Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

Resolution of a Paradox Involving Viscous Dissipation and Nonlinear Drag in a Porous Medium

The modelling of viscous dissipation in a porous medium saturated by an incompressible fluid is discussed, for the case of Darcy, Forchheimer and Brinkman models. An apparent paradox relating to the effect of inertial effects on viscous dissipation is resolved, and some wider aspects of resistance to flow (concerning quadratic drag and cubic drag) in a porous medium are discussed. Criteria are given for the importance or otherwise of viscous dissipation in various situations.     

Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium

Fully developed forced convection in a parallel plate channel filled by a saturated porous medium, with walls held either at uniform temperature or at uniform heat flux, with the effects of viscous dissipation and flow work included, is treated analytically. The Brinkman model is employed. The analysis leads to expressions for the Nusselt number, as a function of the Darcy number and Brinkman number.     

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Effects of viscous dissipation and boundary conditions on forced convection in a channel occupied by a saturated porous medium

Forced convection with viscous dissipation in a parallel plate channel filled by a saturated porous medium is investigated numerically. Three different viscous dissipation models are examined. Two different sets of wall conditions are considered: isothermal and isoflux. Analytical expressions are also presented for the asymptotic temperature profile and the asymptotic Nusselt number. With isothermal walls, the Brinkman number significantly influences the developing Nusselt number but not the asymptotic one. At constant wall heat flux, both the developing and the asymptotic Nusselt numbers are affected by the value of the Brinkman number. The Nusselt number is sensitive to the porous medium shape factor under all conditions considered.     

A Derivation of the Volume-Averaged Boussinesq Equations for Flow in Porous Media with Viscous Dissipation

The volume-averaged equations are derived for convective flow in porous media. In the thermal energy equation viscous dissipation is taken into account, and a suitable form is obtained which is valid when Brinkman effects are significant.     

High-Velocity Laminar and Turbulent Flow in Porous Media

We model high-velocity flow in porous media with the multiple scale homogenization technique and basic fluid mechanics. Momentum and mechanical energy theorems are derived. In idealized porous media inviscid irrotational flow in the pores and wall boundary layers give a pressure loss with a power of 3/2 in average velocity. This model has support from flow in simple model media. In complex media the flow separates from the solid surface. Pressure loss effects of flow separation, wall and free shear layers, pressure drag, flow tube velocity and developing flow are discussed by using phenomenological arguments. We propose that the square pressure loss in the laminar Forchheimer equation is caused by development of strong localized dissipation zones around flow separation, that is, in the viscous boundary layer in triple decks. For turbulent flow, the resulting pressure loss due to average dissipation is a power 2 term in velocity.     

Viscous dissipation effects of power-law fluid flow within parallel plates with constant heat fluxes

Both hydro-dynamically and thermally fully developed laminar heat transfer of non-Newtonian fluids between fixed parallel plates has been analyzed taking into account the effect of viscous dissipation of the flowing fluid. Thermal boundary condition considered is that both the plates kept at different constant heat fluxes. The energy equation, and in turn the Nusselt number, were solved analytically in terms of Brinkman number and power-law index. The findings show that the heat transfer depends on the power-law index of the flowing fluid. Pseudo-plastic and dilatant fluids manifest themselves differently in the heat transfer characteristics under the influence of viscous dissipation. Under certain conditions, the viscous dissipation effects on heat transfer between parallel plates are significant and should not be neglected.     

LAMINAR DISSIPATIVE FLOW IN A POROUS CHANNEL BOUNDED BY ISOTHERMAL PARALI,EL PLATES

The effects of viscous dissipation on thermal entrance heat transfer in a parallel plate channel filled with a saturated porous medium, is investigated analytically on the basis of a Darcy model. The case of isothermal boundary is treated. The local and the bulk temperature distribution along with the Nusselt number in the thermal entrance region were found. The fully developed Nusselt number, independent of the Brinkman number, is found
to be 6. It is observed that neglecting the effects of viscous dissipation would lead to the well-known case of internal flows, with Nusselt number equal to 4.93. A finite difference numerical solution is also utilized. It is seen that the results of these two methods, analytical and numerical, are in good agreement.     

Effects of a magnetic field on the onset of convection in a porous medium

The stability of a conducting fluid saturating a porous medium, in the presence of a uniform magnetic field, is investigated using the Brinkman model. In the first part of the paper constant-flux thermal boundary conditions are considered for which the onset of convection is known to correspond to a vanishingly small wave number. The external magnetic field is assumed to be aligned with gravity. Closed form solutions are obtained, based on a parallel flow assumption, for a porous layer with either rigid-rigid, rigid-free or free-free boundaries. In the second part of the paper, the linear stability of a porous layer, heated isothermally from below, is investigated using the normal mode technique. The external magnetic field is applied either vertically or horizontally. Solutions are obtained for the case of a porous layer with free boundaries. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present analysis as limiting cases.     

