Convective heat transfer in a parallel plate channel with porous lining

The paper proposes a theoretical model for the study of flow and heat transfer in a parallel plate channel, one of whose walls is lined with non-erodible porous material, both the walls being kept at constant temperatures. The analysis uses Brinkman model in the porous medium and employs the velocity and temperature slips at the interface (the so called nominal surface). The influence of the thickness as well as the permeability of the porous medium on the flow field and Nusselt numbers at the walls is investigated.

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Konvektive Wärmeübertragung in einem Parallelplattenkanal mit porösem Überzug

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Vorschlag eines theoretischen Modells, um die Wärmeübertragung in einem Parallelplattenkanal mit unauswaschbarem porösem Überzug zu studieren. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmannschen Gleichung analysiert. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung sind die Geschwindigkeitsgleitung und die Temperaturgleitung benutzt. Die Beeinflussung des Geschwindigkeitsfelds und die Nusseltschen Zahlen an den Wänden in Abhängigkeit von der Dicke und der Durchlässigkeit des porösen Überzugs ist untersucht.

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Nomenclature u streamwise velocity in Zone 1 (Fig. 1) - û streamwise velocity in Zone 2 (Fig. 1) - p pressure - coefficient of viscosity of the fluid - k absolute permeability of the material used for lining - density of the fluid - R Reynolds number - the average velocity in Zone 1 (Fig. 1) - T temperature in Zone 1 (Fig. 1) - T temperature in Zone 2 (Fig. 1) - K thermal conductivity in Zones 1 and 2 (Fig. 1) - M ₁ non-dimensional mass flow rate in Zone 1 (Fig. 1) - M ₂ non-dimensional mass flow rate in Zone 2 (Fig. 1) - (Nu)_U Nusselt number at the upper plate (Fig. 1) - (Nu) _L Nusselt number at the lower plate (Fig. 1) - _E experimental value of the temperature in the channel (with porous lining) at a specified point - _E/* experimental value of the temperature in the channel (without porous lining) at a specified point     

Couette flow of a Bingham plastic in a channel with equally porous parallel walls

In the present paper the flow of a Bingham fluid between two parallel porous walls is studied. One of the walls moves with constant velocity parallel to the other, which is fixed, while a longitudinal pressure gradient exists, as well as a transverse flow field due the porosity of the walls. An exact analytical solution is given for the u-velocity field, which has four different forms depending on the values of the three dimensionless parameters, which are the Bingham, Couette and Reynolds numbers.     

Kinematic analysis of disturbed flow past a plate in a channel with porous walls

Analytic expressions for the complex flow potential are obtained in the linear formulation in the neighborhood of a plate at a small angle of incidence and near porous channel walls. The general solution includes the limiting cases of a plate in a channel with impermeable walls and in a jet. Numerical results concerning the effect of porosity on the flow geometry in the neighborhood of the plate and the channel walls are presented. The disturbed-flow velocity distributions along the channel walls and the flow rate of the fluid sinking at infinity are obtained.Cheboksary. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 13–19, July–August, 1995.     

Laminar flow through a channel with uniformly porous walls of different permeability

Summary The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.     

Characterization of bubble mobility in channel flow with fibrous porous media walls

In composites processing, resin is introduced into a fibrous domain to cover all the empty spaces between the fibers. It is important to extract air bubbles from the domain before the resin solidifies. Failure to do so will entrap these voids in the final part, which is detrimental to its performance. Hence, there is a need to understand bubble motion in a fibrous porous domain in which the bubbles move with the resin in channels surrounded by fibrous walls. A rising bubble model is presented that consists of a single spherical void in a cylindrical axisymmetric two-phase domain of resin and air surrounded by porous media boundaries. The motion of a bubble in a channel flow with porous boundaries is modeled by replacing the walls with a slip velocity. Focus is on how the porous media permeability influences the bubble motion. A parameter called bubble mobility is defined as the ratio of bubble rise velocity to the resin free surface velocity. Results suggest that fabric permeability and fluid properties can be optimized to increase bubble mobility and ultimately lead to reduction in void content during composites processing.     

