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This paper aims to develop a numerical study of the steady natural convection in an inclined square porous cavity filled by a nanofluid with sinusoidal temperature distribution on the side walls and adiabatic conditions on the upper and lower walls.

Governing equations transformed in terms of the dimensionless variables using the Darcy–Boussinesq approximation have been solved numerically using a central finite-difference scheme. The Gaus-Siedel iteration technique was used for the system of discretized equations. The two-phase nanofluid model including the Brownian diffusion and thermophoresis effects has been considered for simulation of nanofluid transport inside the cavity.

The numerical results of streamlines, isotherms and isoconcentrations are investigated and the effect of different important parameters, such as inclination angle of the cavity, amplitude ratio of the sinusoidal temperature or phase deviation, is discussed. The results obtained for no inclination of the cavity are compared and successfully validated with previous reported results of the literature. The important findings of the study are focused on the changes made by the inclination angle and the periodic thermal boundary conditions, on the heat and fluid flow.

The originality of the present study is given by the mathematical model presented for an inclined cavity, the numerical solution with new results for inclined cavity and the applications for design of solar energy devices such as solar collectors in which the boundary conditions vary with time because of changes in weather conditions.

This paper aims to focus on the analysis of the entropy generation in an inclined square cavity filled with a porous media saturated by a nanofluid with sinusoidal temperature distribution on the side walls, adiabatic conditions on the upper wall and a heat source at the lower wall.

The two-phase nanofluid model including the Brownian diffusion and thermophoresis effects has been used for simulation of nanofluid transport inside the porous cavity. The governing equations and the entropy generation owing to fluid friction, heat and mass transfer are transformed in terms of the dimensionless variables, and the results are obtained by using the finite difference method of the second-order accuracy.

The numerical results of the model are investigated, and the effect of different important parameters, such as inclination angle of the cavity, amplitude ratio of the sinusoidal temperature or phase deviation, is discussed. The results for no inclination of the cavity is compared and successfully validated with previous reported results of the literature. The important findings of the study are focused mainly on the existence of the irreversibility phenomena which are affected by the conditions of the model and the values of the studied parameters.

The originality of this work is given by the presented mathematical model, the numerical solution with new results for entropy generation in an inclined porous cavity filled by a nanofluid and the applications for design of electronic or energy devices.

The purpose of this paper is to theoretically study the problem of the unsteady boundary layer flow past a permeable curved stretching/shrinking surface in the presence of a uniform magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically.

The transformed system of ordinary differential equations was solved using a fourth-order Runge-Kutta integration scheme. Results for the reduced skin friction coefficient and velocity profiles are presented through graphs and tables for several sets of values of the governing parameters. The effects of these parameters on the flow characteristics are thoroughly examined.

Results show that for the both cases of stretching and shrinking surfaces, multiple solutions exist for a certain range of the curvature, mass suction, unsteadiness, stretching/shrinking parameters and magnetic field parameter.

The paper describes how multiple (dual) solutions for the flow reversals are obtained. It is shown that the solutions exist up to a critical value of the shrinking parameter, beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations is not possible.

The purpose of this paper is to study the effects of MHD, suction, second-order slip and melting on the stagnation-point and heat transfer of a nanofluid past a stretching/shrinking sheet.

Using appropriate variables, the governing partial differential equations were transformed into ordinary (similarity) differential equations, which are then solved numerically using the function bvp4c from Matlab.

It is found that dual (upper and lower branch) solutions exist for some values of the governing parameters. From the stability analysis, it is found that the upper branch solution is stable, while the lower branch solution is unstable. The sample velocity, temperature and concentration profiles along both solution branches are graphically presented.

The results of the paper are new and original with many practical applications of nanofluids in the modern industry.

This study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition.

Similarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software.

Dual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stability analysis is performed, and it is revealed that the first (upper branch) solution is stable and physically realizable, whereas the second (lower branch) solution is unstable.

The results obtained can be used to explain the characteristics and applications of the generalized slip in boundary layer flow. Such condition is applied for particulate fluids such as foams, emulsions, polymer solutions and suspensions. Furthermore, the phenomenon of stretching/shrinking sheet can be found on the manufacturing of polymer sheets, rising and shrinking balloon or moving and shrinking polymer film.

