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The passive feature of natural convection encourages continuous investigation on this field. This research work studies the natural convection inside a cavity of wavy vertical surfaces, including a heat source block standing on the base of cavity. The employed liquid is a hybrid water–Cu–Al 2 O 3 nanofluid. The vertical wavy surfaces are fixed at low temperature, while the heat source remains constant at a higher temperature. The Galerkin finite element method is applied to obtain the numerical solutions. Besides the common visualization tools, the heatlines tool is used in this paper as well. It is found from the outcomes that the Nusselt number is purely proportional to the length of the heat source. In addition, the hybrid water–Cu–Al 2 O 3 nanofluid exhibits a higher Nusselt number than that corresponding to a pure water and as well as other regular nanofluids. The percentages of the average heat transfer gained by the hybrid nanofluid at ϕ = 0.02 are 13.7% and 5% higher than those for pure water and other nanofluids. [ABSTRACT FROM AUTHOR]

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]

Graphical abstract Highlights • FEM was employed to solve the governing equations subject to the boundary conditions. • Two-phase nanofluid model in a wavy cavity with conductive inner block is considered. • Comparisons with previously experimental and numerical published results are provided. • A selection of optimal number of oscillations can enhance the heat transfer inside the cavity. Abstract By employing the finite element method, thermophoresis and Brownian diffusion are studied numerically relating to the natural convection in a wavy cavity that is filled with an Al 2 O 3 -water nanofluid possessing a central heat-conducting solid block that is influenced by the local heater located on the bottom wall. An isothermal condition is established in the two wavy vertical walls, while adiabatic condition is for the top horizontal wall. Partial heating is applied to the bottom of the horizontal wall, while the remaining part remains in the adiabatic condition. Empirical correlations are employed for the thermal conductivity and dynamic viscosity of the nanofluid. The number of oscillations (1 ⩽ N ≤ 4), Rayleigh number ( 10 3 ⩽ Ra ≤ 10 6 ), nanoparticles volume fraction (0 ⩽ ϕ ≤ 0.04) and dimensionless length of the bottom heater (0.2 ⩽ H ⩽ 0.8) govern the parameters in this study. The grid independency test, as well as experimental and numerical data from other published works, was employed to validate the developed computational code comprehensively. Based on the obtained results, it was found that the heat transfer inside the cavity is enhanced by introducing nanoparticles as well as a selection of optimal number of oscillations. [ABSTRACT FROM AUTHOR]

The present study investigates the conjugate heat transfer in a square cavity heated by a triangular solid and saturated with Al2O3-water nanofluid. Two-phase Buongiorno's model is used for modeling the nanofluid heat transfer. The finite element method is used for numerical solution of the dimensionless governing equations subject to the boundary conditions. Comparisons of the proposed method with previously published experimental and numerical works show a good agreement. The effects of some parameters such as the Rayleigh number, thermal conductivity ratio, dimensionless triangular wall thickness and nanofluid volume fraction on heat transfer and nanoparticle distributions are completely studied and discussed. The results show clockwise rotations for streamlines and nanoparticle migration. Also the Nusselt number increases with the nanofluid volume fraction. A continuous reduction is seen for the mean Nusselt number by increasing the dimensionless triangular wall thickness for all the considered values of the Rayleigh number. [ABSTRACT FROM AUTHOR]

This paper investigates a steady mixed convection in a lid-driven square cavity subjected to an inclined magnetic field and heated by corner heater with an inserted square solid block. Water-Al2O3 nanofluid fills the cavity based on Buongiorno's two-phase model. A corner heater is configured in the left lower corner of the cavity by maintaining 40% of the bottom and vertical walls at constant hot temperature. The top horizontal wall is moving and maintained at a constant low temperature. The remainder walls are thermally insulated. The governing equations are solved numerically using the finite element method. The governing parameters are the nanoparticles volume fraction (0≤ϕ≤0.04), Reynolds number (1≤Re≤500), Richardson number (0.01≤Ri≤100), Hartmann number (0≤Ha≤50) and the size of the inner solid (0.1≤D≤0.7). The other parameters: the Prandtl number, Lewis number, Schmidt number, ratio of Brownian to thermophoretic diffusivity and the normalized temperature parameter, are fixed at Pr=4.623, Le=3.5×105, Sc=3.55×104, NBT=1.1 and δ=155, respectively. The inclination of the magnetic field is fixed at γ=π4. Results show that at low Reynolds number, the increase in nanoparticles loading more the 2% becomes useless. It is also found that a big size of the solid block can augment heat transfer in the case of high values of both the Reynolds and Richardson numbers. [ABSTRACT FROM AUTHOR]

