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The purpose of this paper is to address various works on mixed convection and proposes 10 unified models (Models 1–10) based on various thermal and kinematic conditions of the boundary walls, thermal conditions and/ or kinematics of objects embedded in the cavities and kinematics of external flow field through the ventilation ports. Experimental works on mixed convection have also been addressed.

This review is based on 10 unified models on mixed convection within cavities. Models 1–5 involve mixed convection based on the movement of single or double walls subjected to various temperature boundary conditions. Model 6 elucidates mixed convection due to the movement of single or double walls of cavities containing discrete heaters at the stationary wall(s). Model 7A focuses mixed convection based on the movement of wall(s) for cavities containing stationary solid obstacles (hot or cold or adiabatic) whereas Model 7B elucidates mixed convection based on the rotation of solid cylinders (hot or conductive or adiabatic) within the cavities enclosed by stationary or moving wall(s). Model 8 is based on mixed convection due to the flow of air through ventilation ports of cavities (with or without adiabatic baffles) subjected to hot and adiabatic walls. Models 9 and 10 elucidate mixed convection due to flow of air through ventilation ports of cavities involving discrete heaters and/or solid obstacles (conductive or hot) at various locations within cavities.

Mixed convection plays an important role for various processes based on convection pattern and heat transfer rate. An important dimensionless number, Richardson number (Ri) identifies various convection regimes (forced, mixed and natural convection). Generalized models also depict the role of “aiding” and “opposing” flow and combination of both on mixed convection processes. Aiding flow (interaction of buoyancy and inertial forces in the same direction) may result in the augmentation of the heat transfer rate whereas opposing flow (interaction of buoyancy and inertial forces in the opposite directions) may result in decrease of the heat transfer rate. Works involving fluid media, porous media and nanofluids (with magnetohydrodynamics) have been highlighted. Various numerical and experimental works on mixed convection have been elucidated. Flow and thermal maps associated with the heat transfer rate for a few representative cases of unified models [Models 1–10] have been elucidated involving specific dimensionless numbers.

This review paper will provide guidelines for optimal design/operation involving mixed convection processing applications.

This study aims to numerically scrutinize the entropy generation minimization and mixed convective heat transfer of multi-walled carbon nanotubes–Fe₃O₄/water hybrid nanofluid flow in a lid-driven square enclosure with heat generation in the presence of a porous layer on inner surfaces, considering local thermal non-equilibrium (LTNE) approach and the non-Darcy flow model.

The dimensionless governing equations for hybrid nanofluid and solid phases are solved by applying the finite volume method and semi-implicit method for pressure-linked equations algorithm.

The roles of the internal heat generation in the porous layer, LTNE model and nanoparticles volume fraction on mixed convection phenomenon and entropy generation are introduced for lid-driven cavity hybrid nanofluid flow. Based on the investigation of entropy generation and heat transfer, the minimum total entropy generation and average Nusselt numbers are found at 1 ≤ Ri ≤ 10 where the effect of the forced and free convection flow directions being opposite each other is very significant. When considering various nanoparticle volume fractions, it becomes evident that the minimum entropy generation occurs in the case of φ = 0.1%. The outcomes of LTNE number reveal the operating parameters in which thermal equilibrium occurs between hybrid nanofluid and solid phases.

The analysis of entropy generation under various shear and buoyancy forces plays a significant role in the suitable thermal design and optimization of mixed convective heat transfer applications. This research significantly contributes to the optimization of design and the advancement of innovative solutions across diverse engineering disciplines, such as packed-bed thermal energy storage and thermal insulation.

The aim of this study is to investigate the effects of two-phase nanofluid model on mixed convection in a double lid-driven square cavity in the presence of a magnetic field. The authors believe that this work is a good contribution for improving the thermal performance and the heat transfer enhancement in some engineering instruments.

The current work investigates the problem of mixed convection heat transfer in a double lid-driven square cavity in the presence of magnetic field. The used cavity is filled with water-Al₂O₃ nanofluid based on Buongiorno’s two-phase model. The bottom horizontal wall is maintained at a constant high temperature and moves to the left/right, while the top horizontal wall is maintained at a constant low temperature and moves to the right/left. The left and right vertical walls are thermally insulated. The dimensionless governing equations are solved numerically using the Galerkin weighted residual finite element method.

The obtained results show that the heat transfer rate enhances with an increment of Reynolds number or a reduction of Hartmann number. In addition, effects of thermophoresis and Brownian motion play a significant role in the growth of convection heat transfer.

According to above-mentioned studies and to the authors’ best knowledge, there has no study reported the MHD mixed convection heat transfer in a double lid-driven cavity using the two-phase nanofluid model. Thus, the authors of the present study believe that this work is valuable. Therefore, the aim of this comprehensive numerical study is to investigate the effects of two-phase nanofluid model on mixed convection in a double lid-driven square cavity in the presence of a magnetic field. The authors believe that this work is a good contribution for improving the thermal performance and the heat transfer enhancement in some engineering instruments.

