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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 paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.

In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.

The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.

The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

Compression rings are the main sources of frictional losses in internal combustion engines. The present paper aims to present a thermo-mixed hydrodynamic analysis for coated top compression rings. To understand the coating effects, the main tribological parameters are investigated into a ring-cylinder conjunction in a motorbike engine. Furthermore, flow simulations have been carried out on how different worn profiles on the cylinder inner liner affects friction, lubricant film and localized contact deformation of the coated compression rings.

In this paper, the basic geometrical dimensions of the top compression ring-cylinder system are obtained from a real motorbike engine. A 2D axisymmetric CFD/FLOTRAN model is created for coated compression rings. Flow simulations are performed by solving the Navier-Stokes and the energy equations. The load capacity of the asperities is also taken into account by Greenwood and Tripp contact model. Realistic boundary conditions are imposed to simulate the in-plane ring motion. The simulation model is validated with analytical and experimental data from the literature. Under thermal considerations, the contribution of worn cylinder profiles in conjunction with different coated compression rings is presented.

This research shows that because of thermal effects, the boundary friction is higher at reversals and the viscous friction is lower because of reduced oil viscosity. As regards to the isothermal case, the viscous friction is greater because of a higher lubricant viscosity. In the case of chromium-plated ring, boundary friction was 16 per cent lower than a grey cast iron ring taking into account thermal effects. Regarding the localized contact deformation, the coated compression rings showed lower values under different worn cylinder shapes. In particular, hard wear-resistant (Ni-Cr-Mo) coating showed the slighter local deformation. Therefore, the worn cylinder profiles promote boundary/mixed lubrication regime, whereas the lobed profile of cylinder inner liner becomes more wavy.

The solution of the thermo-mixed lubrication model, concerning the piston ring and worn cylinder tribo pair by taking into account the coating of the top compression ring.

Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years. The finite element method has been widely used to solve contact problems with various grades of complexity. Great progress has been made on both theoretical studies and engineering applications. This paper reviews some of the main developments in contact theories and finite element solution techniques for static contact problems. Classical and variational formulations of the problem are first given and then finite element solution techniques are reviewed. Available constraint methods, friction laws and contact searching algorithms are also briefly described. At the end of the paper, a bibliography is included, listing about seven hundred papers which are related to static contact problems and have been published in various journals and conference proceedings from 1976.

The purpose of this paper is to consider the problem of fully developed laminar mixed convection flow of a couple stress fluid in a vertical channel with the third-kind boundary conditions in the presence or absence of heat source/sink effect.

Through proper choice of dimensionless variables, the governing equations are developed. These governing equations are solved analytically by the differential transform method and numerically by the Runge–Kutta shooting method. Analytical solutions for the velocity and temperature profiles for heat generation and absorption of the problem are reported.

The mass flow rate and Nusselt numbers at both the left and right channel walls on mixed convection parameter, Brinkman number, couple stress parameter and heat generation/absorption parameter for equal and unequal Biot numbers are presented. Favorable comparisons of special cases with previously published work are obtained. It is found that velocity, temperature, mass flow rate and Nusselt number decrease with couple stress parameter and increase with mixed convection parameter and Brinkman number.

The work done in this paper is not done earlier to the authors’ knowledge. This is the first paper in which the sixth-order differential equation is solved using the semi-numerical method, which is a differential method.

The purpose of this paper is to propose some a posteriori residual error estimators (REEs)to evaluate the accuracy of the finite element method for quasi-static electromagnetic problems with mixed boundary conditions. Both classical magnetodynamic A-ϕ and T-Ω formulations in harmonic case are analysed. As an example of application the estimated error maps of an electromagnetic system are studied. At last, a remeshing process is done according to the estimated error maps.

The paper proposes to analyze the efficiency of numerical REEs in the case of magnetodynamic harmonic formulations. The deal is to determine the areas where it is necessary to improve the mesh. Moreover the error estimators are applied for structures with mixed boundary conditions.

The studied application shows the possibilities of the residual error estimators in the case of electromagnetic structures. The comparison of the remeshed show the improvement of the obtained solution when the authors compare with a reference one.

The paper provides some interesting results in the case of magnetodynamic harmonic formulations in terms of potentials. Both classical formulations are studied.

The paper provides some informations to develop the proposed formulations in the software using finite element method.

