The Onset of Convection in a Suspension of Gyrotactic Microorganisms in Superimposed Fluid and Porous Layers: Effect of Vertical Throughflow

The purpose of this paper is to investigate the effect of vertical throughflow on the onset of bioconvection in a suspension of gyrotactic microorganisms. A dilute suspension of gyrotactic microorganisms in a shallow system that consists of superimposed fluid and porous layers is considered. A linear instability analysis of this problem is performed and the Galerkin method is utilized to solve the eigenvalue problem. The analysis leads to an equation for the critical Rayleigh number. It is shown that the vertical throughflow stabilizes the system.     

Stability Analysis of Bioconvection of Gyrotactic Motile Microorganisms in a Fluid Saturated Porous Medium

Despite a large number of publications on bioconvection in suspensions of motile microorganisms, bioconvection in a fluid saturated porous medium is a relatively new area of research. This paper is motivated by experimental research by Kessler (1986) who established that a porous medium prevents the development of convection instability in algal suspensions. This suggests that there may exist a critical value of the permeability of a porous medium. If the permeability is smaller than critical, the system is stable and bioconvection does not develop. If the permeability is larger than critical, bioconvection may develop. This paper presents a model of bioconvection of gyrotactic motile microorganisms in a fluid saturated porous medium. The focus of this research is the determination of the critical value of permeability of a porous medium by a linear stability analysis. A simple but elegant analytical solution for the critical Darcy number is obtained.     

Forced Convection Heat Transfer of Nanofluids in a Porous Channel

This article is concerned with the effects of flow and migration of nanoparticles on heat transfer in a straight channel occupied with a porous medium. Investigation of force convective heat transfer of nanofluids in a porous channel has not been considered completely in the literature and this challenge is generally considered to be an open research topic that may require more study. The fully developed flow and steady Darcy?CBrinkman?CForchheimer equation is employed in porous channel. The thermal equilibrium model is assumed between nanofluid and solid phases. It is assumed that the nanoparticles are distributed non-uniformly inside the channel. As a result the volume fraction distribution equation is also coupled with governing equations. The effects of parameters such as Lewis number, Schmidt number, Brownian diffusion, and thermophoresis on the heat transfer are completely studied. The results show that the local Nusselt number is decreased when the Lewis number is increased. It is observed that as the Schmidt number is increased, the wall temperature gradient is decreased and as a consequence the local Nusselt number is decreased. The effects of Lewis number, Schmidt number, and modified diffusivity ratio on the volume fraction distribution are also studied and discussed.     

Overstability Analysis of Thermo-bioconvection Saturating a Porous Medium in a Suspension of Gyrotactic Microorganisms

A linear stability analysis is performed to analyze bioconvection in a dilute suspension of gyrotactic microorganisms in horizontal shallow fluid layer cooling from below and saturated by a porous medium, in the rigid boundary case. It is established that due to cooling from below thermally stratified layer is stabilized, which opposes the development of bioconvection and the situations for oscillatory convection may take place. The stability criterion is obtained in terms of thermal Rayleigh number, bioconvection Rayleigh number, gyrotactic number, bioconvection Peclet number, measure of cell eccentricity, Prandtl number, and Lewis number. It is observed that the presence of porous medium results in decrease of the magnitude of critical bioconvection Rayleigh number in comparison with its non-existence; hence due to porous effect, the system becomes less stable.     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

Free Convection in Spherical Annular Sectors Filled with a Porous Medium

Calculated free convection flows and heat transfer are presented for concentric spherical annular sectors, filled with a porous medium. Two isothermal walls and an adiabatic radial wall at the sector angle define the sectors. The governing equations (in the stream function and temperature formulation) are solved numerically using ADI (alternative direction implicit) finite-difference method. Over the range of geometric parameters examined, the obtained results for spherical annuli and low Rayleigh number Ra. As Ra increases, multicellular flows develop for small values of the aspect ratio parameter . In addition, analytical solutions of the governing equations were obtained for small values of Ra (1) and it was shown that these solutions agree well with those obtained numerically. Significant differences in the local heat transfer rates on the inner and outer walls of the spherical annuli were observed from these solutions.     

