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The phenomena of buoyant Marangoni natural convection in a right-angled trapezoidal enclosure filled with water-based nanofluids have been studied numerically. The left vertical wall of the trapezoidal cavity is heated uniformly, the right inclined wall is cooled, and the top and bottom walls are insulated (adiabatic). The finite element method (FEM) is implemented to solve the non-dimensional governing equations. The investigation has been done for different values of parameters: the aspect ratio of the enclosure, 0.25 ≤ Ar ≤ 1 , the nanoparticles volume fraction, 0 % ≤ ϕ ≤ 5 % , the Marangoni numbers, 0 ≤ Mabf ≤ 10 3 , and the Rayleigh numbers, 10 3 ≤ Rabf ≤ 10 4. [ABSTRACT FROM AUTHOR]

A numerical study is made on the entropy generation and magnetohydrodynamics natural convection of Al 2 O 3 -water non-homogeneous nanofluid inside a square enclosure equipped with a heated trapezoidal body. The Galerkin weighted residual finite element method is applied to solve the dimensionless governing equations within the utilized computational domain along with the algorithm of Newton–Raphson iteration that is used for simplifying the nonlinear terms in the equations. The characteristics of fluid flow fields, temperature distributions and entropy generation are studied for an enormous range of the Rayleigh number (10 3 ≤ R a ≤ 10 6) , volume fraction of nanoparticles ( 0 ≤ ϕ ≤ 0.04 ), Hartmann number (0 ≤ H a ≤ 50) , thermal conductivity of the trapezoidal solid body ( k w = 0.5 , 0.76, 1.95, 7 and 16) and the height of the trapezoidal solid body ( 0.15 ≤ D ≤ 0.45 ). It is shown that the streamlines pattern is more sensitive to the increase in the Hartmann number in comparison with the augmentation of the volume fraction of nanoparticles. Also, for a more thermodynamically optimized system, the higher Hartmann number at a higher solid volume fraction of nanofluid is recommended as they show less entropy generation. [ABSTRACT FROM AUTHOR]

In this study, the conjugate natural convection in a square cavity filled with Al 2 O 3 –water nanofluid with an inner conducting solid block is studied numerically using nonhomogeneous Buongiorno's two-phase model. The left wall of the cavity is partially heated and the remaining parts of the wall are adiabatic, while the right wall is fully cooled. The top and bottom horizontal walls are adiabatic. The numerical simulations are based on the finite difference method. The results are simulated for various values of the nanoparticle volume fraction (0 ≤ ϕ ≤ 0.04) , Rayleigh number (10 2 ≤ R a ≤ 10 6) , thermal conductivity of the conjugate square (k w = 0.28 , 0.76 , 1.95 , 7.0) and 16.0 (epoxy: 0.28, brickwork: 0.76, granite: 1.95, solid rock: 7, stainless steel: 16), the size of the inner solid (0 ≤ D ≤ 0.7) , and the length of the heater ( 0.1 ≤ H ≤ 1.0 ). The numerical results for the average and local Nusselt numbers, isotherms, distribution of nanoparticles, and streamlines are presented graphically. The findings indicate that increasing the average solid volume fraction and the size of the solid block as well as the thermal conductivity will enhance the rate of the heat transfer at low values of Rayleigh number R a = 10 3 . On the other hand, increasing these parameters at high values of Rayleigh number ( R a > 10 5 ) decreases the average Nusselt number. [ABSTRACT FROM AUTHOR]