A linear stability analysis has been carried out to examine the effect of internal heat source on the onset of Rayleigh–Bénard convection in a rotating nanofluid layer with double diffusive coefficients, namely, Soret and Dufour, in the presence of feedback control. The system is heated from below and the model used for the nanofluid layer incorporates the effects of thermophoresis and Brownian motion. Three types of bounding systems of the model have been considered which are as follows: both the lower and upper bounding surfaces are free, the lower is rigid and the upper is free, and both of them are rigid. The eigenvalue equations of the perturbed state were obtained from a normal mode analysis and solved using the Galerkin method. It is found that the effect of internal heat source and Soret parameter destabilizes the nanofluid layer system while increasing the Coriolis force, feedback control, and Dufour parameter helps to postpone the onset of convection. Elevating the modified density ratio hastens the instability in the system and there is no significant effect of modified particle density in a nanofluid system. [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]

The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-AlO, and water-TiO nanofluids. The governing equations are based on the equations involving the stream function, vorticity, and temperature. The dimensionless forms of the governing equations are solved by the finite difference (FD) scheme consisting of the alternating direction implicit (ADI) method and the tri-diagonal matrix algorithm (TDMA). It is found that the increase in the nanoparticle concentration leads to the decrease in the flow rates in the secondary cells when the convective thermocapillary and the buoyancy force have similar strength. A critical Marangoni number exists, below which increasing the Marangoni number decreases the average Nusselt number, and above which increasing the Marangoni number increases the average Nusselt number. The nanoparticles play a crucial role in the critical Marangoni number. [ABSTRACT FROM AUTHOR]

Natural convection heat transfer in a rotating, differentially heated enclosure is studied numerically in this paper. The rotating enclosure is filled with water-Ag, water-Cu, water-Al2O3, or water-TiO2 nanofluids. The governing equations are in velocity, pressure, and temperature formulation and solved using the staggered grid arrangement together with MAC method. The governing parameters considered are the solid volume fraction, 0.0 = ɸ = 0.05, and the rotational speeds, 3.5 = Ω = 17.5 rpm, and the centrifugal force is smaller than the Coriolis force and both forces were kept below the buoyancy force. It is found that the angular locations of the local maximums heat transfer were sensitive to rotational speeds and nanoparticles concentration. The global quantity of heat transfer rate increases about 1.5%, 1.1%, 0.8%, and 0.6% by increasing 1% ɸ of the nanoparticles Ag, Cu, Al2O3, and TiO2, respectively, for the considered rotational speeds. [ABSTRACT FROM AUTHOR]

The effect of nanofluids on chaotic convection in a fluid layer heated from below was studied in this paper for low Prandtl number based on the theory of dynamical systems. A low-dimensional, Lorenz-like model was obtained using Galerkin-truncated approximations. The fourth-order Runge-Kutta method was employed to solve the nonlinear system. The results show that inhibition of chaotic convection can be observed when using nanofluids. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the flow of nanofluid and heat transfer characteristics between two horizontal plates in a rotating system. The lower plate is a stretching sheet and the upper one is a solid porous plate. Copper (Cu0 as nanoparticle and water as its base fluid have been considered. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions using similarity transformation, which is then solved analytically using the homotopy analysis method (HAM). Comparison between HAM and numerical solutions results showed an excellent agreement. The results for the flow and heat transfer characteristics are obtained for various values of the nanoparticle volume fraction, suction/injection parameter, rotation parameter, and Reynolds number. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable of causing change in the flow pattern. It is found that for both suction and injection, the heat transfer rate at the surface increaseswith increasing the nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter. [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]

The aim of the present study is to analyze the convective flow and heat transfer of nanofluid in an inclined cavity filled with heat generating porous medium using the local thermal non-equilibrium model. The horizontal bottom wall of the cavity is maintained at a constant higher temperature while the left and right vertical walls are maintained at a constant lower temperature. The top wall is to be kept as adiabatic. The Darcy model is adopted for flow through the porous medium and the Boussinesq approximation is taken into account. The pores are saturated by a water-based nanofluid consisting of Cu nanoparticles. The governing equations are solved iteratively by finite difference based Alternating Direction Implicit (ADI) method. The results are discussed by various combinations of parameters involved in the study. The average heat transfer rate decreases when increasing the values of nanoparticle volume fraction for the case of high Rayleigh numbers. The heat transfer rate is enhanced by increasing values of the modified conductivity ratio and the porosity of the media. The results have possible applications in the heat-storage nanofluid-saturated porous systems. [ABSTRACT FROM AUTHOR]