The purpose of this study is to determine interfacial convective heat transfer coefficient numerically, for a porous media consisting of square blocks in inline arrangement under mixed convection heat transfer.

The continuity, momentum and energy equations are solved in dimensionless form for a representative elementary volume of porous media, numerically. The velocity and temperature fields for different values of porosity, Ri and Re numbers are obtained. The study is performed for the range of Ri number from 0.01 to 10, Re number from 100 to 500 and porosity value from 0.51 to 0.96. Based on the obtained results, the value of the interfacial convective heat transfer coefficient is calculated by using volume average method.

It was found that at low porosities (such as 0.51), the interfacial Nusselt number does not considerably change with Ri and Re numbers. However, for porous media with high Ri number and porosity (such as 10 and 0.51, respectively), secondary flows occur in the middle of the channel between rods improving heat transfer between solid and fluid, considerably. It is shown that the available correlations of interfacial heat transfer coefficient suggested for forced convection can be used for mixed convection for the porous media with low porosity (such as 0.51) or for the flow with low Ri number (such as 0.01).

To the best of the authors’ knowledge, there is no study on determination of interfacial convective heat transfer coefficient for mixed convection in porous media in literature. The present study might be the first study providing an accurate idea on the range of this important parameter, which will be useful particularly for researchers who study on mixed convection heat transfer in porous media, macroscopically.

The purpose of this paper is to investigate numerically mixed convection of Al₂O₃-water nanofluids flowing through a horizontal ventilated cavity heated from below by a temperature varying sinusoidally along its lower wall. The simulations focus on the effects of different key parameters, such as Reynolds number (200 ≤ Re ≤ 5,000), nanoparticles’ concentration (0 ≤ ϕ ≤ 0.1) and phase shift of the heating temperature (0 ≤ γ ≤ π), on flow and thermal patterns and heat transfer performances.

The Navier–Stokes equations describing the nanofluid flow were discretized using a finite difference technique. The vorticity and energy equations were solved by the alternating direction implicit method. Values of the stream function were obtained by using the point successive over-relaxation method.

The simulations were performed for two modes of imposed external flow (injection and suction). The main findings are that the dynamical and thermal fields are affected by the parameters Re, ϕ, γ and the applied ventilation mode; the addition of nanoparticles leads to an improvement of heat transfer rate and an increase of mean temperature inside the enclosure; the heat exchange performance and the better cooling are more pronounced in suction mode; the phase shift of the heating temperature may lead to periodic solutions for weaker values of Re and contributes to an increase or a decrease of heat transfer depending on the value of ϕ and the convection regime.

To the best of the authors’ knowledge, the problem of mixed convection of a nanofluid inside a vented cavity using the injection or suction technics and submitted to non-uniform heating conditions has not been treated so far.

This study aims to investigate the numerical analysis of mixed convection within the horizontal annulus in the presence of water-based fluid with nanoparticles of aluminum oxide, copper, silver and titanium oxide. Numerical solution is performed using a finite-volume method based on the SIMPLE algorithm, and the discretization of the equations is generally of the second order. Inner and outer cylinders have a constant temperature, and the inner cylinder temperature is higher than the outer one. The two cylinders can be rotated in both directions at a constant angular velocity. The effect of parameters such as Rayleigh, Richardson, Reynolds and the volume fraction of nanoparticles on heat transfer and flow pattern are investigated. The results show that the heat transfer rate increases with the increase of the Rayleigh number, as well as by increasing the volume fraction of the nanoparticles, the heat transfer rate increases, and this increase is about 8.25 per cent for 5 per cent volumetric fraction. Rotation of the cylinders reduces the overall heat transfer. Different directions of rotation have a great influence on the flow pattern and isotherms, and ultimately on heat transfer. The addition of nanoparticles does not have much effect on the flow pattern and isotherms, but it is quantitatively effective. The extracted results are in good agreement with previous works.

Studying mixed convection heat transfer in the horizontal annulus in the presence of a water-based fluid with aluminum oxide, copper, silver and titanium oxide nanoparticles is carried out quantitatively using a finite-volume method based on the SIMPLE algorithm.

Increasing the Rayleigh number increases the Nusselt number. Increasing the Richardson number increases heat transfer. Adding nanoparticles does not have much effect on the flow pattern but is effective quantitatively on heat transfer parameters. The addition of nanoparticles sometimes increases the heat transfer rate by about 8.25 per cent. In constant Rayleigh numbers, increasing the Reynolds number reduces heat transfer. The Rayleigh and Reynolds numbers greatly affect the isotherms and streamlines. In addition to the thermal conductivity of nanoparticles, the thermo-physical properties of nanoparticles has great effect in the formation of isotherms and streamlines and ultimately heat transfer.

Studying the effect of different direction of rotation on the isotherms and streamlines, as well as the comparison of different nanoparticles on mixed convection heat transfer in annulus.

