Mixed convection with viscous dissipation in a vertical channel filled with a porous medium

Summary The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux ÷ isothermal wall conditions is investigated assuming that (i) the Darcy law and the Boussinesq approximation hold, (ii) the effect of viscous dissipation is significant, and (iii) the average flow velocity U _m (as an experimentally accessible quantity) is prescribed. It is shown that under these conditions both upward (U _m > 0) and downward (U _m < 0) laminar flow solutions may exist as long as U _m does not exceed a maximum value U _{m, max}. The velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for U _m = 0. A remarkable feature of the problem is that for U _m < U _{m, max} even two solution branches (dual solutions) exist, which merge when U _m approaches its maximum value U _{m, max}. The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in detail.     

Natural convection heat transfer of a viscous fluid in a vertical porous channel

Approximate analytic solutions to second-order nonlinear systems arising in natural convection flow and heat transfer in vertical porous channels are obtained via the Galerkin–Legendre Spectral Method. Furthermore, existence, uniqueness, and concavity results are established using Green’s functions and degree theory. We find that an increase in either the Darcy number or the quadratic density temperature variation results in an increase in the velocity and the temperature of a Newtonian fluid. Finally, parametric zones for the occurrence of reverse flow are considered, and the resulting influences on the obtained approximate solutions are analyzed.     

Non-Darcy mixed convection flow with thermal dispersion on a vertical cylinder in a saturated porous medium

Summary The non-Darcy mixed convection flow on a vertical cylinder embedded in a saturated porous medium has been studied taking into account the effect of thermal dispersion. Both forced flow and buoyancy force dominated cases with constant wall temperature condition have been considered. The governing partial differential equations have been solved numerically using the Keller box method. The results are presented for the buoyancy parameter which cover the entire regime of mixed convection flow ranging from pure forced convection to pure free convection. The effect of thermal dispersion is found to be more pronounced on the heat transfer than on the skin friction and it enhances the heat transfer but reduces the skin friction.     

The similarity regime for natural convection in a vertical cylindrical well filled with an anisotropic porous medium

An integral method based on Lighthill’s analysis (Q J Mech Appl Math 6 (1953) 398–439) is carried out to study the similarity regime for penetration of convective heat transfer in a vertical cylindrical well filled with an anisotropic porous medium. The porous medium is anisotropic in permeability with its principal axes oriented in a direction that is oblique to the gravity vector. In the limit of the slenderness of the porous matrix, the penetration length of the convective flow and the heat-transfer rate are expressed in terms of the anisotropic properties of the porous medium, the modified Darcy–Rayleigh number and the aspect ratio of the geometrical configuration. A scale analysis is applied to predict the order of magnitudes involved in the similarity regime of the phenomenon. The conditions of existence of the similarity pattern is found to be dependent on the anisotropic parameters. It is demonstrated that both the anisotropic permeability ratio and the orientation angle of the principal axes have a strong influence on the heat-transfer rate and on the vertical penetration length into the well.     

Non-Darcy regime mixed convection on vertical plates in saturated porous media with lateral mass flux

Summary A unified treatment is presented of mixed convection on vertical plates embedded in fluid saturated porous media with prescribed variable plate temperature or surface heat flux for the case of non-Darcy limiting regime. The plates are permeable with lateral mass flux. By suitable similarity transformations, it is shown that the two problems of prescribed temperature and prescribed heat flux lead to identical differential equations with two common boundary conditions and third boundary condition differing in the two cases. The effect of lateral mass flux and the free stream on the Nusselt number and the energy transport by the boundary layer is investigated. The unified approach to the mixed convection problem includes free convection problem as a special case. Exact analytical solutions are obtained for two cases of free convection problem.Notation b inertial coefficient - c specific heat - D _p pore diameter - E rate of upward energy transport - Ê dimensionless rate of energy transport - f dimensionless stream function - f _w mass flux parameter - g acceleration due to gravity - k thermal conductivity - K permeability of the porous medium - m exponent in the variation of heat flux - M mixed convection parameter - Nu _x Nusselt number - Pe _x Peclet number - q _w surface heat flux - Ra _x local Rayleigh number - Ra _x ^* modified local Rayleigh number - T temperature - T _e ambient temperature - T _w plate temperature - T _w temperature difference=T _w-T_e - u velocity in thex-direction - u _e free stream velocity - v velocity in they-direction - v _w lateral velocity at the plate - x coordinate along the plate in the upward direction - y coordinate normal to the plate - equivalent thermal diffusivity =k/c _e - coefficient of thermal expansion - porosity - dimensionless variable - dimensionless temperature - viscosity - fluid density - _e ambient density - exponent in the variation of plate temperature - stream function     

