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The present study deals with computational investigation of natural convection in a porous chamber having a copper finned heat sink and a thermally producing element of finite thickness and heat conductivity. The enclosure is cooled from the external vertical and upper horizontal walls, whilst the remaining borders are considered to be thermally insulated. Newtonian and heat-conducting liquid with temperature-dependent viscosity is assumed as a working medium. Governing equations are written in non-dimensional variables «stream function—vorticity—temperature» and they are worked out by the finite difference technique. The temperature distribution, streamlines and the dependences of the integral characteristics on the governing characteristics, such as the Darcy and Ostrogradsky numbers and geometric characteristics of the radiator have been illustrated. Presented results can be used for the development of passive cooling systems.

Purpose The purpose of this study is a numerical analysis of transient natural convection in a square partially porous cavity with a heat-generating and heat-conducting element using the local thermal non-equilibrium model under the effect of cooling from the vertical walls. It should be noted that this research deals with a development of passive cooling system for the electronic devices. Design/methodology/approach The domain of interest is a square cavity with a porous layer and a heat-generating element. The vertical walls of the cavity are kept at constant cooling temperature, while the horizontal walls are adiabatic. The heat-generating solid element is located on the bottom wall. A porous layer is placed under the clear fluid layer. The governing equations, formulated in dimensionless stream function, vorticity and temperature variables with corresponding initial and boundary conditions, are solved using implicit finite difference schemes of the second order accuracy. The governing parameters are the Darcy number, viscosity variation parameter, porous layer height and dimensionless time. The effects of varying these parameters on the average total Nusselt number along the heat source surface, the average temperature of the heater, the fluid flow rate inside the cavity and on the streamlines and isotherms are analyzed. Findings The results show that in the case of local thermal non-equilibrium the total average Nusselt number is an increasing function of the interphase heat transfer coefficient and the porous layer thickness, while the average heat source temperature decreases with the Darcy number and viscosity variation parameter. Originality/value An efficient numerical technique has been developed to solve this problem. The originality of this work is to analyze unsteady natural convection within a partially porous cavity using the local thermal non-equilibrium model in the presence of a local heat-generating solid element. The results would benefit scientists and engineers to become familiar with the analysis of convective heat transfer in enclosures with local heat-generating heaters and porous layers, and the way to predict the heat transfer rate in advanced technical systems, in industrial sectors including transportation, power generation, chemical sectors and electronics.

Numerical study of unsteady mixed convection in an open partially porous horizontal channel with a heatgenerating source was performed. The outer surfaces of horizontal walls of finite thickness were adiabatic. In the channel there was a Newtonian heat-conducting fluid with a temperature-dependent viscosity. The discrete heatconducting and heat-generating source is located inside the bottom wall. The temperature of the fluid phase was equal to the temperature of the porous medium, and calculations were performed using the local thermal equilibrium model. The porous insertion is isotropic, homogeneous and permeable to fluid. The Darcy-Brinkman model was used to simulate the transport process within the porous medium. Governing equations formulated in dimensionless variables "stream function - vorticity - temperature" using the Boussinesq approximation were solved numerically by the finite difference method. The vorticity dispersion equation and energy equation were solved using locally one-dimensional Samarskii scheme. The diffusive terms were approximated by central differences, while the convective terms were approximated using monotonic Samarskii scheme. The difference equations were solved by the Thomas algorithm. The approximated Poisson equation for the stream function was solved separately by successive over-relaxation method. Optimal value of the relaxation parameter was found on the basis of computational experiments. The developed computational code was tested using a set of uniform grids and verified by comparing the results obtained of other authors. Numerical analysis of unsteady mixed convection of variable viscosity fluid in the horizontal channel with a heat-generating source was performed for the following parameters: $\mathrm{Pr} = 7.0$, $\varepsilon = 0.8$, $\mathrm{Gr} = 10^5$, $C = 0-1$, $10^{-5} \mathrm{Da} 10^{-1}$, $50 \mathrm{Re} 500$, $\delta = l/H = 0.6-3$. Distributions of the isolines of the stream function, temperature and the dependences of the average Nusselt number and the average temperature inside the heater were obtained in a steady-state regime, when the stationary picture of the flow and heat transfer is observed. As a result we showed that an addition of a porous insertion leads to an intensification of heat removal from the surface of the energy source. The increase in the porous insertion sizes and the use of working fluid with different thermal characteristics, lead to a decrease in temperature inside the source.

