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The aim of this paper is to investigate the onset of bio‐thermal convection in a shallow fluid layer; the convection is thus driven by the combined effect of swimming of oxytactic microorganisms and inclined temperature gradient.

Linear stability analysis of the basic state is performed; the numerical problem is solved using the collocation method.

The most interesting outcome of this analysis is the correlation between three Rayleigh numbers, two traditional, “thermal” Rayleigh numbers, which are associated with the vertical and horizontal temperature gradients in the fluid layer, and the bioconvection Rayleigh number, which is associated with the density variation induced by the upswimming of microorganisms.

Further research should address the application of weakly nonlinear analysis to this problem.

The increase of the horizontal thermal Rayleigh number stabilizes the basic flow. The effect of increasing the horizontal thermal Rayleigh number is to distort the basic temperature profile away from the linear one. The increase of the Schmidt number stabilizes the basic flow. The increase of the Prandtl number first causes the bioconvection Rayleigh number to decrease and then to increase.

To the best of the authors’ knowledge, this is the first research dealing with the effect of inclined temperature gradient on the stability of bioconvection.

The purpose of this paper is to examine numerically the three natural convection of air induced by temperature difference between a cold outer cubic enclosure and a hot inner cylinder. Simulations have been carried out for Rayleigh numbers ranging from 103 to 107 and titled angle of the enclosure from 0° to 90°. The developed mathematical model is governed by the coupled equations of continuity, momentum and energy, and is solved by finite volume method. The effects of cylinder inclination and Rayleigh number on fluid flow and heat transfer are presented. The distribution of isocontours of temperature and isosurfaces of velocity eventually reaches a steady state in the range of Rayleigh numbers between 103 and 107 for titled inclination of 90°; however, for the remaining inclinations, Rayleigh number must be in the range 103-106 to avoid unsteady state, which is manifested by the division of the area containing the maximum local heat transfer rate into three parts for a Rayleigh number equal to 107 and an inclination of 90°. We mention that instability study is not included in the present paper, which is solely devoted to three-dimensional calculations. Results also indicate that optimal average heat transfer rate is obtained for both high Rayleigh number of 106 and high inclination of 90° for the two cases of the inner cylinder and cubical enclosure.

The manuscript deals with prediction of the three-dimensional natural convection phenomena in a cubical cavity induced by an isothermal cylinder at the center with different inclinations by simulating the flow using highly numerical methods such as finite volume method.

It is found that the local Nusselt number through active walls for titled inclination set at 90°, the symmetry of the flow is conserved and the area containing the maximum heat transfer is divided into three smaller areas situated near the upper portion of the wall, taking the maximum value. That may be due to the preparation of local occurrence of instabilities and bifurcation phenomena that appear for Ra > 107, which is not included in the present paper to save journal space. It was found also that an optimal heat transfer appears when the cylinder orientation becomes vertical (a = 90°). For this inclination, buoyancy forces act upward, corresponding to an aiding situation. In addition, heat transfer rate is increasing with Rayleigh numbers, so correlations of average Nusselt through the cubical cavity and the cylinder are established as function of two parameters (Ra, a). Comparisons of the numerical results with those obtained from all correlations show good agreements.

To the author’s knowledge, studies have thus far adressed three-dimensional cuboids enclosures induced by an inner shape which the location is changed. However, no study has examined three-dimensional natural convection between the inner isothermal cylinder and outer cooled cubical enclosure when the outer enclosure is tilted.

The post-critical convective state for Rayleigh-Benard (RB) convection is studied using a nonlinear spectral-amplitude-perturbation approach in a fluid layer heated from below. The paper aims to discuss these issues.

In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A combined amplitude-perturbation approach is developed to solve the nonlinear spectral system in the post-critical range, even far from the linear stability threshold. Also, to leading order, the Lorenz model is recovered.

It is found that very small Prandtl numbers (Pr < 0.1) can change the Nusselt number, when terms to O(ε5/2) and higher are considered. However, to lower orders the Prandtl number does not affect the results. Variation of the Nusselt number to different orders is found to be highly consistent. Comparison with experimental results is made and a very good qualitative agreement is observed, even far from the linear threshold.

Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection. The number and type of modes to be included are directly related to the post-critical Rayleigh number. The method is not limited to the weakly nonlinear range.