Abstract: Convective heat transfer in a differentially heated square enclosure with an inner rotating cylinder is studied theoretically. The free space between the cylinder and the enclosure walls is filled with water–Ag, water–Cu, water–Al2O3 or water–TiO2 nanofluids. The governing equations are formulated for velocity, pressure and temperature formulation and are modeled in COMSOL, a partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). The governing parameters considered are the solid volume fraction, 0.0⩽ ϕ ⩽0.05, the cylinder radius, 0⩽ R ⩽0.3 and the angular rotational velocity, −1000⩽Ω⩽1000. The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the strength of the flow circulation is much stronger for a higher nanoparticle concentration, a better thermal conductivity value and a smaller cylinder with a faster, negative rotation. The maximum heat transfer are obtained at a high nanoparticle concentration with a good conductivity value, a slow positive rotation and a moderate cylinder size located in the center of the enclosure. [Copyright &y& Elsevier]

Abstract: Heat transfer enhancement utilizing nanofluids in a trapezoidal enclosure is investigated for various pertinent parameters. Transport equations are modelled by a stream-vorticity formulation and solved numerically by finite difference approach. The inclined sloping boundaries is treated by adopting staircase-like zigzag lines. Based upon the numerical predictions, the effects of Grashof number, inclination angle of the sloping wall, volume fraction of nanoparticles on flow and temperature patterns as well as the heat transfer rate within the enclosure are presented. Water–Cu and water–Al2O3 nanofluids were tested. We found that acute sloping wall and Cu nanoparticles with high concentration are effective to enhance the rate of heat transfer. We also developed a new correlation for the average Nusselt number as a function of the angle of the sloping wall, effective thermal conductivity and viscosity as well as Grashof number. [ABSTRACT FROM AUTHOR]

Highlights • Transient mixed convection in a double lid-driven wavy cavity was considered in this work. • The Galerkin weighted residual finite-element method is applied to solve the dimensionless governing equations. • Two phase Buongioenos model is used for modeling the nanofluid heat transfer within the wavy cavity. • A growth of the average Nusselt number at hot wall was depending on the constant moving parameter. Graphical abstract Abstract Heat transfer enhancement in engineering systems can be reached using nanofluids. Very often, technical devices are chambers with moving borders of a flat or a wavy shape having internal heat-conducting blocks. Therefore, the present study is devoted to study computationally the problem of time-dependent heat transfer of alumina-water nanoliquid within a differentially heated chamber with isothermal moving vertical walls and adiabatic horizontal ones under the impact of an inner solid cylinder. The upper wall of the chamber is assumed to have a wavy shape. Basic equations written in non-dimensional primitive variables using the two-component nonhomogeneous equilibrium model for transport phenomena in nanofluids incorporating the effects of Brownian diffusion and thermophoresis with Corcione empirical correlations for the viscosity and thermal conductivity combined with heat conduction equation for the centered solid cylinder has been resolved by the Galerkin finite-element method. The influences of the dimensionless time (0 ≤ τ ≤ 120), Reynolds number (R e = 10 and 100), Richardson number (0.01 ≤ Ri ≤ 100), constant moving parameter [(λ l = 1 , λ r = − 1), (λ l = 1 , λ r = 1), (λ l = − 1 , λ r = 1), (λ l = − 1 , λ r = − 1)], nanoparticle volume fraction (0 ≤ ϕ ≤ 0.04), number of undulations (0 ≤ N ≤ 4) and the dimensionless radius of solid cylinder (0.05 ≤ S ≤ 0.25) on the isolines of stream function, temperature and nanoparticles concentrations, as well as the local and average Nusselt numbers have been investigated. It has been found that a rise of the average Nusselt number at the hot wall depends on the moving parameter and the thermal transmission intensity diminishes with the undulations number and the inner solid cylinder diameter. At the same time, the wavy shape of the border, the characteristics of internal cylinder and the properties of nanofluid are very good control parameters for the heat transfer rate and the fluid flow rate. [ABSTRACT FROM AUTHOR]