Cooling of heat-generating elements is a challenging problem in engineering. In this article, the transient free convection of a temperature-dependent viscosity liquid inside the porous cavity with copper radiator and the heat-generating element is studied using mathematical modeling techniques. The vertical and top walls of the chamber are kept at low constant temperature, while the bottom wall is kept adiabatic. The working fluid is a heat-conducting liquid with temperature-dependent viscosity. A mathematical model is developed based on dimensionless stream function, vorticity, and temperature variables. The governing properties are the variable viscosity, geometric parameters of the radiator, and size of thermally insulated strip on vertical surfaces of the cavity. The effect of these parameters on the energy transport and circulation patterns are analyzed numerically. Based on the numerical results obtained, recommendations are given on the optimal values of the governing parameters for the effective operation of the cooling system. It is shown that the optimal number of radiator fins for the cooling system configuration under consideration is 3. In addition, the thermal insulation of the vertical walls and the increased thickness of the radiator fins have a negative effect on the operation of the cooling system.

Purpose The purpose of this paper is a numerical analysis of natural convection in a square porous cavity filled with a water-based magnetic fluid of geothermal viscosity under the effect of inclined uniform magnetic field. Design/methodology/approach The domain of interest includes the square porous cavity filled with a water-based magnetic fluid (W40). Horizontal walls are supposed to be adiabatic, while right vertical wall is kept at constant low temperature and left vertical wall is kept at constant high temperature. An inclined uniform magnetic field affects the fluid flow and heat transfer inside the cavity. The viscosity of the working fluid is proportional to the linearly decreasing function of depth (vertical coordinate) and inversely proportional to the linear function of temperature. It is assumed in the analysis that the flow is laminar. The fluid is Newtonian and the Boussinesq approximation is valid. The governing equations have been discretized using the finite difference method with the uniform grid. Simulations have been carried out for different values of the Rayleigh number, Hartmann number, Darcy number, magnetic field inclination angle and viscosity variation parameters. Findings It has been revealed that an increase in the viscosity parameters leads to the heat transfer enhancement and convective flow intensification. At the same time, this intensification is more essential for high values of the Rayleigh number. Originality/value The originality of this work is to analyze MHD natural convection in a square porous cavity filled with a water-based magnetic fluid of geothermal viscosity. The results would benefit scientists and engineers to become familiar with the analysis of convective heat and mass transfer in nanofluids, and the way to predict the properties of nanofluid convective flow in advanced technical systems, in industrial sectors including transportation, power generation, chemical sectors and electronics.

In this study, mixed convection of Al2O3-water nanofluid in a lid-driven cavity under the effect of two porous layers is numerically studied. Porous layers of different thermal properties, permeability and porosity are located on the bottom wall. This bottom wall of the cavity is kept at hot temperature Th, while upper moved wall is kept at constant cold temperature Tc and other walls of the cavity are supposed to be adiabatic. Governing equations with corresponding boundary conditions formulated in dimensionless stream function and vorticity using Brinkman-extended Darcy model for porous layers have been solved numerically using finite difference method. Numerical analysis has been carried out for a wide range of the Richardson number (Ri = 0.01–10.0), the Darcy number for the porous layer I (Da1 = 10–7–10–3), porous layers thickness (δ = 0.1–0.3) and nanoparticles volume fraction (ϕ = 0–0.04). It has been found that in the natural convection regime an addition of nanoparticles leads to the heat transfer enhancement, while for mixed convection and forced convection regimes an increase in nanoparticles volume fraction leads to the heat transfer reduction.

In this study, natural convection combined with entropy generation of Fe3O4–water nanofluid within an open trapezoidal cavity filled with a porous layer and a ferrofluid layer under the effect of uniform inclined magnetic field is numerically analyzed. Porous layer is located on the bottom wall and heated from the left inclined wall. Bottom wall, right and left tilted walls of the cavity are adiabatic except for the active part along the left inclined wall where hot temperature Th is constant, upper open boundary is kept at constant cold temperature Tc. Governing equations with corresponding boundary conditions formulated in dimensionless stream function and vorticity using Brinkman-extended Darcy model for porous layer have been solved numerically using finite difference method. Numerical analysis has been carried out for a wide range of Hartmann number, magnetic field inclination angle, height of the porous layer and nanoparticles volume fraction. It has been found that an increase in Hartmann number leads to a growth of oscillations amplitude for average Nusselt number and average entropy generation. At the same time inclination angle α = π/2 illustrates unstable behavior of heat and fluid flow.