The purpose of this paper is to investigate a combined bioconvection and thermal instability problem in a horizontal layer of finite depth with a basic temperature gradient inclined to the vertical. The basic flow, driven by the horizontal component of temperature gradient, is the Hadley circulation, which becomes unstable when the vertical temperature difference and density stratification induced by upswimming of microorganisms that are heavier than water become sufficiently large.

Linear stability analysis of the basic state is performed; the numerical problem is solved using the collocation method.

The steady‐state solution of this problem is obtained. Linear stability analysis of this steady‐state solution for the case of three‐dimensional disturbances is performed; the numerical problem is solved using the collocation method. The stability problem is governed by three Rayleigh numbers: the bioconvection Rayleigh number and two thermal Rayleigh numbers characterizing temperature gradients in the vertical and horizontal directions, respectively.

Further research should address the application of weakly non‐linear analysis to this problem.

The dependence of the critical bioconvection Rayleigh number on the two thermal Rayleigh numbers and other relevant parameters is investigated.

This paper presents what is believed to be the first research dealing with the effect of inclined temperature gradient on the stability of bioconvection in a suspension of gyrotactic microorganisms.

According to the research, viscoplastic fluids are sensitive to slipping. The purpose of this study is to determine whether slip affects the Rayleigh–Bénard convection of viscoplastic fluids in cavities and, if so, under what conditions.

The wall slip was evaluated using a model created for viscoplastic (Bingham) fluids. The coupled conservation equations were solved numerically using the finite element method. Simulations were performed for various parameters: the Rayleigh number, yield number, slip yield number and friction number.

Wall slip determines two essential yield stresses: a specific yield stress value beyond which wall slippage is impossible (S_Yc); and a maximum yield stress beyond which convective flow is impossible (Y_c). At low Rayleigh numbers, Y_c is smaller than S_Yc. Hence, the flow attained a stable (conduction) condition before achieving the no-slip condition. However, for more significant Rayleigh numbers Y_c exceeded S_Yc. Thus, the flow will slip at low yield numbers while remaining no-slip at high yield numbers. The possibility of slipping on the wall increases the buoyancy force, facilitating the onset of Rayleigh–Bénard convection.

An essential aspect of this study lies in its comprehensive examination of the effect of slippage on the natural convection flow of viscoplastic materials within a cavity, which has not been previously investigated. This research contributes to a new understanding of the viscoplastic fluid behavior resulting from slipping.

The purpose of this paper is to evaluate the thermal and fluid dynamic behaviors of natural convection in a vertical channel‐chimney system heated symmetrically at uniform heat flux in order to detect the different fluid motion structures inside the chimney, such as the cold inflow from the outlet section of the chimney and the reattachment due to the hot jet from the channel, for different extension and expansion ratios of the adiabatic extensions.

The model is constituted by two‐dimensional steady‐state fully elliptic conservation equations which are solved numerically in a composite three‐part computational domain by means of the finite‐volume method.

Stream function and temperature fields in the system are presented in order to detect the different fluid motion structures inside the chimney, for different extension and expansion ratios of the adiabatic extensions. The analysis allows to evaluate the effect of the channel aspect ratio on the thermal and fluid dynamic behaviors on a channel‐chimney system and thermal and geometrical conditions corresponding to a complete downflow. Guidelines to estimate critical conditions related to the beginning of flow separation and complete downflow are given in terms of order of magnitude of Rayleigh and Froude numbers.

The hypotheses on which the present analysis is based are: two‐dimensional, laminar and steady‐state flow, constant thermophysical properties with the Boussinesq approximation. The investigation is carried out in the following ranges: from 100 to 100,000 for the Rayleigh number, from 5.0 to 20 for the aspect ratio, from 1.0 to 4.0 for the expansion ratio and from 1.5 to 4 for the extension ratio.

Thermal design of heating systems in different technical fields, such as in electronic cooling and in building ventilation and houses solar components, evaluation of heat convective coefficients and guidelines to estimate critical conditions related to the beginning of flow separation and complete downflow.

The paper is useful to thermal designers because of its evaluation of the thermal and velocity fields, correlation for the Nusselt number and guidelines criteria in terms of Rayleigh and Froude numbers to evaluate conditions of flow separation and complete downflow in natural convection in air for vertical channels‐chimney systems.