The problem of darcian natural convection in inclined square cavity partially filled between the central square hole filled with fluid and inside a square porous cavity filled with nanofluid is numerically studied using the finite element m ethod. The left vertical wall is maintained at a constant hot temperature Th and the right vertical wall is maintained at a constant cold temperature Tc, while the horizontal walls are adiabatic. The governing equations are obtained by applying the Darcy model and Boussinesq approximation. COMSOL's finite element method is used to solve the non-dimensional governing equations together with the specified boundary c onditions. The governing parameters of this study are the Rayleigh number (103 ⩽ Ra ⩽ 107), the Darcy number (10-5 ⩽ Da ⩽ 10-3), the fluid layer thickness (0:4 ⩽ S ⩽ 0:8) and the inclination angle of the cavity ( 0° ⩽ ω ⩽ 6 0°). The results for the values of the governing parameters in terms of the streamlines, isotherms and average Nusselt number will be presented. The convection is shown to be inhibited by the presence of the hole insert. The thermal property of the insert and the size have opposite influence on the convection. The results have possible applications in heat-removal and heat-storage fluid-saturated porous systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the combined effect of internal-heating and time-periodic gravity modulation on thermal instability in a viscous fluid layer, heated from below. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the fluid layer. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. Effects of various parameters such as internal Rayleigh number, Prandtl number, and amplitude and frequency of gravity modulation have been analysed on heat transport. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further, it is found that the heat transport can be controlled by suitably adjusting the external parameters of the system. [Copyright &y& Elsevier]

The effects of time-periodic boundary temperatures and internal heating on Nusselt number in the Bénard-Darcy convective problem has been considered. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small. By performing a weakly non-linear stability analysis, the Nusselt number is obtained in terms of the amplitude of convection, which is governed by the non-autonomous Ginzburg-Landau equation, derived for the stationary mode of convection. The effects of internal Rayleigh number, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. Increasing values of internal Rayleigh number results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. [ABSTRACT FROM AUTHOR]

This investigation reports the magnetic field effect on natural convection heat transfer in a curved-shape enclosure. The numerical investigation is carried out using the control volume-based-finite element method (CVFEM). The numerical investigations are performed for various values of Hartmann number and Rayleigh number. The obtained results are depicted in terms of streamlines and isotherms which show the significant effects of Hartmann number on the fluid flow and temperature distribution inside the enclosure. Also, it was found that the Nusselt number decreases with an increase in the Hartmann number. [ABSTRACT FROM AUTHOR]

In this article we study the combined effect of internal heating and time-periodic gravity modulation on thermal instability in a closely packed anisotropic porous medium, heated from below and cooled from above. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the porous medium. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg-Landau equation derived for the stationary mode of convection. The effects of various parameters such as; internal Rayleigh number, amplitude and frequency of gravity modulation, thermo-mechanical anisotropies, and Vadász number on heat transport has been analyzed. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further it is found that the heat transport can also be controlled by suitably adjusting the external parameters of the system. [ABSTRACT FROM AUTHOR]

A chaotic system arising from double-diffusive convection in a fluid layer is investigated in this paper based on the theory of dynamical systems. A five-dimensional model of chaotic system is obtained using the Galerkin truncated approximation. The results showed that the transition from steady convection to chaos via a Hopf bifurcation produced a limit cycle which may be associated with a homoclinic explosion at a slightly subcritical value of the Rayleigh number. [ABSTRACT FROM AUTHOR]

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ≤ Ra ≤ 1000), the internal heating and the local exponent parameters (0 ≤ γ ≤ 5), (1 ≤ λ ≤ 3), the wall to porous thermal conductivity ratio (0.44 ≤ Kr ≤ 9.9) and the ratio of wall thickness to its width (0.02 ≤ D ≤ 0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio. [ABSTRACT FROM AUTHOR]

In this paper, a non-standard finite difference scheme is proposed for solving a steady finite Rayleigh number convection in a porous cavity with an inclined magnetic field and non-uniform internal heating. Numerical results are compared with the classical finite difference scheme. [ABSTRACT FROM AUTHOR]