A numerical study of the natural convection combined with thermal radiation inside a square porous cavity filled with a fluid of temperature- dependent viscosity is carried out. The side horizontal walls are assumed to be adiabatic while both the left and right vertical walls are kept at constant but different temperatures. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The governing equations formulated in dimensionless stream function, vorticity and temperature variables are solved using finite difference method. A parametric analysis illustrating the effects of the radiation parameter (0 ≤ Rd ≤ 10), Darcy number (10–5 ≤ Da ≤ 10–2) and viscosity variation parameter (0 ≤ C ≤ 6) on fluid flow and heat transfer is implemented. The results show an essential intensification of convective flow with an increase in the radiation parameter.

Thermal convection is one of the main mechanisms that are implemented in different cooling systems. Most often this phenomenon occurs in such practical areas as microelectronics, instrumentation and building construction. In this article, the numerical simulation of thermogravitational convection of medium having dependent viscosity inside closed enclosure with isothermal heating/cooling from vertical walls has been conducted. Other borders of the chamber are heat insulated. The control relations have been written employing non-dimensional vector potential functions, vorticity vector and temperature. The finite difference procedures are applied for solving of the considered differential equations. Obtained outcomes have been compared with numerical data for thermal convection within a differentially heated square cabinet. An influence of third coordinate, Rayleigh number and viscosity variation parameter on thermal distribution and flow structures combined with mean Nusselt number has been demonstrated. Obtained outcomes have shown that for the considered 3D problem an evaluation of the mean Nusselt number can be performed by employing 2D data when the aspect ratio is greater than or equal to unity.

Convective heat transfer is a major phenomenon in different technical and natural systems. This process can be found in enclosures under the different temperature distributions at bounded walls. In this research, the thermogravitation energy transport of a non-constant viscosity medium inside a 3D region under the non-uniform or uniform temperature profile at one of the vertical walls is addressed. The opposite border is at a low constant temperature, and the remaining borders are well insulated. The control equations with the border restrictions are reduced into a non-dimensional form and then integrated by the finite difference procedures. The control characteristics are the Rayleigh number and viscosity changeable magnitude. The impacts of these characteristics on the energy transference and flow structures are studied.

Convective energy transport can be found in various engineering and nature systems. From practical point of view this transport phenomenon within engineering cabinets can be progressed in the presence of temperature modulation. In this research, the transient natural convective heat transference of a temperature-dependent viscosity liquid inside an inclined cube in the presence of time-dependent temperature profile at one vertical bounded surface is investigated. The opposite vertical border is kept at permanent low temperature, whilst other surfaces are thermally insulated. Partial differential equations formulated using dimensionless non-primitive characteristics with appropriate restrictions are worked out by the finite difference technique. Impacts of the domain tilted angle, and wall temperature oscillation frequency on energy transport and flow structures are scrutinized.

Numerical investigation of natural convective heat transfer and fluid flow in a differentially heated square cavity filled with a CuO-water nanofluid having variable properties is performed. Governing partial differential equations formulated in nondimensional stream function, vorticity, temperature, and nanoparticles volume fraction are solved by the second-order accurate finite difference method and taking into account the Brownian diffusion and thermophoresis. The effects of Rayleigh number (Ra = 104–106), initial nanoparticles volume fraction (C0 = 0–0.09), location of the heater (Δ = 0.0–0.9), and dimensionless time (τ = 0–300) on flow patterns, isotherms, and concentration fields as well as the local and average Nusselt numbers at the heater surface are studied. The isoconcentrations reveal that for most of the cavity domain the nanoparticle concentration is around the initial average concentration of nanoparticles except for a very limited variation in a region close to the cavity walls that experiences minor deviation from the initial concentration. It was found that the flow strength within the cavity (i.e., ψmaxRa⋅Pr) is inversely proportional to the heater location Δ and is directly proportional to the Rayleigh number. Also, it was found that the best location of the heater, from a heat transfer perspective, is placing it entirely at the left wall of the cavity where a maximum average Nusselt number is registered. This study revealed that for all heater locations there is always an adverse impact of nanoparticles on the heat transfer and the worst case is registered for the Δ = 0 and the least deterioration is noticed for Δ = 0.9.

