A numerical study of micropolar flow inside a lid-driven triangular enclosure

A study is carried out to analyze the mixed convection flow and heat transfer inside a lid-driven triangular conduit under the effects of micro-gyration boundary conditions. The micropolar constitutive equation characterizes the fluid inside the cavity. The lower boundary is at a uniform temperature and sliding in its plane with constant velocity u₀, while the inclined walls are cold. Dual cases are considered here, namely the intense concentration (d) and the weak concentration of microelements (\(m = 0.5\)). The governing nonlinear equations are simulated employing the Galerkin finite element method, where the pressure term is handled via the Penalty approach. Using the numerical data, graphical results are produced to illustrate the effects of physical parameters. Specifically, this refers to the effects of the Grashof number (Gr), Prandtl number (Pr), Reynolds number (Re) and vortex viscosity parameter (K) on the streamlines, mid-section velocity profiles, temperature contours, and local and average Nusselt numbers on the cold and heated boundaries of the conduit. Particular emphasis is given on the identification of the set of parameters for which simultaneous symmetry in streamlines and isotherms prevails. The grid independence test is also performed by comparing the average Nusselt numbers (on the hot and cold boundaries of the conduit) for various mesh sizes, and the optimal solution is found. Moreover, the results are also benchmarked with the previously published data.     

Simultaneous impacts of MHD and variable wall temperature on transient mixed Casson nanofluid flow in the stagnation point of rotating sphere

A numerical analysis is provided to scrutinize time-dependent magnetohydrodynamics(MHD) free and forced convection of an electrically conducting non-Newtonian Casson nanofluid flow in the forward stagnation point region of an impulsively rotating sphere with variable wall temperature. A single-phase flow of nanofluid model is reflected with a number of experimental formulae for both effective viscosity and thermal conductivity of nanofluid. Exceedingly nonlinear governing partial differential equations(PDEs)subject to their compatible boundary conditions are mutated into a system of nonlinear ordinary differential equations(ODEs). The derived nonlinear system is solved numerically with implementation of an implicit finite difference procedure merging with a technique of quasi-linearization. The controlled parameter impacts are clarified by a parametric study of the entire flow regime. It is depicted that from all the exhibited nanoparticles,Cu possesses the best convection. The surface heat transfer and surface shear stresses in the x-and z-directions are boosted with maximizing the values of nanoparticle solid volume fraction ? and rotation λ. Besides, as both the surface temperature exponent n and the Casson parameter γ upgrade, an enhancement of the Nusselt number is given.     

Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

Two-component modeling for non-Newtonian nanofluid slip flow and heat transfer over sheet: Lie group approach

The present paper deals with the multiple solutions and their stability analysis of non-Newtonian micropolar nanofluid slip flow past a shrinking sheet in the presence of a passively controlled nanoparticle boundary condition. The Lie group transformation is used to find the similarity transformations which transform the governing transport equations to a system of coupled ordinary differential equations with boundary conditions. These coupled set of ordinary differential equation is then solved using the RungeKutta-Fehlberg fourth-fifth order(RKF45) method and the ode15 s solver in MATLAB.For stability analysis, the eigenvalue problem is solved to check the physically realizable solution. The upper branch is found to be stable, whereas the lower branch is unstable. The critical values(turning points) for suction(0 sc s) and the shrinking parameter(χc χ 0) are also shown graphically for both no-slip and multiple-slip conditions. Multiple regression analysis for the stable solution is carried out to investigate the impact of various pertinent parameters on heat transfer rates. The Nusselt number is found to be a decreasing function of the thermophoresis and Brownian motion parameters.     

Effect of surface permeability on the structure of a separated turbulent flow and heat transfer behind a backward-facing step

The structure and heat transfer in a turbulent separated flow in a suddenly expanding channel with injection (suction) through a porous wall are numerically simulated with the use of two-dimensional averaged Navier–Stokes equations, energy equations, and v ²–f turbulence model. It is shown that enhancement of the intensity of the transverse mass flux on the wall reduces the separation region length in the case of suction and increases the separation region length in the case of injection up to complete boundary layer displacement. The maximum heat transfer coefficient as a function of permeability is accurately described by the asymptotic theory of a turbulent boundary layer.     

Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation

We consider unsteady laminar natural convection flow of water subject to density inversion in a rectangular cavity formed by isothermal vertical walls with internal heat generation. The top and bottom horizontal walls are considered to be adiabatic, whereas the temperature of the left vertical wall is assumed to be greater than that of the right vertical wall. The equations are non-dimensionalized and are solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of both heat generation and variations in the aspect ratio on the streamlines, isotherms and the rate of heat transfer from the walls of the enclosure are presented. Investigations are performed for water taking Prandtl number to be Pr=11.58 and the Rayleigh number to be Ra=10⁵.     

Combined radiation and natural convection in a rectangular cavity with a transparent wall and containing a non-participating fluid

This paper describes a numerical method for the study of combined natural convection and radiation in a rectangular, two-dimensional cavity containing a non-participating (i.e. transparent) fluid. One wall of the cavity is isothermal, being heated either by solar radiation or independently. The opposite wall is partially transparent, permitting radiation exchanges between the cavity and its surroundings and/or the Sun; that wall also exchanges heat by convection from its external surface to the surroundings. The other two walls are adiabatic: convection and radiation there are balanced, so that there is no heat transfer through those walls. The equations of motion and energy are solved by finite difference methods. Coupled to these equations are the radiative flux boundary conditions which are used to determine the temperature distribution along the non-isothermal walls. A two-band radiation model has been employed. Results are presented for a square cavity with a vertical hot wall at 150 °C, the ambient at 20 °C and 10⁴ ? Ra ? 3 × 10⁵, in the absence of direct insolation. The effects on the flow and heat transfer in the cavity of radiation and external convection have been examined. More extensive results will be presented in subsequent papers.     

Influence of heating load on heat transfer characteristics in micro-pin-fin arrays

Experimental investigations were carried out to explore the convective heat transfer in micro pin-fins with different aspect ratios, and the influence of heating load on Nusselt numbers in micro pin-fins with liquid water as working fluid were investigated. The mechanism of convective heat transfer in micro pin-fins at different heating load were studied by 3-D numerical investigations, and the relationships of thermal physical properties change, the end wall effect and axial thermal conduction with Nu numbers in micro pin-fins were analysed. It was found that the thickness of boundary layer was decreased as much as 33.3 % attributed to the destructive effect of thermal physical properties change, and convective heat transfer in the micro pin-fin channel was more than 20 % enhanced by the flow disturbance caused by the increase of temperature difference. The discrepancy of Nu in micro pin-fin channel with different aspect ratios reached 34.59 %, and this discrepancy was reduced by the increase of heating load. The maximum value of impact factors of dynamic viscosity and thermal conductivity on the Nu in micro-pin-fins reached 25.02 and 7.68 %, respectively.     

Integral boundary layer relations in the theory of wave flows for capillary liquid films

A generalized method of deriving the model equations is considered for wave flow regimes in falling liquid films. The viscous liquid equations are used on the basis of integral boundary layer relations with weight functions. A family of systems of evolution differential equations is proposed. The integer parameter n of these systems specifies the number of a weight function. The case n = 0 corresponds to the classical IBL (Integral Boundary Layer) model. The case n ≥ 1 corresponds to its modifications called the WIBL (Weighted Integral Boundary Layer) models. The numerical results obtained in the linear and nonlinear approximations for n = 0, 1, 2 are discussed. The numerical solutions to the original hydrodynamic differential equations are compared with experimental data. This comparison leads us to the following conclusions: as a rule, the most accurate solutions are obtained for n = 0 in the case of film flows on vertical and inclined solid surfaces and the accuracy of solutions decreases with increasing n. Hence, the classical IBL model has an advantage over the WIBL models.     

Heat transfer characteristics of internally heated liquid pools at high Rayleigh numbers

Detailed numerical analysis is presented for buoyancy driven flow of a Newtonian fluid contained in a square enclosure for high Rayleigh (Ra) numbers. Natural convection is due to internal heating sources, which are assumed to be uniformly distributed within the enclosure. All walls of the cavity are maintained at constant temperature. Flow and heat transfer characteristics are investigated for a Ra number range of 10⁷ to 10¹² while Prandtl (Pr) number is taken to be 7.0. Governing equations (in primitive variables) are discretised using control volume technique based on staggered grid formulation. These equations are solved using SIMPLER algorithm of Patankar. Flow and heat transfer characteristics, streamlines, isotherms and average wall Nusselt (Nu) number, are presented for whole range of Ra number considered. Finally, present results for average wall Nu numbers are compared with experimental observations obtained from open literature. It is concluded that both results are in very good agreement, which confirmed the accuracy of the scaling used for present investigation. Received on 15 November 1999     

