 times in the AP domain and only by 3.5 times in the PA domain. The destabilizing effect enhances with increases in the solid fraction.

Gill's stability problem may be unstable with horizontal heterogeneity in permeability

The linear stability of thermal buoyant flow in a fluid-saturated vertical porous slab is studied under the assumption of weak and strong horizontal heterogeneities of the permeability. The two end vertical isothermal boundaries are impermeable and some paradigmatic cases of linear, quadratic and exponential heterogeneity models are deliberated. The stability/instability of the basic flow is examined by carrying out a numerical solution of the governing equations for the disturbances as Gill's proof (A.E. Gill, J. Fluid Mech, vol. 35, 1969, pp. 545–547) of linear stability is found to be ineffective. The possibilities of base flow becoming unstable due to heterogeneity in permeability are recognized, in contrast to Gill's stability problem. The neutral stability curves are presented and the critical Darcy–Rayleigh number for the onset of convective instability is computed for different values of the variable permeability constant. The similarities and differences between different heterogeneity models on the stability of fluid flow are clearly discerned.

A study on the effect of temperature modulation on Darcy–Bénard convection using a local thermal non-equilibrium model

The criterion for the onset of Darcy–Bénard convection is analyzed when the fluid and porous medium are out of thermal equilibrium and the temperatures of the boundaries vary sinusoidally with time in either a synchronous or an asynchronous manner. A stability analysis of the linearized governing equations is performed by using the matrix differential operator theory. The shift in the critical Darcy–Rayleigh number is evaluated in terms of system parameters, and the effect of those on the shift is depicted graphically to realize the significant effect of temperature modulation on the onset, especially when the thermal non-equilibrium effects are prominent.

Instability of Vertical Throughflows in Bidisperse Porous Media

In this paper, the instability of a vertical fluid motion, or throughflow, is investigated in a horizontal bidisperse porous layer that is uniformly heated from below. By means of the order-1 Galerkin approximation method, the critical Darcy–Rayleigh number for the onset of steady instability is determined in closed form. The coincidence between the linear instability threshold and the global nonlinear stability threshold, in the energy norm, is shown.

Magnetic field and Rotation effect on thermal stability in an Anisotropic couple-stress fluid saturated Porous Layer under presence of Cross-Diffusion

The present paper is based on the investigation of thermal stability problem under the influence of rotation, solute, heat and magnetic field on electrically conducting fluid saturated rotating horizontal porous media under local thermal non-equilibrium effect. We have considered a rotating horizontal anisotropic porous medium saturated with binary fluid. Process of Heating and salting is made from above to the fluid layer.Governing equations is followed by Darcy Brinkman model for characterizing the convection through porous media under the above said physical configuration. Normal mode technique and Application of Linear stability analysis has been made under the consideration of free-free boundary conditions. Thereafter the characteristic graphs between critical Darcy-Rayleigh number and corresponding wave number for beginning of convection in stationary case has been drawn and the effect of rotation, magnetic and couple stress parameters has been illustrated.At last it has been observed the non-dimensonalise parametres regarding singnificant components like rotation magnetic field and anisotropic properties of the physical configuration in present study has stabilizing effect.

Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection

We investigate the onset of convection in an inclined Darcy-Bénard layer. When such a layer is unbounded in the spanwise direction it is generally known that longitudinal rolls comprise the most unstable planform. On the other hand, when a layer has a sufficiently small spanwise width, then transverse rolls form the most unstable planform. However, the layer remains stable to transverse roll disturbances when the inclination is above roughly 31 degrees from the horizontal. This paper considers the transition between these two extreme cases where the spanwise width takes moderate values and where rectangular cells are considered. It is found that the most unstable planform is quite strongly sensitive to the magnitude of the spanwise width and that there are large regions of parameter space within which three-dimensional convection patterns have the smallest critical Darcy-Rayleigh number.

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.

The Horton–Rogers–Lapwood problem for an inclined porous layer with permeable boundaries

A formulation of the Horton-Rogers-Lapwood problem for a porous layer inclined with respect to the horizontal and characterized by permeable (isobaric) boundary conditions is presented. This formulation allows one to recover the results reported in the literature for the limiting cases of horizontal and vertical layer. It is shown that a threshold inclination angle exists which yields an upper bound to a parametric domain where the critical wavenumber is zero. Within this domain, the critical Darcy-Rayleigh number can be determined analytically. The stability analysis is performed for linear perturbations. The solution is found numerically, for the inclination angles above the threshold, by employing a Runge-Kutta method coupled with the shooting method.

Stability of Penetrative Natural Convection in a Non-Newtonian Fluid-Saturated Vertical Porous Layer

The stability of natural penetrative convection arising due to a uniform internal heat source in a vertical porous layer saturated with an Oldroyd-B fluid is investigated. The vertical walls of the porous layer are impermeable and maintained at different uniform temperatures. The energy stability analysis performed reveals that the system is unconditionally stable even in the presence of internal heating in the case of Newtonian fluids, while for viscoelastic fluids the base flow is found to be unstable. As the energy stability analysis of Gill type is unable to decide the stability of the system, the Galerkin method is used to solve the complex eigenvalue problem. The internal heating introduces asymmetry in the basic flow and amounts to the existence of different set of onset modes. The internal heating and stress relaxation parameter facilitates instability of the system while increasing strain retardation parameter discloses stabilizing effect on the system. Moreover, the critical Darcy–Rayleigh number, wave number and wave speed become invariant as Ns becomes large. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state.