Instability Analysis of Gyrotactic Microorganisms: A Combined Effect of High-Frequency Vertical Vibration and Porous Media

A linear stability analysis is carried out to predict the instability analysis in a dilute suspension of gyrotactic microorganisms in horizontal fluid-saturated porous layer influenced by high-frequency vertical vibration. The governing equations, describing the mean flow, are the time-averaged Boussinesq equations and the analytical solution of the problem has been obtained using Galerkin method. A secular relation involving bioconvection Rayleigh number and its vibrational analogs and other parameters have been established. The graphical interpretations for dependence of bioconvection Rayleigh–Darcy number and corresponding wave number, on gyrotactic number and bioconvection Péclet number in the presence of vibration are utilized to understand the problem.     

Mixed Convection in Non-Newtonian Fluids Along an Isothermal Vertical Cylinder in a Porous Medium

Mixed convection in power-law type non-Newtonian fluids along an isothermal vertical cylinder in porous media is studied. The problem is solved by means of a finite difference method for the case of uniform wall temperature. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5–1.5.     

Nonsimilar Solutions for Mixed Convection in Non-Newtonian Fluids Along Horizontal Surfaces in Porous Media

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along horizontal surfaces with variable heat flux distribution. The mixed convection regime is divided into two regions, namely, the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Transient Free Convection Fluid Flow in Domains Partially Filled with Porous Media

The problem of transient free convection in domains partly filled with porous substrates is investigated analytically using Laplace transformation technique. Four configurations are considered which are subject to an isothermal heating boundary condition. The Brinkman-extended Darcy model is adopted to describe the hydrodynamics behavior of the porous domain.     

Free Convection in a Square Cavity Filled with a Tridisperse Porous Medium

This work is dealing with the natural convection heat transfer in a square filled with porous medium that has been extended according to the Nield and Kuznetsov model to tridisperse porous medium. Considering impermeable walls which the horizontal ones are insulated and vertical ones are assumed to be isothermal, the governing equations are set as the three equations for momentum and three equations for energy for three phases of porosity and are numerically solved utilizing finite element method. In this study isothermal contours, streamlines and Nusselt number values are foremost criteria which are presented for three levels of porosity. The influence of various governing parameters on the heat transfer is investigated.     

Combined Free and Forced Convection in Vertical Channels of Porous Media

In this paper we investigate the combined free and forced convection of a fully developed Newtonian fluid within a vertical channel composed of porous media when viscous dissipation effects are taken into consideration. The flow is analysed in the region of a first critical Rayleigh number in order to interpret the multiple-valued solutions and discuss their validity. The governing fourth-order, ordinary differential equation, which contains the Darcy and the viscous dissipation terms, is solved analytically using perturbation techniques and numerically using D02HBF NAG Library. A detailed investigation of the governing O.D.E. is performed on both clear fluid and porous medium for various values of the viscous dissipation parameter, , when the wall temperature decreases linearly with height, and the pressure gradient is both above and below its hydrostatic value. Although mathematically the results in all cases show that there are two solution branches, producing four possible solutions, the study of the velocity and buoyancy profiles together with the Darcy effect indicate that only one of the two solutions at any value of the Rayleigh number appears to be physically acceptable. It is shown that the effect of the Darcy number decreases as the critical Rayleigh numbers increase.     

Validation of the Local Thermal Equilibrium Assumption in Forced Convection of Non-Newtonian Fluids through Porous Channels

In this paper, we assess the validity of the local thermal equilibrium assumption in the non-Newtonian forced convection flow through channels filled with porous media. For this purpose, the problem is solved numerically using local thermal non-equilibrium and non-Darcian models. Numerical solutions obtained over broad ranges of representative dimensionless parameters are utilized to map conditions at which the local thermal equilibrium assumption can or cannot be employed. The circumstances of a higher modified Peclet number, a lower modified Biot number, a lower fluid-to-solid thermal conductivity ratio, a lower power-law fluid index, and a lower microscopic and macroscopic frictional flow resistance coefficients, are identified as unfavorable circumstances for the local thermal equilibrium (LTE) condition to hold. Quantitative LTE validity maps that reflect the proportional effect of each parameter as related to others are presented.     

