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The purpose of this paper is to investigate the effects of magnetic field and viscous dissipation on mixed convection heat transfer, fluid flow and entropy generation in a porous media filled square enclosure heated with corner isothermal heater.

Finite volume method has been used to solve governing equations. A code is developed by FORTRAN and entropy generation is calculated from the obtained results of velocities and temperature. Results are presented via streamlines, isotherms, local and mean Nusselt number for different values of Richardson number (0.001=Ri=100), Hartmann number (0.001=Ha=100), Darcy number (0.001=Da=0.1), length of heaters (0.25=hx=hy=0.75) and viscous dissipation factors (10−4=ε=10−6).

It is observed that entropy is generated mostly due to lid-driven wall and right side of the heater. Entropy generation decreases with increasing of Hartmann number and heat transfer increases with decreasing of viscous parameter.

The originality of this work is to application of magnetic field and viscous dissipation on entropy generation in a lid-driven cavity with corner heater. Here, both corner heater and the external forces are original parameters.

This paper presents a boundary element method (BEM) based on a subdomain approach for the solution of non‐Newtonian fluid flow problems which include thermal effects and viscous dissipation. The volume integral arising from non‐linear terms is converted into equivalent boundary integrals by the multi‐domain dual reciprocity method (MD‐DRM) in each subdomain. Augmented thin plate splines interpolation functions are used for the approximation of field variables. The iterative numerical formulation is achieved by viewing the material as divided into small elements and on each of them the integral representation formulae for the velocity and temperature are applied and discretised using linear boundary elements. The final system of non‐linear algebraic equations is solved by a modified Newton's method. The numerical examples include non‐Newtonian problems with viscous dissipation, temperature‐dependent viscosity and natural convection due to bouyancy forces.

To investigate the effect of viscous dissipation on unsteady, combined convective heat transfer to water near its density maximum in a rectangular cavity.

The upwind finite difference scheme along with successive over relaxation iteration technique is used to solve the governing equations for mixed convection flow of water with density maximum inversion in a rectangular cavity.

The effect of viscous dissipation was to increase the fluid temperature and resulted in the formation of vortex motion near the lower part of the cavity in an opposite direction to the central vortex. An increase in the Eckert number and Reynolds number of the flow resulted in augmented surface heat transfer rates from the top heated surface.

The analysis is valid for unsteady, two dimensional laminar flow. Isothermal conditions are assumed for the top and bottom walls. An extension to unsteady three dimensional flow case is left for future work.

The method is very useful to analyze nuclear reactor thermal/hydraulic loss of coolant transients, energy conservation, ventilation of rooms, solar energy collection, cooling of electronic equipment, dispersion of waste heat in estuaries and crystal growth in liquids.

The results of this study may be of interest to engineers interested in heat transfer augmentation of mixed convection in window cavities.

Efficient thermal management of central processing unit (CPU) cooling systems is vital in the context of advancing information technology and the demand for enhanced data processing speeds. This study aims to explore the thermal performance of a CPU cooling setup using a cylindrical porous metal foam heat sink.

Nanofluid flow through the metal foam is simulated using the Darcy–Brinkman–Forschheimer equation, accounting for magnetic field effects. The temperature distribution is modeled through the local thermal equilibrium equation, considering viscous dissipation. The problem’s governing partial differential equations are solved using the similarity method. The CPU’s hot surface serves as a solid wall, with nanofluid entering the heat sink as an impinging jet. Verification of the numerical results involves comparison with existing research, demonstrating strong agreement across numerical, analytical and experimental findings. Ansys Fluent® software is used to assess temperature, velocity and streamlines, yielding satisfactory results from an engineering standpoint.

Investigating critical parameters such as Darcy number (10⁻⁴ ≤ Da_D ≤ 10⁻²), aspect ratio (0.5 ≤ H/D ≤ 1.5), Reynolds number (5 ≤ Re_D,bf ≤ 3500), Eckert number (0 ≤ EC_bf ≤ 0.1) , porosity (0.85 ≤ ε ≤ 0.95), Hartmann number (0 ≤ Ha_D,bf ≤ 300) and the volume fraction of nanofluid (0 ≤ φ ≤ 0.1) reveals their impact on fluid flow and heat sink performance. Notably, Nusselt number will reduce 45%, rise 19.2%, decrease 14.1%, and decrease 0.15% for Reynolds numbers of 600, with rising porosity from 0.85 to 0.95, Darcy numbers from 10⁻⁴ to 10⁻², Eckert numbers from 0 to 0.1, and Hartman numbers from 0 to 300.