The paper deals with the possibility to improve the determination of the meshes in the analysis of electromagnetic structure with the finite element method. The proposed method can be a good solution to obtain an optimal mesh for a given numerical error.

The paper proposes some elements of solution for the numerical analysis of electromagnetic structures. More particularly the results can be used to determine the good meshes of the finite element method.

The purpose of this paper is to extend the penalty concept to treat partial slip, free surface, contact and related boundary conditions in viscous flow simulation.

The penalty partial‐slip formulation is analysed and related to the classical Navier slip condition. The same penalty scheme also allows partial penetration through a boundary, hence the implementation of porous wall boundaries. The finite element method is used for investigating and interpreting penalty approaches to boundary conditions.

The generalised penalty approach is verified by means of a novel variant of the circular‐Couette flow problem, having partial slip on one of the cylindrical boundaries, for which an analytic solution is derived. Further verificationis provided by consideration of viscous flow over a sphere with partial slip on the surface, and comparison of numerical and classical solutions. Numerical studies illustrate the versatility of the approach.

The penalty approach is applied to some different boundaries: partial slip and partial penetration with no/full slip/penetration as limiting cases; free surface; space‐ and time‐varying boundary conditions which allow progressive contact over time. Application is made to curved and inclined boundaries. Sensitivity of flow to penalty parameters is an avenue for continued research, as is application of the penalty approach for non‐Newtonian flows.

This is the first work to show the relation between penalty formulation of boundary conditions and physical boundary conditions. It provides a method that overcomes past difficulties in implementing partial slip on boundaries of general shape, and which handles progressive contact. It also provides useful benchmark problems for future studies.

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

It is well known that the mixed convection process is the combined effect of the presence of both the forced and the free convection processes. In several applications such as environmental chambers, IC engines, etc. the forced convection is brought in by multiple suction/injection (S/I) effect. Study of mixed convection in a vertical square fluid saturated porous cavity with multiple S/I effect greatly contributes to such an understanding. So far, not much research work has reported in this direction. Hence, the purpose of this paper is to investigate such a mixed convection process in a fluid saturated vertical porous square cavity.

In this study, the authors numerically solved the couple partial differential equations governing the mixed convection process in a fluid saturated vertical square porous cavity by finite element method. The study is parametric in nature wherein the authors cover a large range of values for different parameters arising the mathematical model governing the problem under consideration.

The influence of multiple S/I effects on mixed convection is analyzed for a wide range of controlling parameters such as S/I flow velocities (a), S/I window size (D/H) and Rayleigh number (Ra). Both the flow and temperature fields are highly sensitive to magnitude of S/I velocity, S/I window slit size and “Ra”. While heat fluxes along the isothermal left vertical wall decrease with increasing S/I velocities they are formed to increase with increasing “(D/H)” and “Ra”. Nusselt numbers increase with increasing “Ra” and increasing size of S/I window slit size. Multi‐cellular circulation pattern and thermal boundary layers are seen to manifest in flow and temperature fields, respectively.

The study is based on 2D model, but the model is generic in nature; also it is fully numerical in nature. Due to lack of apt literature no experimental support is provided. The mathematical model used in the study is based on certain assumptions such as isotropic porous medium, fluid is viscous in nature and follows Newtonian laws and the porous structure is saturated with fluid, etc. Regarding future work, 3D modelling and simulation is in progress and attempts are also being made to collaborate with experimental groups on the problem under investigation.

The results from the work are relevant to the context of heat and fluid flow studies in IC engines, influence of mixed convection process on bacterial growth process in environmental chambers and cooling of electronic devices, etc.

The paper describes a mathematical model, especially the boundary treatment, for describing the influence of multiple S/I effects on mixed convection flow in a vertical square enclosure filled with a Darcian fluid saturated homogeneous porous medium. To understand the physics behind the mixed convection process in the proposed configuration, extensive numerical simulations have been carried out for the first time for different values of the important governing parameters arising from the model.

Using the topological degree of Leray‐Shauder, and Grisvard's results for elliptic equations with mixed boundary conditions, we extend Mock's results for the steady‐state Van Roosbroeck system, with the change from Neuman to Dirichlet boundary conditions occuring at a flat angle. Similar results are obtained for continuity equations that include a general recombination rate.