Natural convection nanofluid heat transfer enhancement in a partially heated cavity is considered under the effect of an external Lorentz force exerted through interaction of the nanoparticles and the applied constant magnetic field. The aluminum oxide (Al2O3 or alumina) nanofluid is considered to be with variable properties (i.e. thermal conductivity, viscosity and electric conductivity) and the cavity is partially heated from its left top corner. The effect of the inclination angle of the applied magnetic field is studied and analyzed. The Nusselt number is calculated at the heater to probe the heat transfer enhancement. Both effective thermal and electric conductivities have been investigated in their respective theoretical and experimental correlations. Numerical experiments are presented to show the discrepancy in heat transfer with the use of such correlations. A substantial difference in the heat transfer is noticed for the use of different correlations. An adverse effect is identified and analyzed with the increase of Hartmann number, the nanoparticle volume fraction, and the position of the heater within the cavity.

Laminar natural convection in a square cavity having two centered adherent porous blocks filled with an alumina/water nanofluid under the effect of horizontal temperature gradient is studied numerically. Each porous block has the unique values of the porosity and permeability. Water-based nanofluids with alumina nanoparticles are chosen for investigation. The control characteristics of this study are the Darcy number of the first porous block ( $10^{-7}\le Da_{1}\le 10^{-3}$ ), the dimensionless porous blocks size ( $0.1\le\delta\le 0.4$ ) and nanoparticles volume fraction ( $0\le\phi\le 0.04$ ). The developed computational code has been validated comprehensively using the grid independency test and experimental data of other authors. The obtained results revealed the heat transfer enhancement at the hot wall with the Darcy number, while a growth of the porous layers size reduces the heat transfer rate at this hot wall. The behavior of the average Nusselt number at the right cold wall is opposite.

Development of electronic devices demands to create an effective cooling system using optimal working liquid and porous insertions near the heat-generating elements. Such features allow intensifying the heat disposal from the heated element area. In this study, the unsteady convection of a variable viscosity liquid within a chamber having a heat-generating element under the impact of porous insertion is investigated. A porous layer surrounds the energy source located on the lower surface. The horizontal and vertical walls are thermally insulated, kept at constant cooling temperatures, respectively. Partial differential equations written in non-dimensional non-primitive variables with appropriate additional conditions are worked out by the finite difference method. The governing characteristics are the non-dimensional medium permeability, volumetric heat flux, variable viscosity and thickness of the porous insertion. The influence of these properties on local and mean fields’ characteristics for the energy transport and flow structures are studied in details. The calculations demonstrate that the inclusion of a porous insertion and the variation of its properties lead to more essential cooling of the heater.

Paper presents the numeral results of transient natural convection in a partially porous enclosure with a heat-conducting and heat-generating element. The horizontal walls of the cavity are adiabatic, while the vertical ones are isothermal and have low temperature. A mathematical model formulated in dimensionless variables such as vorticity, stream function and temperature have been solved by means of the finite difference method. The governing parameters are the porous layer thickness and dimensionless time. The effects of these parameters are shown using streamlines, isotherms and average temperature of the heat-conducting element. The results demonstrate an opportunity to intensify the passive cooling system using the porous layer.

Unsteady natural convection inside of a differentially-heated square enclosure filled with a fluid of temperature-dependent viscosity has been numerically studied. A mathematical model formulated in the dimensionless stream function and vorticity has been solved by a finite difference method of the second order accuracy. The effect of dimensionless time and Prandtl number on streamlines and isotherms has been investigated for Ra = 105. The results clearly demonstrate an evolution of fluid flow and heat transfer in the case of variable viscosity fluid.

A numerical investigation is implemented on the unsteady natural convection with a temperature-dependent viscosity inside a square porous cavity. The vertical walls of the cavity are kept at constant but different temperatures, while the horizontal walls are adiabatic. The mathematical model formulated in dimensionless stream function, vorticity and temperature variables is solved using implicit finite difference schemes of the second order. The governing parameters are the Rayleigh number, Darcy number, viscosity variation parameter and dimensionless time. The effects of these parameters on the average Nusselt number along the hot wall as well as on the streamlines and isotherms are analyzed. The results show an intensification of convective flow and heat transfer with an increase in the viscosity variation parameter for the porous media, while in the case of pure fluid, the effect is opposite.