Generalised Couette flow of two immiscible viscous fluids with heat transfer using Brinkman model

The present study deals with generalised Couette flow of two viscous, incompressible, immiscible fluids with heat transfer in presence of heat source through two straight parallel horizontal walls. The lower wall is bounded below, by a naturally permeable material of high porosity and the flow inside the porous medium is assumed to be of moderate permeability, modelled by Brinkman equation. The flow domain is divided into three zones to obtain analytical solutions of the momentum and energy equations. To link various flow regions, appropriate matching conditions have been used. The effects of permeability parameter, Reynolds number and viscous parameter on velocity field and the effects of Reynolds number, viscous parameter, permeability parameter, constant heat source and Brinkman number on temperature distribution in different zones are discussed graphically. The mass flow rate, skin-friction factor and rates of heat transfer at the upper boundary and porous interface are discussed with the help of tables.     

The Opposing Effect of Viscous Dissipation Allows for a Parallel Free Convection Boundary-Layer Flow Along a Cold Vertical Flat Plate

External free convection boundary-layer flows are usually treated by neglecting the effect of viscous dissipation. This assumption always results in a non-parallel flow, besides a strong parallel component also a weak transversal component of the (steady) velocity field occurs. The present paper shows, however, that the weak opposing effect of the buoyancy forces due to heat release by viscous dissipation, can give rise along a cold vertical plate adjacent to a fluid saturated porous medium to a strictly parallel steady free convection flow. This boundary-layer flow is described by an algebraically decaying exact analytical solution of the basic balance equations.     

Thermal Instability in a Horizontal Porous Channel with Horizontal Through Flow and Symmetric Wall Heat Fluxes

The linear thermoconvective instability of the basic parallel flow in a plane and horizontal porous channel is investigated. The boundary walls are assumed to be impermeable and subject to symmetric and uniform heat fluxes. The wall heat fluxes produce either a net heating or a net cooling of the fluid saturated porous medium. A horizontal mass flow rate is externally impressed leading to a stationary basic state with a temperature gradient inclined to the vertical. A region of possibly unstable thermal stratification exists either in the lower half-channel (boundary heating), or in the upper half-channel (boundary cooling). The convective instability of the basic flow is governed by the Rayleigh number and by the Péclet number. In the case of boundary heating, the thermal instability arises when the Rayleigh number exceeds its critical value, that depends on the Péclet number. The change of the critical Rayleigh number as a function of the Péclet number is determined numerically for arbitrary normal modes oblique to the basic flow direction. The most dangerous modes are the longitudinal rolls, with a wave vector perpendicular to the basic velocity. There exists a minimum value of the Péclet number, 19.1971, below which no linear instability is detected.     

LAMINAR DISSIPATIVE FLOW IN A POROUS CHANNEL BOUNDED BY ISOTHERMAL PARALLEL PLATES

IntroductionTheproblemofforcedconvectioninaporousmediumchannelorductisaclassicalone (atleastforthecaseofslugflow (Darcymodel) .Therehasrecentlybeenrenewedinterestintheproblembecauseoftheuseofhyperporousmediainthecoolingofelectronicequipment.Recently ,NieldandBejan[1]refertomorethan 3 0papersonthetopic ,butnoneofthemdealsexplicitlywiththecaseofthermaldevelopment.ThisgapintheliteraturehasbeenpartlyfilledbyNieldetal.[2 - 4 ].Lahjomrietal.[5 ,6 ]havesolvedmathematicallysimilarproblemsbyusingthe…     

Flow instability in a curved porous channel formed by two concentric cylindrical surfaces

Stability of laminar flow in a curved channel formed by two concentric cylindrical surfaces is investigated. The channel is occupied by a fluid saturated porous medium; the flow in the channel is driven by a constant azimuthal pressure gradient. The momentum equation takes into account two drag terms: the Darcy term that describes friction between the fluid and the porous matrix, and the Brinkman term, which allows imposing the no-slip boundary condition at the channel walls. An analytical solution for the basic flow velocity is obtained. Numerical analysis is carried out using the collocation method to investigate the onset of instability leading to the development of a secondary motion in the form of toroidal vortices. The dependence of the critical Dean number on porosity and the channel width is analyzed.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.     

Generalized Plain Couette Flow and Heat Transfer in a Composite Channel

An analytical study of fluid flow and heat transfer in a composite channel is presented. The channel walls are maintained at different constant temperatures in such a way that the temperatures do not allow for free convection. The upper plate is considered to be moving and the lower plate is fixed. The flow is modeled using Darcy–Lapwood–Brinkman equation. The viscous and Darcy dissipation terms are included in the energy equation. By applying suitable matching and boundary conditions, an exact solution has been obtained for the velocity and temperature distributions in the two regions of the composite channel. The effects of various parameters such as the porous medium parameter, viscosity ratio, height ratio, conductivity ratio, Eckert number, and Prandtl number on the velocity and temperature fields are presented graphically and discussed.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.