The present numerical results are original and new for the study of unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity.

The purpose of this study is to analyze numerically the steady axisymmetric rotational stagnation point flow impinging on a radially permeable stretching/shrinking sheet in a nanofluid.

Similarity transformation is used to convert the system of partial differential equations into a system of ordinary (similarity) differential equations. This system is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in MATLAB software.

Dual solutions exist when the surface is stretched, as well as when the surface is shrunk. For these solutions, a stability analysis is carried out revealing that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable and therefore not physically realizable.

The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Weidman (2016) to the case of nanofluids.

This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions.

The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions.

The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases.

The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

The purpose of this paper is to study the effects of vertical throughflow on the boundary layer flow and heat transfer of a nanofluid driven by a permeable stretching/shrinking surface.

Similarity transformation is used to convert the system of boundary layer equations into a system of ordinary differential equations. The system of governing similarity equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. The generated numerical results are presented graphically and discussed based on some governing parameters.

It is found that dual solutions exist in both cases of stretching and shrinking sheet situations. Stability analysis is performed to determine which solution is stable and valid physically.

Dual solutions are found for positive and negative values of the moving parameter. A stability analysis has also been performed to show that the first (upper branch) solutions are stable and physically realizable, while the second (lower branch) solutions are not stable and, therefore, not physically possible.

The present study aims to address the flow and heat transfer of MgO-MWCNTs/EG hybrid nanofluid in a complex shape enclosure filled with a porous medium. The enclosure is subject to a uniform inclined magnetic field and radiation effects. The effect of the presence of a variable magnetic field on the natural convection heat transfer of hybrid nanofluids in a complex shape cavity is studied for the first time. The geometry of the cavity is an annular space with an isothermal wavy outer cold wall. Two types of the porous medium, glass ball and aluminum metal foam, are adopted for the porous space. The governing equations for mass, momentum and heat transfer of the hybrid nanofluid are introduced and transformed into non-dimensional form. The actual available thermal conductivity and dynamic viscosity data for the hybrid nanofluid are directly used for thermophysical properties of the hybrid nanofluid.

The governing equations for mass, momentum and heat transfer of hybrid nanofluid are introduced and transformed into non-dimensional form. The thermal conductivity and dynamic viscosity of the nanofluid are directly used from the experimental results available in the literature. The finite element method is used to solve the governing equations. Grid check procedure and validations were performed.

The effect of Hartmann number, Rayleigh number, Darcy number, the shape of the cavity and the type of porous medium on the thermal performance of the cavity are studied. The outcomes show that using the composite nanoparticles boosts the convective heat transfer. However, the rise of the volume fraction of nanoparticles would reduce the overall enhancement. Considering a convective dominant regime of natural convection flow with Rayleigh number of 107, the maximum enhancement ratio (Nusselt number ratio compared to the pure fluid) for the case of glass ball is about 1.17 and for the case of aluminum metal foam is about 1.15 when the volume fraction of hybrid nanoparticles is minimum as 0.2 per cent.

The effect of the presence of a variable magnetic field on the natural convection heat transfer of a new type of hybrid nanofluids, MgO-MWCNTs/EG, in a complex shape cavity is studied for the first time. The results of this paper are new and original with many practical applications of hybrid nanofluids in the modern industry.

The purpose of this study is to investigate the influence of the second order slip velocity on the boundary layer stagnation point flow of a nanofluid past a non-aligned stretching/shrinking sheet.

Proper similarity variables are used to transform the system of partial differential equations into a system of ordinary (similarity) differential equations. This system is then solved numerically using the bvp4c solver in MATLAB software. As in the papers by Kuznetsov and Nield (2010, 2013) and Fang et al. (2009), the authors considered the stretching/shrinking parameter λ, the first-order (a₁, a₂) and second-order (b₁) slip parameters and the Lewis number Le, Nb the Brownian parameter and Nt the thermophoresis parameter fixed at Le = 10, Nb = Nt = 0.5 when the Prandtl number Pr is fixed at Pr = 1.

Dual solutions are found as the sheet is shrunk in the horizontal direction. Stability analysis shows that the first solution is physically realizable, whereas the second solution is not practicable.

The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Wang (2008) and Lok et al. (2011) to the case of nanofluids.