This study aims to numerically analyse the impact of an inclined magnetic field and Joule heating on the conjugate heat transfer because of the mixed convection of an Al₂O₃–water nanofluid in a thick wall enclosure.

A horizontal temperature gradient together with the shear-driven Flow creates the mixed convection inside the enclosure. The nonhomogeneous model, in which the nanoparticles have a slip velocity because of thermophoresis and Brownian diffusion, is adopted in the present study. The thermal performance is evaluated by determining the entropy generation, which includes the contribution because of magnetic field. A control volume method over a staggered grid arrangement is adopted to compute the governing equations.

The Lorentz force created by the applied magnetic field has an adverse effect on the flow and thermal field, and consequently, the heat transfer and entropy generation attenuate because of the presence of magnetic force. The Joule heating enhances the fluid temperature but attenuates the heat transfer. The impact of the magnetic field diminishes as the angle of inclination of the magnetic field is increased, and it manifests as the volume fraction of nanoparticles is increased. Addition of nanoparticles enhances both the heat transfer and entropy generation compared to the clear fluid with enhancement in entropy generation higher than the rate by which the heat transfer augments. The average Bejan number and mixing-cup temperature are evaluated to analyse the thermodynamic characteristics of the nanofluid.

This literature survey suggests that the impact of an inclined magnetic field and Joule heating on conjugate heat transfer based on a two-phase model has not been addressed before. The impact of the relative slip velocity of nanoparticles diminishes as the magnetic field becomes stronger.

The purpose of this paper is to numerically investigate steady, laminar natural and mixed convection heat transfer in a two-dimensional cavity by using a finite volume method with a fourth-order approximation of convective terms, with and without the presence of nanoparticles. Highly accurate benchmark results are also provided.

A finite volume method on a non-uniform staggered grid is used for the solution of two-dimensional momentum and energy conservation equations. Diffusion terms, in the momentum and energy equations, are approximated using second-order central differences, whereas a non-uniform four-point fourth-order interpolation (FPFOI) scheme is developed for the convective terms. Coupled mass and momentum conservation equations are solved iteratively using a semi-implicit method for pressure-linked equation method.

For the case of natural convection problem at high-Rayleigh numbers, grid density must be sufficiently high in order to obtain grid-independent results and capture reality of the physics. Heat transfer enhancement for natural convection is observed up to a certain value of the nanoparticle volume fraction. After that value, heat transfer deterioration is found with increasing nanoparticle volume fraction.

Developed a non-uniform FPFOI scheme. Highly accurate benchmark results for the heat transfer of Al₂O₃-water nanofluid in a cavity are provided.

This paper's aim is to examine flow and heat transfer through vertical channels between parallel plates, which is of prime importance in the design of cooling systems for electronic equipment such as that of finned cold plates in general, plate‐and‐frame heat exchangers, etc.

Numerical and analytical solutions are presented to investigate the heat transfer enhancement and the pressure drop reduction due to buoyancy effects (for buoyancy‐aided flow) for the developing laminar mixed convection in vertical channel between parallel plates in the vicinity of the critical values of the buoyancy parameter (Gr/Re)crt that are obtained analytically. The numerical solutions are presented for a wide range of the buoyancy parameters Gr/Re that cover both of buoyancy‐opposed and buoyancy‐aided flow situations under each of the isothermal boundary conditions under investigation.

Buoyancy parameters greater than the critical values result in building‐up the pressure downstream of the entrance such that the vertical channel might act as a thermal diffuser with possible incipient flow reversal. Locations at which the pressure gradient vanishes and the locations at which the pressure‐buildup starts have been numerically obtained and presented for all the investigated cases.

The study is limited to the laminar flow situation.

The results clearly show that for buoyancy‐aided flow, the increase of the buoyancy parameter enhances the heat transfer and reduces the pressure drop across the vertical channel. These findings are very useful for cooling channel or chimney designs.

The study is original and presents new findings, since none of the previous studies reported the conditions for which pressure buildup might take place due to mixed convection in vertical channels between parallel plates.

The main objective of the present study is to investigate the effects of various lengths and different locations of the heater on the left sidewall in a square lid‐driven cavity.

The non‐dimensional equations are discretized by the finite‐volume method. The upwind scheme and the central difference scheme are implemented for the convection and the diffusion terms, respectively.

On increasing the Richardson number, the overall heat transfer is increased whether the length and the location of the heater is considered or not. Among the various lengths of the heater considered, the total heat transfer is better only for the length LH=1/3 of the heater if it is extended from top or bottom of the cavity. In the case of location of the heater, the average heat transfer enhances for center location of the heater. Existence of the magnetic field suppresses the convective heat transfer and the fluid flow.

The results can be used in the cooling of electronic devices and heat transfer improvement in heat exchangers.

The numerical results obtained here focus on the detailed investigation of flow and temperature field in a discretely heated lid‐driven square cavity. The findings will be helpful in many applications such as heat exchangers and cooling of electronic devices.