Transient natural convection in a square cavity filled with a porous medium is studied numerically. The cavity is assumed heated from one vertical wall and cooled at the top, while the other walls are kept adiabatic. The governing equations are solved numerically by a finite difference method. The effects of Rayleigh number on the initial transient state up to the steady state are investigated for Rayleigh number ranging from 10 to 2 x 10². The evolutions of flow patterns and temperature distributions were presented for Rayleigh numbers, Ra = 10² and 10³. It is observed that the time taken to reach the steady state is longer for low Rayleigh number and shorter for high Rayleigh number. [ABSTRACT FROM AUTHOR]

Heat transfer performance utilizing nanofluids in a trapezoidal enclosure is investigated taking into account variable thermal conductivity and viscosity. Transport equations are modelled by a stream-vorticity formulation, and are solved numerically by the finite difference method. The effects of the Rayleigh number, base angle, volume fraction, and size of nanoparticles on flow and temperature patterns as well as the heat transfer rate are presented. We found that the effect of the viscosity was more dominant than the thermal conductivity, and there is almost no improvement in heat transfer performance utilizing nanofluids. [ABSTRACT FROM PUBLISHER]

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure. [ABSTRACT FROM AUTHOR]

The problem of unsteady natural convection in a square region filled with a fluid-saturated porous medium having non-uniform internal heating and heated laterally is considered. The heated wall surface temperature varies sinusoidally with the time about fixed mean temperature. The opposite cold wall is maintained at a constant temperature. The top and bottom horizontal walls are kept adiabatic. The flow field is modelled with the Darcy model and is solved numerically using a finite difference method. The transient solutions obtained are all periodic in time. The effect of Rayleigh number, internal heating parameters, heating amplitude and oscillating frequency on the flow and temperature field as well as the total heat generated within the convective region are presented. It was found that strong internal heating can generate significant maximum fluid temperatures above the heated wall. The location of the maximum fluid temperature moves with time according to the periodically changing heated wall temperature. The augmentation of the space-averaged temperature in the cavity strongly depends on the heating amplitude and rather insensitive to the oscillating frequency. [ABSTRACT FROM AUTHOR]

Abstract: Using the theory of dynamical systems, this study investigated the effects of a uniform internal heat generation on chaotic behaviour in thermal convection in a fluid-saturated porous layer subject to gravity and heated from below for low Prandtl number. A low-dimensional, Lorenz-like model was obtained using Galerkin truncated approximation. The fourth-order Runge–Kutta method was employed to solve the nonlinear system. We found that there is an inverse proportional relation between the level of internal heat G and the scaled Rayleigh number R, and consequently the porous media gravity-related Rayleigh number Ra. [Copyright &y& Elsevier]

Abstract: This study investigated the instability of Bénard–Marangoni convection in a horizontal layer of ferrofluid under the influence of a linear feedback control and cubic temperature profile. A linear stability analysis was performed. A single-term Galerkin technique was used to obtain the critical Marangoni number and critical Rayleigh number. The possibility of delaying the onset of convection by the application of linear feedback control is demonstrated. [Copyright &y& Elsevier]

The effects of nonuniform heating and a finite wall thickness on natural convection in a square porous cavity based on the local thermal nonequilibrium (LTNE) model are studied numerically using the finite difference method (FDM). The finite-thickness horizontal wall of the cavity is heated either uniformly or nonuniformly, and the vertical walls are maintained at constant cold temperatures. The top horizontal insulated wall allows no heat transfer to the surrounding. The Darcy law is used along with the Boussinesq approximation for the flow. The results of this study are obtained for various parametric values of the Rayleigh number, thermal conductivity ratio, ratio of the wall thickness to its height, and the modified conductivity ratio. Comparisons with previously published work verify good agreement with the proposed method. The effects of the various parameters on the streamlines, isotherms, and the weighted-average heat transfer are shown graphically. It is shown that a thicker bottom solid wall clearly inhibits the temperature gradient which then leads to the thermal equilibrium case. Further, the overall heat transfer is highly affected by the presence of the solid wall. The results have possible applications in the heat-storage fluid-saturated porous systems and the applications of the high power heat transfer. [ABSTRACT FROM AUTHOR]