Unstable buoyant flow in a vertical porous layer with convective boundary conditions

The instability of natural convection in a vertical porous layer is analysed. The plane parallel boundaries of the vertical layer are modelled as open and subject to Robin-type temperature conditions. The latter conditions describe heat transfer to the external environment, with a finite conductance. The basic state is given by a stationary fully-developed flow with linear velocity and temperature profiles. Instability arises when the Darcy-Rayleigh number exceeds its critical value. This value depends on the Biot number associated with the temperature boundary conditions. The most unstable normal modes turn out to be transverse. By solving numerically the stability eigenvalue problem, it is shown that the critical Darcy-Rayleigh number is a decreasing function of the Biot number when the Biot number is sufficiently small. For larger Biot numbers, a minimum is attained, and then the critical Darcy-Rayleigh number becomes an increasing function of the Biot number.

The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium

In this study, the linear stability of an unsteady thermal boundary layer in a semi-infinite porous medium is considered. This boundary layer is induced by varying the temperature of the horizontal boundary sinusoidally in time about the ambient temperature of the porous medium; this mimics diurnal heating and cooling from above in subsurface groundwater. Thus if instability occurs, this will happen in those regions where cold fluid lies above hot fluid, and this is not necessarily a region that includes the bounding surface. A linear stability analysis is performed using small-amplitude disturbances of the form of monochromatic cells with wavenumber, k. This yields a parabolic system describing the time-evolution of small-amplitude disturbances which are solved using the Keller box method. The critical Darcy-Rayleigh number as a function of the wavenumber is found by iterating on the Darcy-Rayleigh number so that no mean growth occurs over one forcing period. It is found that the most dangerous disturbance has a period which is twice that of the underlying basic state. Cells that rotate clockwise at first tend to rise upwards from the surface and weaken, but they induce an anticlockwise cell near the surface at the end of one forcing period, which is otherwise identical to the clockwise cell found at the start of that forcing period.

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter $$\Lambda _2 $$ portrays stabilizing influence on the system while increasing stress relaxation parameter $$\Lambda _1$$ displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.

Stability of natural convection in a vertical layer of Brinkman porous medium

A classical linear stability theory is applied to emphasize the effect of inertia on the stability of buoyancy-driven parallel shear flow in a vertical layer of porous medium. The Lapwood–Brinkman model with fluid viscosity different from effective viscosity is used to describe the flow in a porous medium. The resulting eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Darcy–Rayleigh number \(R_\mathrm{Dc} \), the critical wave number \(a_\mathrm{c}\) and the critical wave speed \(c_\mathrm{c}\) are computed over a wide range of values of the Darcy–Prandtl number \(Pr_\mathrm{D}\) and the Darcy number \({\tilde{D}}a\). Depending on the choice of physical parameters, instability occurs due to the presence of inertia. The value of \(Pr_\mathrm{D}\) at which the transition from stationary to traveling-wave mode instability takes place increases with decreasing \({\tilde{D}}a\). Besides, the effect of decreasing \({\tilde{D}}a\) shows destabilizing effect if the instability is via stationary mode, and on the contrary, it exhibits a dual behavior if the instability is through traveling-wave mode. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state. In the energy spectrum, transition of instability from one type to another is found to take place as a function of \(Pr_\mathrm{D}\). The disturbance kinetic energy due to surface drag and viscous force plays no significant role in the stability of flow throughout the domain of \(Pr_\mathrm{D}\) considered.

A study on the onset of thermally modulated Darcy–Bénard convection

A stability analysis of linearized Rayleigh–Benard convection in a densely packed porous layer was performed using a matrix differential operator theory. The boundary temperatures were assumed to vary periodically with time in a sinusoidal manner. The correction in the critical Darcy–Rayleigh number was computed and depicted graphically. It was shown that the phase difference between the boundary temperatures rather than the frequency of modulated temperatures decides the nature of influence of modulation on the onset of convection. Conclusions were drawn regarding the possible transitions from harmonic to subharmonic solutions. The results on the onset of thermally modulated convection in a rectangular porous enclosure were obtained using those on the modulated Darcy–Benard convection.

On the horizontal throughflow across a vertical porous wall

The stationary two-dimensional mixed convection flow in a vertical porous wall subject both to a temperature and to a pressure gap across its boundaries is studied. The instability of this flow is investigated by adopting a linear modal analysis. The two-dimensional transverse rolls are shown to be the preferred pattern at onset of instability. Transverse rolls turn out to be travelling modes. They are driven by the net vertical flow predicted by the basic solution. A numerical procedure based on the joint use of the Runge-Kutta method and on the shooting method is employed to solve the perturbation eigenvalue problem. This numerical solution yields both the neutral stability curves, defining the threshold of linear instability, and the critical Darcy-Rayleigh number yielding the onset condition of instability.