Penetrative convection in anisotropic porous media with variable permeability

The onset of penetrative convection in an anisotropic porous medium, for fluids with quadratic density law, is studied. In particular, the effects of anisotropic permeability and thermal diffusivity have been taken into account.     

Network numerical analysis of magneto-micropolar convection through a vertical circular non-Darcian porous medium conduit

The fully developed, mixed convection heat transfer of a magneto-micropolar fluid in a Darcy–Forchheimer porous medium containing heat sources contained in a vertical circular conduit is investigated in this article. The conservation equations for mass, linear momentum, micro-inertia, angular momentum (micro-rotation) and energy are presented in a cylindrical coordinate system (r, θ, z) with appropriate boundary conditions. A Darcy–Forchheimer drag force model is employed to simulate the effects of bulk linear porous impedance and second order porous resistance. The governing partial differential equations are non-dimensionalized into a set of ordinary differential equations in a single independent variable (η) and solved using the Network Simulation Method. Benchmark solutions are compared with earlier computations using the finite element method, showing excellent agreement. The influence of Darcy number, Forchheimer number, Grashof number, Hartmann number, geometric scale ratio (conduit radius to length ratio), Eringen parameter (ratio of vortex viscosity to Newtonian viscosity) and heat source/sink parameter on the linear velocity, angular velocity (micro-rotation) and temperature functions are studied in detail. Flow i.e. linear (translational) velocity, f, is seen to be inhibited with increasing magnetic field (Hartmann number), Forchheimer number and Eringen parameter, but accelerated with increasing Darcy number. Micro-rotation (g) is decreased with increasing Forchheimer number and Hartmann number, but increased with a rise in Grashof number, Darcy number, geometric scale ratio and Eringen parameter. Both velocity (f) and micro-rotation (g) are increased in the presence of a heat source but decreased with a heat sink. Several special cases of the flow regime are also documented. Applications of the problem include the cooling of porous combustion chambers, geophysical transport in electrically-conducting zones, exhaust nozzles of porous walled flow reactors, hydromagnetic control processes in nuclear engineering and magnetic materials processing (ceramic foams).     

Combined radiation and mixed convection from a vertical wall with suction/injection in a non–Darcy porous medium

Summary. Mixed convection flow of an absorbing fluid up a uniform non–Darcy porous medium supported by a semi-infinite ideally transparent vertical flat plate due to solar radiation is considered. The external flow field is assumed to be uniform, the effect of the radiation parameter in the boundary layer adjacent to the vertical flat plate with fluid suction/injection through it is analyzed in both aiding and opposing flow situations. It is observed that the similarity solution is possible only when the fluid suction/injection velocity profile varies as x^–1/2. The velocity and temperature profiles in the boundary layer and the heat transfer coefficient are presented for selected values of the parameters. It is observed that the Nusselt number increases with the increase in the radiation parameter and also when the value of the surface mass flux parameter moves from the injection to the suction region.     

Natural convection between coaxial vertical cylinders partially filled with a porous material

This paper presents laminar fully developed free convection flow between two coaxial vertical cylinders partially filled with a porous matrix when the cylinders are kept at different temperatures. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous region. The two regions are coupled by equating the velocity and by considering shear stress jump condition at the interface. The effect of Darcy number on the velocity throughout porous domain and adjustable coefficient in the stress jump condition at the interface is shown graphically. It is observed that velocity is influenced by the shear stress jump condition at the interface.     