Lattice Boltzmann simulations of turbulent shear flow between parallel porous walls

The effects of two parallel porous walls are investigated, consisting of the Darcy number and the porosity of a porous medium, on the behavior of turbulent shear flows as well as skin-friction drag. The turbulent channel flow with a porous surface is directly simulated by the lattice Boltzmann method （LBM）. The Darcy-Brinkman- Forcheimer （DBF） acting force term is added in the lattice Boltzmann equation to simu- late the turbulent flow bounded by porous walls. It is found that there are two opposite trends （enhancement or reduction） for the porous medium to modify the intensities of the velocity fluctuations and the Reynolds stresses in the near wall region. The parametric study shows that flow modification depends on the Darcy number and the porosity of the porous medium. The results show that, with respect to the conventional impermeable wall, the degree of turbulence modification does not depend on any simple set of param- eters obviously. Moreover, the drag in porous wall-bounded turbulent flow decreases if the Darcy number is smaller than the order of O（10-4） and the porosity of porous walls is up to 0.4.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Homotopy analysis solutions for the asymmetric laminar flow in a porous channel with expanding or contracting walls

In this paper, the asymmetric laminar flow in a porous channel with expanding or contracting walls is investigated. The governing equations are reduced to ordinary ones by using suitable similar transformations. Homotopy analysis method (HAM) is employed to obtain the expres- sions for velocity fields. Graphs are sketched for values of parameters and associated dynamic characteristics, especially the expansion ratio, are analyzed in detail.     

The freeze-shut of a convectively cooled parallel plate channel subjected to laminar internal liquid flow

The paper presents an approximative solution for the time dependent development of the ice layers at the cooled walls inside a parallel plate channel. The upper and the lower wall of the channel are cooled by an uniform external convection. By assuming a constant pressure drop across the channel, the freeze-shut of the planar channel could be calculated approximately. It was found out that the origin of the freezing fronts moves upstream during the ice layer growth. Furthermore a simple criterion is presented to predict whether a given system will lead to blockade.     

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient freezing of liquids in forced laminar flow inside a parallel plate channel

A simple analytical approximative solution was given for calculating the time dependent development of the ice-layers at the cooled walls inside a parallel plate channel. By ignoring the effect of acceleration, resulting from converging ice-layers in the axial direction, an analytical solution for the variation of the ice-layer thickness with time and axial position could be obtained. The approximative solution was checked by numerical calculations and good agreement was found.Es wurde ein analytisches Näherungsverfahren entwickelt, das es ermöglicht, die zeitliche Entwicklung der Erstarrungsfronten im gekühlten, ebenen Kanal zu bestimmen. Die Methode liefert unter Vernachlässigung der Beschleunigungsterme durch die konvergenten Eisschichten eine exakte Lösung der Phasengrenzbeziehung. Das Näherungsverfahren wurde mittels numerischer Berechnungen überprüft und stimmt bis zu Wandunterkühlungsverhältnissen vonB=10 sehr gut mit der numerischen Lösung überein.     

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.     

On the flow of a viscous fluid driven along a channel by suction at porous walls

This paper is a theoretical treatment of the flow of a viscous incompressible fluid driven along a channel by steady uniform suction through porous parallel rigid walls. Many authors have found such flows when they are symmetric, steady and two-dimensional, by assuming a similarity form of solution due to Berman in order to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation. We generalise their work by considering asymmetric flows, unsteady flows and three-dimensional perturbations. By use of numerical calculations, matched asymptotic expansions for large values of the Reynolds number, and the theory of dynamical systems, we find many more exact solutions of the Navier-Stokes equations, examine their stability, and interpret them. In particular, we show that most previously found steady solutions are unstable to antisymmetric two-dimensional disturbances. This leads to a pitchfork bifurcation, stable asymmetric steady solutions, a Hopf bifurcation, stable time-periodic solutions, stable quasi-periodic solutions, phase locking and chaos in succession as the Reynolds number increases.     