The effects of forced boundary conditions on flow within a cubic cavity using digital particle image thermometry and velocimetry (DPITV)

Buoyancy-driven convection within a cavity, whose sidewalls are heated and cooled, is a problem of great interest, because it has applications in heat transfer and mixing. Most studies to date have studied one of two cases: the steady-state case or the development of the transient flow as it approaches steady state. Our main concern was to study the response of the cavity to time-varying thermal boundary conditions. We therefore decided to observe the flow phenomena within a convection cavity under sinusoidal thermal forcing of the sidewalls. To map the flow properly, it is necessary to have simultaneous kinematic and thermal information. Therefore, the digital particle image thermometry and velocimetry (DPITV) is used to acquire data. Implementing this technique requires seeding the flow with encapsulated liquid crystal particles and illuminating a cross section of the flow with a sheet of white light. Extraction of the thermal and kinematic content is in two parts. For the first, the liquid crystals will reflect different colors of the visible spectrum, depending on the temperatures to which they are subjected. Therefore, calibrating their color reflection with temperature allows for the extraction of the thermal content. For the second part, the kinematic information is obtained through the use of a digital cross-correlation particle image velocimetry technique. With the use of DPITV, the flow within a convection cavity is mapped and studied under steady forcing and sinusoidally forced boundary conditions at the Brunt-Väisälä frequency. For the sinusoidally forced case, three cases are studied. In the first, the heating between the two walls is in phase. In the second, the heating between the two walls is 180° out of phase. In the third, the heating between the two walls is 90° out of phase. For steady forcing, the thermal plots show that the flow develops a linearly stratified profile within the center of the cell. At the sidewalls, however, owing to forcing, hot/cold thermal boundary layers develop at the left/right walls. These hot/cold thermal boundary layers then turn around the upper-left/lower-right corners and develop into intrusion layers that extend across the top and bottom walls. The vorticity and streamlines show that the bulk of the fluid motion is concentrated around the walls, whereas the fluid within the center of the cell remains stationary. For the sinusoidally forced cases, the thermal plots show the existence of many thermal “islands,” or pockets of fluid where the temperature is different with respect to its surroundings. The vorticity plots show that the center of the cell is mostly devoid of vorticity and that the vorticity is mainly confined to the sidewalls, with some vorticity at the top and bottom walls. For the 0° forcing, the streamlines show the development of two counterrotating rollers. For the 180° forcing, the streamlines show the development of only one roller. Finally, for the 90° forcing, the streamlines show the development of both a two-roller and a one-roller system, depending on the position within the forcing cycle.     

A numerical study of mixed convection in a horizontal channel with a discrete heat source in an open cavity

This article aims to numerically investigate mixed convection heat transfer in a two-dimensional horizontal channel with an open cavity. A discrete heat source is considered to be located on one of the walls of the cavity. Three different heating modes are considered which relate to the location of the heat source on three different walls (left, right and bottom) of the cavity. The analysis is carried out for a range of Richardson numbers and cavity aspect ratios. The results show that there are noticeable differences among the three heating modes. When the heat source is located on the right wall, the cavity with an aspect ratio of two has the highest heat transfer rate compared to other cavity heating modes. Moreover, when the heat source is located on the bottom wall, the flow field in the cavity with an aspect ratio of two experiences a fluctuating behaviour for Richardson number of 10. The results also show that at a fixed value of Richardson number, all three different heating modes show noticeable improvements in the heat transfer mechanism as the cavity aspect ratio increases.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

Numerical heat transfer in a cavity with a solar control coating deposited to a vertical semitransparent wall