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow

Summary Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.     

g-Jitter mixed convection adjacent to a vertical stretching sheet

This paper studies the effect of periodical gravity modulation, or g-jitter induced mixed convection, on the flow and heat transfer characteristics associated with a stretching vertical surface in a viscous and incompressible fluid. The velocity and temperature of the sheet are assumed to vary linearly withx, wherex is the distance along the sheet. It is assumed that the gravity vector modulation is given byg*(t)=g_o [1+ɛ cos(πωt)]k, and the resulting non-similar boundary layer equations are solved numerically using an implicit finite-difference scheme. The effects of the amplitude of modulation, frequency of the single-harmonic component of oscillation, mixed convection parameter and Prandtl number on the skin friction and Nusselt number are discussed in detail.     

Effects of chemical reaction on free convection flow through a porous medium bounded by a vertical surface

The effect of a chemical reaction on a free convection flow through a porous medium bounded by a vertical infinite surface has been studied. Velocity, temperature, and concentration profiles have been obtained for different values of parameters like the Grashof number, Prandtl number, and the chemical reaction parameter in the presence of homogeneous chemical reaction of first order. It is observed that the velocity and concentration increase during a generative reaction and decrease in a destructive reaction. The same is true for the behavior of the fluid temperature. The presence of the porous media diminishes the temperature.     

Natural convection in a cavity with a sloping upper surface filled with an anisotropic porous material

Steady natural convection in a viscous incompressible fluid flowing in a vertical permeable non-rectangular cavity is considered. The permeability of the porous medium and the thermal diffusivity of the fluid are assumed to be anisotropic in nature. Using the Boussinesq approximation and the Darcy law to model the porous medium, the governing equations—recast in terms of a stream function—are solved numerically. An algebraic grid generation technique is employed to transform the non-rectangular domain to a square domain. The influence of a number of parameters, arising due to geometrical and physical aspects, is discussed in relation to the flow and heat transfer.     

Free convection in hydromagnetic flows in a vertical wavy channel

A study is made of the free convection in hydromagnetic flows in a vertical wavy channel in the presence of heat source or sink. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved subject to the relevant boundary conditions with the assumption that the solution consists of a mean part and a perturbed part. The zeroth-order, the first order and the total solution of the problem are numerically evaluated for various values of the magnetic parameter, heat source/sink parameter, wall-waviness parameter, and free convection parameter. The velocity and the temperature profiles are graphically represented for these parameters. The qualitative features of the hydromagnetic solution are discussed. A comparison is made between the hydromagnetic and the hydrodynamic solutions. The numerical values of the skin friction and the Nusselt number are tabulated for various parameters involved in the analysis. Special attention is given on the characteristic features of the flow, heat transfer, skin friction and the Nusselt numbers at the walls.     

Natural convection in a vertical channel in a nonuniform magnetic field

It is shown that a nonuniform magnetic field suppresses the natural convection and shapes the specific MHD flow.Translated from Inzhenerno-Fizicheskii Zhurnal. Vol. 21, No. 3, pp. 510–517, September, 1971.The authors express their sincere thanks to Academician A. V. Lykov for scientific guidance and for constant attention to the work.     

Degenerate onset of convection in vertical porous cylinders

In this work we analyze the performance of a novel conceptual design for water-wave energy converters. The model system in question consists of a water tank with two hinged side walls that oscillate when the water surface within the tank is subjected to a temporally periodic, spatially distributed pressure variation. Then, through transmissions, the two oscillating walls are connected to electric generators. A linearized two-dimensional potential flow problem is formulated for this model system, and the steady periodic system response is solved for analytically. A comprehensive parameter study then is carried out so as to clarify how the system’s performance is affected by its design and operation parameters. It is found that, in addition to a series of resonant forcing frequencies that produce large electric power output, there also exist certain anti-resonant forcing frequencies that generate zero power. Moreover, with optimally tuned parameters, the maximized electric power output of the model system studied in this work is comparable with, and even higher than, that of preexisting systems of similar nominal size.