Despite notable progress in studying thermal management in CPU cooling systems using porous media and nanofluids, there are still significant gaps in the existing literature. First, few studies have considered the Darcy–Brinkman–Forchheimer equation, which accounts for non-Darcy effects and the flow and geometric interactions between coolant and porous medium. The influence of viscous dissipation on heat transfer in this specific geometry has also been largely overlooked. Additionally, while nanofluids and impinging jets have demonstrated potential in enhancing thermal performance, their utilization within porous media remains underexplored. Furthermore, the unique thermal and structural characteristics of porous media, along with the incorporation of a magnetic field, have not been fully investigated in this particular configuration. Consequently, this study aims to address these literature gaps and introduce novel advancements in analytical modeling, non-Darcy flow, viscous dissipation, nanofluid utilization, impinging jets, porous media characteristics and the impact of a magnetic field. These contributions hold promising prospects for improving CPU cooling system thermal management and have broader implications across various applications in the field.

This study aims to investigate the micropolar fluid flow through a moving flat plate containing CoFe₂O₄-TiO₂ hybrid nanoparticles with the substantial influence of thermophoresis particle deposition and viscous dissipation.

The partial differential equations are converted to the similarity equations of a particular form through the similarity variables. Numerical outcomes are computed by applying the built-in program bvp4c in MATLAB. The process of flow, heat and mass transfers phenomena are examined for several physical aspects such as the hybrid nanoparticles, micropolar parameter, the thermophoresis particle deposition and the viscous dissipation.

The friction factor, heat and mass transfer rates are higher with an increment of 1.4%, 2.2% and 1.4%, respectively, in the presence of the hybrid nanoparticles (with 2% volume fraction). However, they are declined because of the rise of the micropolar parameter. The imposition of viscous dissipation reduces the heat transfer rate, significantly. Meanwhile, thermophoresis particle deposition boosts the mass transfer. Multiple solutions are developed for a certain range of physical parameters. Lastly, the first solution is shown to be stable and reliable physically.

As far as the authors have concerned, no work on thermophoresis particle deposition of hybrid nanoparticles on micropolar flow through a moving flat plate with viscous dissipation effect has been reported in the literature. Most importantly, this current study reported the stability analysis of the non-unique solutions and, therefore, fills the gap of the study and contributes to new outcomes in this particular problem.

In this paper the effects of viscous dissipation and ohmic heating on the fluid flow and resulting heat and mass transfer caused by vertically moving rotating disk are explored with magnetic field acting perpendicular to disk rotation. The flow regime is also under the influence of Dufour and Soret effects.

An approach of similarity transformation is used to transform the governing set of equations into non-linear ordinary differential equations. Numerical simulations are carried out in Maple software to study the influence of incorporated non-dimensional parameters viz. disk movement parameter (−0.3 < S < 0.2), magnetic parameter (0.1 < M < 0.4), Eckert number (0.1 < Ec < 1), Schmidt number (0.1 < Sc < 1), Soret parameter (0.1 < Sr < 1) and Dufour number (0.1 < Du < 1) on velocity, temperature and concentration profiles.

The upward/downward motion of the disk along with rotation set up a three-dimensional flow over the disk surface and exerts the same effects as injection/suction through the wall. It is also observed that incorporated parameters along with disk movement greatly affect the flow regime and associated heat and mass transfer.

The present study examines the heat and mass transfer characteristics of incompressible Newtonian fluid over an impermeable rotating disk moving vertically. The effect of viscous dissipation and ohmic heating is considered. To the best of the authors’ knowledge, such consideration is yet to be published in the literature.

The purpose of this paper is to investigate two-dimensional viscous incompressible magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting fluid over a continuous moving flat surface considering the viscous dissipation and Joule heating.

Suitable similarity variables are introduced to reduce the governing nonlinear boundary layer partial differential equations to ordinary differential equations. A numerical solution of the resulting two-point boundary value problem is carried out by using the finite element method with the help of Gauss elimination technique.

A comparison of obtained results is made with the previous work under the limiting cases. Behavior of flow and thermal fields against various governing parameters like mass transfer parameter, moving flat surface parameter, magnetic parameter, Prandtl number and Eckert number are analyzed and demonstrated graphically. Moreover, shear stress and heat flux at the moving surface for various values of the physical parameters are presented numerically in tabular form and discussed in detail.

The work is relatively original, as very little work has been reported on magnetohydrodynamic flow and heat transfer over a continuous moving flat surface. Viscous dissipation and Joule heating are neglected in most of the previous studies. The numerical method applied to solve governing equations is finite element method which is new and efficient.