A transient two‐dimensional computational model of combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one, containing air as a laminar and non‐participating fluid, is presented. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall, and an opposite semitransparent wall, which consists of a 6‐mm glass sheet with a solar control coating of SnS–Cu_xS facing the cavity. The semitransparent wall also exchanges heat by convection and radiation from its external surface to the surroundings and allows solar radiation pass through into the interior of the cavity. The momentum and energy equations in the transient state were solved by finite differences using the alternating direction implicit (ADI) technique. The transient conduction equation and the radiative energy flux boundary conditions are coupled to these equations. The results in this paper are limited to the following conditions: 10⁴≤Gr≤10⁶, an isothermal vertical cold wall of 21°C, outside air temperatures in the range 30°C≤T₀≤40°C and incident solar radiation of AM2 (750 W m⁻²) normal to the semitransparent wall. The model allows calculation of the redistribution of the absorbed component of solar radiation to the inside and outside of the cavity. The influences of the time step and mesh size were considered. Using arguments of energy balance in the cavity, it was found that the percentage difference was less than 4 per cent, showing a possible total numerical error less than this number. For Gr=10⁶ a wave appeared in the upper side of the cavity, suggesting the influence of the boundary walls over the air flow inside the cavity. A Nusselt number correlation as a function of the Rayleigh number is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of MHD natural convection flow in a wavy cavity filled by a hybrid Cu-Al_2O_3-water nanofluid with discrete heating

We consider the combined effect of the magnetic field and heat transfer inside a square cavity containing a hybrid nanofluid(Cu-Al_2O_3-water). The upper and bottom walls of the cavity have a wavy shape. The temperature of the vertical walls is lower,the third part in the middle of the bottom wall is kept at a constant higher temperature,and the remaining parts of the bottom wall and the upper wall are thermally insulated.The magnetic field is applied under the angle γ, an opposite clockwise direction. For the numerical simulation, the finite element technique is employed. The ranges of the characteristics are as follows: the Rayleigh number(10~3≤Ra≤10~5), the Hartmann number(0≤Ha≤100), the nanoparticle hybrid concentration(?_(Al_2O_3),?_(Cu) = 0, 0.025, 0.05),the magnetic field orientation(0≤γ≤2π), and the Prandtl number P_r, the amplitude of wavy cavity A, and the number of waviness n are fixed at P_r = 7, A = 0.1, and n = 3, respectively. The comparison with a reported finding in the open literature is done,and the data are observed to be in very good agreement. The effects of the governing parameters on the energy transport and fluid flow parameters are studied. The results prove that the increment of the magnetic influence determines the decrease of the energy transference because the conduction motion dominates the fluid movement. When the Rayleigh number is raised, the Nusselt number is increased, too. For moderate Rayleigh numbers, the maximum ratio of the heat transfer takes place for the hybrid nanofluid and then the Cu-nanofluid, followed by the Al_2O_3-nanofluid. The nature of motion and energy transport parameters has been scrutinized.     

Mixed convection in an inclined lid-driven cavity with non-uniform heating on both sidewalls

The present work reports a numerical simulation of mixed convection in an inclined square cavity. The vertical sidewalls are assumed to have a nonuniform temperature distribution. The finite volume method is used to solve dimensionless governing equations. Simulations are performed for different Richardson numbers, amplitude ratios, phase deviations, and cavity inclination angles. The results are presented graphically. The mean heat transfer significantly increases in the buoyancy-dominated mode on increasing cavity inclination angle if both walls have identical heating and cooling zones.     

Bistable Boundary Reactions in Two Dimensions

In a bounded domain \({\Omega \subset \mathbb R^2}\) with smooth boundary we consider the problem

$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$

where ν is the unit normal exterior vector, ε > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(π u) or f(u) = (1 ? u ²)u. We construct solutions that develop multiple transitions from ?1 to 1 and vice-versa along a connected component of the boundary ?Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(π u).

     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

Analysis of Sakiadis flow of nanofluids with viscous dissipation and Newtonian heating

The combined effects of viscous dissipation and Newtonian heating on boundary layer flow over a moving flat plate are investigated for two types of water-based Newtonian nanofluids containing metallic or nonmetallic nanoparticles such as copper (Cu) and titania (TiO₂). The governing partial differential equations are transformed into ordinary differential equations through a similarity transformation and are solved numerically by a Runge-Kutta-Fehlberg method with a shooting technique. The conclusions are that the heat transfer rate at the moving plate surface increases with the increases in the nanoparticle volume fraction and the Newtonian heating, while it decreases with the increase in the Brinkmann number. Moreover, the heat transfer rate at the moving plate surface with Cu-water as the working nanofluid is higher than that with TiO₂-water.     

A method of solving the diffusion problem for a vortex layer in a viscoplastic half-plane

A method is proposed to reduce the classical formulation of the problem to a system of two functional equations whose solution can be found numerically. A number of assertions that characterize the behavior of a rigid zone are proved. In particular, the lower estimate h ⁰(t) = 2b√t for the boundary motion is obtained; an explicit expression for b is given as a boundary stress function.