The effect of magnetic field dependent (MFD) viscosity on the onset of convection in a ferromagnetic fluid layer heated from below saturating rotating porous medium in the presence of vertical magnetic field is investigated theoretically by using Darcy model. The resulting eigen value problem is solved using the regular perturbation technique. Both stationary and oscillatory instabilities have been obtained. It is found that increase in MFD viscosity and increase in magnetic Rayleigh number is to delay the onset of ferroconvection, while the nonlinearity of fluid magnetization has no influence on the stability of the system.

The thermal perturbation method is employed for analytical solution. A theory of linear stability analysis and normal mode technique have been carried out to analyze the onset of convection for a fluid layer contained between two impermeable boundaries for which an exact solution is obtained.

The conditions for the system to stabilize both by stationary and oscillatory modes are studied. Even for the oscillatory system of particular frequency dictated by physical conditions, the critical Rayleigh numbers for oscillatory mode of the system were found to be greater than for the stationary mode. The system gets destabilized for various physical parameters only through stationary mode. Hence, the analysis is restricted to the stationary mode. To the Coriolis force, the Taylor number T_a is calculated to discuss the results. It is found that the system stabilizes through stationary mode for values of and for oscillatory instability is favored for Ta > 104. Therefore the Taylor number T_a leads to stability of the system. For larger rotation, magnetization leads to destabilization of the system. The MFD viscosity is found to stabilize the system.

This research paper is new and original.

Thermally driven flow in a tall inclined cavity bounded by porous layers is studied analytically and numerically. A constant heat flux is applied for heating and cooling of two opposing walls of the cavity, while the other two are insulated. The Beavers—Joseph slip condition on velocity is applied at the interface between the fluid and porous layers. An analytical solution is obtained by assuming parallel flow in the core region of the cavity and a numerical solution by solving the complete governing equations. The flow and heat transfer variables are obtained in terms of the Rayleigh number, Ra, slip condition parameter N and angle of inclination of the cavity Φ. The critical Rayleigh numbers for the onset of convection in a layer heated from below are predicted for various hydrodynamic boundary conditions. The results for a fluid layer bounded by solid walls (N → ∞) and by free surfaces (N → 0) emerge from the present analysis as limiting cases.

In general, the bifurcation phenomenon of the natural convection has largely been studied. But the bifurcation of natural convection under magnetic conditions has not been studied as per the authors’ knowledge. This paper aims to investigate the changes in bifurcation phenomenon by the self-induced circular magnetic field.

The authors numerically solved the natural convection in an annulus. The SIMPLE algorithm was adopted for pressure-momenturm coupling. The Boussinesq approximation was used for numerical modeling of natural convection. Finally, the Lorentz force effect by the magnetic field was considered through the source terms in the momentum conservation equation.

It was determined that the heat-transfer rate changes by 17% owing to the applied magnetic effect, and the range of the Rayleigh number for flow bifurcation is changed by the magnetic effect. Moreover, under the strong magnetic condition, the flow bifurcation continues even at very high Ra. Previously, flow bifurcation has been understood as a flow instability phenomena, and the Lorentz force was regarded as a flow-damping effect; however, in this study, it was found that the magnetic field can boost the flow instability and induce flow bifurcation even in the Rayleigh number region where the bifurcation does not appear.

This paper is dealing with the bifurcation phenomenon in MHD natural convection problems. In the past, the electromagnetic forces were regarded as always acting to damp out the existing flows; herewith, the authors first investigated that the magnetic effect can boost the bifurcation of a kind of flow instability phenomenon.

The purpose of this paper is to explore a model for thermal convection in a plane layer when the density-temperature relation in the buoyancy term is quadratic. A heat source/sink varying in a linear fashion with a vertical height expressed as z was allowed, functioning as a heat sink in an area of the layer and as a heat source in the remainder.

First, the authors present the governing equations of motion and derive the associated perturbation equations. Second, the authors introduce the linear and nonlinear analysis of the system. Third, the authors transform the system to velocity-vorticity-potential formulation and introduce a numerical study of the problem in three dimensions.

First, the linear instability and nonlinear stability thresholds are derived. Second, the linear instability thresholds accurately predict the onset of instability. Third, the required time to arrive at the steady state increases as Ra tends to Ra_L . Fourth, the authors find that the convection has three different interesting patterns.

With the modernday need for heat transfer or insulation devices in industry, particularly those connected with nanotechnology, the usefulness of a mathematical analysis of such resonance became apparent. Thus, this study is believed to be of value.