Flow of a suspension in a flat channel with porous walls

In the flow of a suspension in a channel with porous walls, when the size of particles of a suspended phase is much less than the width of the channel but greatly exceeds the size of the pores (in particular, in the flow of blood in the plasma separator used in an artificial kidney system [1, 2]), phenomena are observed which apparently cannot be satisfactorily explained by means of the well-known solutions of problems on the motion of a Newtonian fluid [3]. For example, the flow rate of the liquid phase of the suspension through the walls depends on the velocity of the main flow and does not depend on the pressure drop on the wall at fairly high values of it [1, 2]. The present study gives below the formulation and an approximate solution, which explains this effect, of the problem of an incompressible two-phase suspension in a long slit with porous walls which are impermeable in relation to the suspended phase and through which the liquid phase is pumped. Certain effects are taken into account which are caused by the high volume concentration of the suspended phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 37–43, November–December, 1987.     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

The effect of Hall currents on MHD flow and heat transfer between two parallel porous plates

The effect of Hall currents on magneto hydrodynamic (MHD) flow of an incompressible viscous electrically conducting fluid between two non-conducting porous plates in the presence of a strong uniform magnetic field is studied. The flow is generated by a small uniform suction at the plates. Solutions are obtained for suction Reynolds number R1, considering two cases for the imposed magnetic field, viz. (i) when the magnetic field is perpendicular to the plates (parallel to y-axis), and (ii) when the magnetic field is parallel to the plates and perpendicular to the primary flow direction (parallel to z-axis). The effect of the Hall currents on the flow as well as on the heat transfer is studied. It is observed that in the absence of Hall currents, the change of the direction of the applied magnetic field does not affect the primary flow.Nomenclature B total magnetic induction vector - V velocity vector - E electric field vector - J current density vector - U ₀ suction velocity - T temperature of the fluid at any point - B ₀ imposed magnetic field - u x-component of fluid velocity - v y-component of fluid velocity - w z-component of fluid velocity - density of the fluid - kinematic viscosity of the fluid - c _p specific heat at constant pressure - p fluid pressure - electrical conductivity of the fluid - K coefficient of thermal conductivity - _e magnetic permeability - n _e number density of electrons - e electric charge - dimensionless distance (=y/h) - f(), g(), Q(), () dimensionless functions defined in (14) - R suction Reynolds number (=U ₀ h/) - M Hartmann number (=B ₀ h(/)^1/2) - m Hall parameter (=B ₀/en _e) - Pr Prandtl number of the fluid (=c _p/K) - s dimensionless quantity defined as s=(T ₁–T ₀)/[vU ₀/(hc _p)]     

Homotopy analysis solution for micropolar fluid flow through porous channel with expanding or contracting walls of different permeabilities

The flow of a micropolar fluid through a porous channel with expanding or contracting walls of different permeabilities is investigated. Two cases are considered, in which opposing walls undergo either uniform or non-uniform motion. In the first case,the homotopy analysis method (HAM) is used. to obtain the expressions for the velocity and micro-rotation fields. Graphs are sketched for some parameters. The results show that the expansion ratio and the different permeabilities have important effects on the dynamic characteristics of the fluid. Following Xu's model, in the second case which is more general, the wall expansion ratio varies with time. Under this assumption, the governing equations are transformed into nonlinear partial differential equations that can also be solved analytically by the HAM. In the process, both algebraic and exponential models are considered to describe the evolution of α(t) from the initial state α0 to the final state α1. As a result, the time-dependent solutions are found to approach the steady state very rapidly. The results show that the time-dependent variation of the wall expansion ratio can be ignored because of its limited effects.