The problem of conjugate natural convection of Al 2 O 3 –water nanofluid in a square cavity with concentric solid insert and isothermal corner boundaries using non-homogenous Buongiorno’s two-phase model is studied numerically by the finite difference method. An isothermal heater is placed on the left bottom corner of the square cavity while the right top corner is maintained at a constant cold temperature. The remainder parts of the walls are kept adiabatic. Water-based nanofluids with Al 2 O 3 nanoparticles are chosen for the investigation. The governing parameters of this study are the nanoparticle volume fraction (0 ≤  ϕ  ≤ 0.04), the Rayleigh number (10 2  ≤  Ra  ≤ 10 6 ), thermal conductivity of the solid block ( k w = 0.28 , 0.76, 1.95, 7 and 16) (epoxy: 0.28, brickwork: 0.76, granite: 1.95, solid rock: 7, stainless steel: 16) and dimensionless solid block thickness (0.1 ≤  D  ≤ 0.7). Comparisons with previously experimental and numerical published works verify good agreement with the proposed method. Numerical results are presented graphically in the form of streamlines, isotherms and nanoparticles volume fraction as well as the average Nusselt number and fluid flow rate. The results show that the thermal conductivity ratio and solid block size are very good control parameters for an optimization of heat transfer inside the partially heated and cooled cavity. [ABSTRACT FROM AUTHOR]

The effects of finite wall thickness and sinusoidal heating on convection in a nanofluid-saturated local thermal non-equilibrium (LTNE) porous cavity are studied numerically using the finite difference method. The finite thickness vertical wall of the cavity is maintained at a constant temperature and the right wall is heated sinusoidally. The horizontal insulated walls allow no heat transfer to the surrounding. The Darcy law is used along with the Boussinesq approximation for the flow. Water-based nanofluids with Cu nanoparticles are chosen for investigation. The results of this study are obtained for various parameters such as the Rayleigh number, periodicity parameter, nanoparticles volume fraction, thermal conductivity ratio, ratio of wall thickness to its height and the modified conductivity ratio. Explanation for the influence of the various above-mentioned parameters on the streamlines, isotherms, local Nusselt number and the weighted average heat transfer is provided with regards to the thermal conductivities of nanoparticles suspended in the pure fluid and the porous medium. It is shown that the overall heat transfer is significantly increased with the relative non-uniform heating. Further, the convection heat transfer is shown to be inhibited 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 conjugate natural convection in a square cavity filled with a nanofluid with sinusoidal temperature variations on both horizontal walls is visualized by heatlines. Water-based nanofluids with Ag, Cu, Al 2 O 3 , or TiO 2 nanoparticles are chosen for investigation. The governing equations together with the specified boundary conditions are solved numerically using the finite difference method over a wide range of Rayleigh number ( 10 5 ⩽ Ra ⩽ 10 8 ) , nanoparticle volume fractions ( 0 ⩽ ϕ ⩽ 0.2 ), phase deviations ( 0 ⩽ γ ⩽ π ), amplitude ratios ( 0 ⩽ ε ⩽ 1 ), wall to nanofluid thermal conductivity ratios ( 0.44 ⩽ K r ⩽ 23.8 ) and wall thickness to height ratios ( 0 ⩽ S ⩽ 0.7 ). Comparisons with previously published work verify good agreement with the proposed method. Detailed computational results for the influence of the various parameters on streamlines, heatlines, isotherms, and the overall heat transfer are shown graphically. It is found that the heat transfer rate is significantly enhanced by incrementing the solid wall thickness. Different values of the thermal conductivity ratio are shown to depict a variety of enhancements for the heat transfer rate. [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]

Development of modern heat exchangers or solar collectors is related to the analysis of working fluid flow and heat transfer within different channels. The energy transport enhancement can be reached by including nanofluids as working media and irregular channels to intensify the heat removal. The present research is devoted to computational analysis of nanosuspension forced convection in a horizontal wavy channel under the impact of heating from the upper wavy surface. The single-phase nanofluid approach with experimentally-based correlations for viscosity and thermal conductivity holds implemented for an investigation in combination with Newton's second law for the description of the motion of the nanoparticle within the channel. The formulated boundary-value problem has been worked out by the finite element technique. Rules of Reynolds number, number of channel waviness, and dimensionless time on nanoliquid flow, energy transport and nanoparticles motion within the channel as well as average parameters. It has occurred that a rise from Reynolds number characterizes a narrowing of the fluid tube within the channel with an improvement of the average velocity and average Nusselt number. Augmentation of the channel waviness number results in an increment of the average particles velocity and average temperature. [Display omitted] [ABSTRACT FROM AUTHOR]