The problem of transient natural convection heat transfer in a nanoliquid-saturated porous oblique cavity is studied numerically using the finite difference method. The hot left wall of the cavity is maintained at a constant temperature and so is the cold right wall. The horizontal walls allow no heat transfer to the surrounding. The Darcy law is used along with the Boussinesq approximation for the flow. The heat transfer equations are those of a two-phase medium, one for the nanoliquid and another for the porous medium. Water-based nanoliquids with Ag or Cu or Al 2 O 3 or TiO 2 nanoparticles are chosen for investigation. The governing parameters of this study are the Rayleigh number ( 10 2 ≤ Ra ⩽ 10 4 ) , modified conductivity ratio ( 0.1 ≤ γ ⩽ 1000 ) , inter-phase heat transfer ( 0.1 ≤ H ⩽ 1000 ) , inclination angle of the sloping walls ( − π 3 ≤ ω ≤ π 3 ) , volume fraction of nanoparticles ( 0 ≤ φ ≤ 0.2 ) and dimensionless time ( 0 ≤ τ ⩽ 0.1 ) . The assumption of local thermal non-equilibrium leads to an enhanced heat transfer situation in nanoliquids saturating a porous medium compared to that in a porous medium with local thermal equilibrium. Explanation for the observed influences of various parameters on streamlines, isotherms and weighted average Nusselt number is given in terms of thermophysical properties of four nanoparticles, water and porous medium. It is shown that the strength of the flow circulation increases for the relative concentration of nanoliquid with the increment of the inclination angle to the positive direction. [ABSTRACT FROM AUTHOR]

• FEM and Arbitrary Lagrangian-Eulerian (ALE) procedure are used in the numerical analysis. • Two solid elastic thin fins are attached on the top cold wall. • Elasticity of the flexible fin can highly improve the heat transfer. • The results revealed an essential effect of the fin direction on the flow and heat transfer. The problem of unsteady mixed convection in a square enclosure containing a heated circular cylinder is studied in this paper. Two solid elastic thin fins are attached on the top cold wall. The governing equations for fluid flow and energy in the fluid and structures domains are written in Arbitrary Lagrangian-Eulerian (ALE) formulation. Finite element method was used to obtain the approximate solutions of the governing equation. The governing parameters considered are the oscillating fin direction, fin amplitude and length with various elasticity values and cylinder sizes. The results show that the instantaneous heat transfer characteristics of the contrary oscillation differs greatly from the identical oscillation. The influence of the elasticity value on the heat transfer rate is significant at a sufficiently small amplitude oscillation. [ABSTRACT FROM AUTHOR]

The problem of natural convection in a trapezoidal cavity partly filled with nanofluid porous layer and partly with non-Newtonian fluid layer is visualized by heatline. Water-based nanofluids with silver or copper or alumina or titania nanoparticles are chosen for investigation. The governing equations are solved numerically using the Finite Volume Method (FVM) over a wide range of Rayleigh number ( Ra = 10 5 and 10 6 ), Darcy number ( 10 - 5 ⩽ Da ⩽ 10 - 1 ), nanoparticle volume fraction ( 0 ⩽ ϕ ⩽ 0.2 ), power-law index ( 0.6 ⩽ n ⩽ 1.4 ), porous layer thickness ( 0.3 ⩽ S ⩽ 0.7 ), the side wall inclination angle ( 0 ° ⩽ φ ⩽ 21.8 ° ) and the inclination angle of the cavity ( 0 ° ⩽ ω ⩽ 90 ° ). Explanation for the influence of various above mentioned parameters on streamlines, isotherms and overall heat transfer is provided on the basis of thermal conductivities of nanoparticles, water and porous medium. It is shown that convection increases remarkably by the addition of silver–water nanofluid and the heat transfer rate is affected by the inclination angle of the cavity variation. The results have possible applications in heat-removal and heat-storage fluid-saturated porous systems. [ABSTRACT FROM AUTHOR]