Non-Darcian and Anisotropic Effects on Free Convection in a Porous Enclosure

The free convective flow and heat transfer, within the framework of Boussinesq approximation, in an anisotropic fluid filled porous rectangular enclosure subjected to end-to-end temperature difference have been investigated using Brinkman extended non-Darcy flow model. The studies involve simultaneous consideration of hydrodynamic and thermal anisotropy. The flow and temperature fields in general are governed by, Ra, the Rayleigh number, AR, the aspect ratio of the slab, K*, the permeability ratio and k*, the thermal conductivity ratio, and Da, Darcy number. Numerical solutions employing the successive accelerated replacement (SAR) scheme have been obtained for 100 ≤ Ra ≤ 1000, 0.5 ≤ AR ≤ 5, 0.5 ≤ K* ≤ 5, 0.5 ≤ k* ≤ 5, and 0 ≤ Da ≤ 0.1. It has been found that [`(Nu)]{\overline {Nu}}, average Nusselt number increases with increase in K* and decreases as k* increases. However, the magnitude of the change in [`(Nu)]{\overline {Nu}} depends on the parameter Da, characterizing the Brinkman extended non-Darcy flow.     

Thermal Dispersion Effects on Combined Convection in Non-Newtonian Fluids Along a Nonisothermal Vertical Plate in a Porous Medium

The method of non-similarity solution is used to study the influence of thermal dispersion on combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

Unsteady Free Convection Boundary Layer Flows of a Bingham Fluid in Cylindrical Porous Cavities

Transport in Porous Media - We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is...     

On a Free Convection Problem Over a Vertical Flat Surface in a Porous Medium

The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T₁ at the leading edge, where T₁ is above the ambient temperature, and takes a value T₂ at a given distance L along the surface, varying linearly between these two values and remaining constant afterwards. Numerical solutions of the boundary-layer equations are obtained as well as solutions valid for both small and large distance along the surface. Results are presented for the three cases, when the temperature T₂ is greater, equal or less than the ambient temperature T_∞. In the first case, T₂ > T_∞, a boundary-layer flow develops along the surface starting with a flow associated with the temperature difference T₁ − T_∞ at the leading edge and approaching a flow associated with the temperature difference T₂ − T_∞ at large distances. In the second case, T₂ = T_∞, the convective flow set up on the initial part of the surface drives a wall jet in the region where the surface temperature is the same as ambient. In the final case, T₂ < T_∞, a singularity develops in the numerical solution at the point where the surface temperature becomes T_∞. The nature of this singularity is discussed.     

The Effect of Double Dispersion on Natural Convection Heat and Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

The effect of power law index parameter of the non-Newtonian fluid on free convection heat and mass transfer from a vertical wall is analyzed by considering double dispersion in a non-Darcy porous medium with constant wall temperature and concentration conditions. The Ostwald–de Waele power law model is used to characterize the non-Newtonian fluid behavior. In this case a similarity solution is possible. The variation of heat and mass transfer coefficients with the governing parameters such as power law index, thermal and solutal dispersion parameters, inertia parameter, buoyancy ratio, and the Lewis number is discussed for a wide range of values of these parameters.     

Laminar Free Convection Over a Vertical Wavy Surface Embedded in a Porous Medium Saturated with a Nanofluid

Numerical analysis is performed to examine laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. In this pioneering study, we have considered the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Non-similar transformations are presented for the governing equations and the obtained PDE are then solved numerically employing a fourth order Runge–Kutta method with shooting technique. A detailed parametric study (nanofluid parameters) is performed to access the influence of the various physical parameters on the local Nusselt number and the local Sherwood number. The results of the problem are presented in graphical forms and discussed.