The purpose of this work is to study the flow of mixed convection and mass transfer of a steady laminar boundary layer about an isothermal vertical flat plate embedded in a non‐Darcian porous medium in the presence of chemical reaction and viscous dissipation effects.

The governing partial differential equations are converted into ordinary differential equations by similarity transformation, which are solved numerically by employing the fourth‐order Runge‐Kutta integration scheme with Newton‐Raphson shooting technique.

It was found that the local Nusselt number was predicted to decrease as either of the chemical reaction parameter or the Eckert number increased. On the other hand, the local Sherwood number was predicted to increase as a result of increasing either of the chemical reaction parameter or the Eckert number. Also, in the absence of viscous dissipation, both the local Nusselt and Sherwood numbers increased as the mixed convection increased.

The paper illustrates chemical reaction and viscous dissipation effects on Darcy‐Forchheimer mixed convection in a fluid saturated porous media.

The purpose of this study is to provide that porous media and viscous dissipation are crucial considerations when working with hybrid nanofluids in various applications.Recent years have witnessed significant progress in optimizing these fluids for enhanced heat transfer within porous (Darcy–Forchheimer) structures, offering promising solutions for various industries seeking improved thermalmanagement and energy efficiency.

The first step is to transform the original partial differential equations into a system of first-order ordinary differential equations (ODEs). The fourth-order Runge–Kutta method is chosen for its accuracy in solving ODEs. The present study investigates the free convective boundary layer flow of hybrid nanofluids over a moving thin inclined needle with the slip flow brought about by inclined Lorentz force and Darcy–Forchheimer porous matrix, viscous dissipation.

It is found that slip conditions (velocity and Thermal) exist for a range of the natural convection boundary layer flow. In the hybrid nanofluid flow, which consists of Al₂O₃ and Fe₃O₄ are nanoparticles, H₂O − C₂H₆O₂ (50:50) are considered as the base fluid. The consequence of the governing parameter on the momentum and temperature profile distribution is graphically depicted. The range of the variables is 1 ≤ M ≤ 4, 1 ≤ d ≤ 2.5, 1 ≤ δ ≤ 4, 1 ≤ Fr ≤ 7, 1 ≤ Kr ≤ 7 and 0.5≤λ ≤ 3.5. The Nusselt number and skin friction factors are used to calculate the numerical values of various parameters, which are displayed in Table 4. These analyses elucidate that upsurges in the value of the Fr noticeably diminish the momentum and temperature. It is investigated to see if the contemporary results are in outstanding promise with the outcomes reported in earlier works.

The results can be very helpful to improve the energy efficiency of thermal systems.

The hybrid nanofluids in heat transfer have the potential to improve the energy efficiency and performance of a wide range of systems.

This study proposes that in the combined effects of hybrid nanofluid properties, the inclined Lorentz force, the Darcy–Forchheimer model for porous media and viscous dissipation on the boundary layer flow of a conducting fluid over a moving thin inclined needle. Assessing the potential practical applications of the hybrid nanofluids in inclined needles, this could involve areas such as biomedical engineering, drug delivery systems or microfluidic devices. In future should explore the benefits and limitations of using hybrid nanofluids in these applications.

The purpose of this paper is to consider unsteady free convection flow of a dissipative fluid past an exponentially accelerated infinite vertical porous plate in the presence of Newtonian heating and mass diffusion.

The problem is governed by coupled non-linear partial differential equations with appropriate boundary conditions. A Galerkin finite element numerical solution is developed to solve the resulting well-posed two-point boundary value problem. It is a powerful, stable technique which provides excellent convergence and versatility in accommodating coupled systems of ordinary and partial differential equations.

It is found that the skin friction coefficient increases with increases in either of the Eckert number, thermal Grashof number, mass Grashof number or time whereas it decreases with increases in either of the suction parameter, Schmidt number or the acceleration parameter for both air and water. The skin friction coefficient is also found to decrease with increases in the values of the Prandtl number. In addition, it is found that the rate of heat transfer increases with an increase in the suction parameter and decreases with an increase in the Eckert number for both air and water. Lastly, it is found that the rate of heat transfer increases with increasing values of the Prandtl number and decreases with increasing time for all values of the Prandtl number.

The present study has considered only Newtonian fluids. Future studies will address non-Newtonian liquids.

A very useful source of information for researchers on the subject of free convective flow over the surface when the rate of heat transfer from the surface is proportional to the local surface temperature.

This paper is relatively original and illustrates the effects of viscous dissipation on free convective flow past an exponentially accelerated infinite vertical porous plate with Newtonian heating and mass diffusion.