Steady Natural Convection Flow of a Particulate Suspension Through a Parallel-Plate Channel

A continuum model for two-phase (fluid/particle) flow induced by natural convection is developed and applied to the problem of steady natural convention flow of a particulate suspension through an infinitely long channel. The walls of the channel are maintained at constant but different temperatures. The two-phase model accounts for particle-phase viscous effects. Boundary conditions borrowed from rarefied gas dynamics are employed for the particle-phase wall conditions. Various closed-form solutions for different special cases are obtained. A parametric study of the physical parameters involved in the problem are performed to illustrate the influence of these parameters on the flow and heat transfer aspects of the problem.     

MHD Peristaltic Transport of a Jeffery Fluid in a Channel With Compliant Walls and Porous Space

Here an attempt has been made to investigate the magnetohydrodynamic (MHD) flow of a non-Newtonian fluid filling the porous space in a channel with compliant walls. Constitutive equations of a Jeffery fluid are used in the mathematical modeling. The flow is created due to sinusoidal traveling waves on the channel walls. The resulting problem is solved analytically and series solution for a stream function is derived. The effects of pertinent flow parameters are discussed through graphs.     

Flow through a porous annulus

Summary The problem of laminar flow through a porous annulus with constant velocity of suction at the walls and with swirl is reduced to the solution of four non-linear differential equations. The significance of each of these equations is discussed. By taking the swirl to be zero series solutions are obtained for (i) small suction or blowing (ii) when the total flow into the channel through the walls is small. Finally the asymptotic behaviour of the flow for large suction or blowing is discussed.     

Calculation of laminar flow of compressible gas in the presence of heat transfer in two-dimensional curvilinear channels

Numerical solution of the complete system of Navier-Stokes equations is used to investigate laminar (Re ? 1000) subsonic flows of a compressible gas in the presence of heat transfer (cooled walls) in two-dimensional channels containing a bend section (for different curvature parameters). The appearance of closed separation regions of the flow on the channel walls, their deformation as the parameters of the problem are changed, and the loss of pressure are studied. The sections of the channel walls with maximal and minimal heat fluxes are determined, and the connection between these sections and the separation regions is elucidated.     

Thermal nonequilibrium,non-darcian forced convection in a channel filled with a fluid saturated porous medium—A perturbation solution

This paper concentrates on the analysis of the thermal nonequilibrium effects during forced convection in a parallel-plate channel filled with a fluid saturated porous medium. The flow in a channel is described by the Brinkman-Forchheimer-extended Darcy equation and the thermal nonequilibrium effects are accounted for by utilizing the two energy equations model. Applying the perturbation technique, an analytical solution of the problem is obtained. It is established that the temperature difference between the fluid and solid phases for the steady fully developed flow is proportional to the ratio of the flow velocity to the mean velocity. This results in a local thermal equilibrium at the walls of the channel if the Brinkman term which allows for the no-slip boundary condition at the walls is included into the momentum equation.     

Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel

Mixed convective flow and heat transfer in a vertical channel with one region filled with conducting fluid and another region with non-conducting fluid is analyzed. The viscous and Ohmic dissipation terms are 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 prescribed. Analytical solutions are found for the governing equations using the regular perturbation method. A selected set of graphical results illustrating the effects of various parameters involved in the problem are presented and discussed.     

Numerical study of particulate suspension flow through wavy‐walled channels

The particulate suspension flow in a channel whose walls describe a travelling wave motion is examined numerically. A perturbation method is employed and the primitive variables are expanded in a series with the wall amplitude as the perturbation parameter. The boundary conditions are applied at the mean surface of the channel and the first‐order perturbation quantities are numerically determined by solving the governing system of ordinary differential equations by shooting technique. The present approach does not impose any restriction on the Reynolds number of the flow and the wave number and frequency of the wavy‐walled channel, although it is limited by the linear analysis. The wall shear stress and the positions of flow separation and reattachment points are computed and the influence of the volume fraction density of the particles is examined. The variations of velocity and pressure of the particulate suspension flow with frequency of excitation are also presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

Conjugate heat transfer inside a porous channel

Analytical and numerical analyses have been performed for fully developed forced convection in a fluid-saturated porous medium channel bounded by two parallel plates. The channel walls are assumed to be finite in thickness. Conduction heat transfer inside the channel wall is also accounted and the full problem is treated as a conjugate heat transfer problem. The flow in the porous material is described by the Darcy–Brinkman momentum equation. The outer surfaces of the solid walls are treated as isothermal. A temperature dependent volumetric heat generation is considered inside the solid wall only. Analytical expressions for velocity, temperature, and Nusselt number are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show an excellent agreement with the analytical results.     

Entropy generation during fluid flow in a channel under the effect of transverse magnetic field

Entropy generation due to fluid flow and heat transfer inside a horizontal channel made of two parallel plates under the effect of transverse magnetic field is numerically investigated. The flow is assumed to be steady, laminar, hydro-dynamically and thermally fully developed of electrically conducting fluid. Both horizontal walls are maintained at constant temperatures higher than that of the fluid. The governing equations in Cartesian coordinate are solved by an implicit finite difference technique. After the flow field and the temperature distributions are obtained, the entropy generation profiles are computed and presented graphically. The factors, which were found to affect the problem under consideration are the magnetic parameter, Eckert number, Prandtl number, and the temperature parameter (θ_∞). It was found that, entropy generation increased as all parameters involved in the present problem increased.     

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.     

Chemically reactive hydromagnetic flow of a second-grade fluid in a semi-porous channel

An analysis of a second-grade fluid in a semi-porous channel in the presence of a chemical reaction is carried out to study the effects of mass transfer and magnetohydrodynamics. The upper wall of the channel is porous, while the lower wall is impermeable. The basic governing flow equations are transformed into a set of nonlinear ordinary differential equations by means of a similarity transformation. An approximate analytical solution of nonlinear differential equations is constructed by using the homotopy analysis method. The features of the flow and concentration fields are analyzed for various problem parameters. Numerical values of the skin friction coefficient and the rate of mass transfer at the wall are found.     

Unsteady flow and heat transfer of porous media sandwiched between viscous fluids

The problem of unsteady oscillatory flow and heat transfer of porous medin sandwiched between viscous fluids has been considered through a horizontal channel with isothermal wall temperatures. The flow in the porous medium is modeled using the Brinkman equation. The governing partial differential equations are transformed to ordinary differential equations by collecting the non-periodic and periodic terms. Closed-form solutions for each region are found after applying the boundary and interface conditions. The influence of physical parameters, such as the porous parameter, the frequency parameter, the periodic frequency parameter, the viscosity ratios, the conductivity ratios, and the Prandtl number, on the velocity and temperature fields is computed numerically and presented graphically. In addition, the numerical values of the Nusselt number at the top and bottom walls are derived and tabulated.     

Unsteady two-fluid flow and heat transfer in a horizontal channel

The problem of unsteady oscillatory flow and heat transfer of two viscous immiscible fluids through a horizontal channel with isothermal permeable walls has been considered. The partial differential equations governing the flow and heat transfer are solved analytically using two-term harmonic and non-harmonic functions in both fluid regions of the channel. Effects of physical parameters such as viscosity ratio, conductivity ratio, Prandtl number and frequency parameter on the velocity and temperature fields are shown graphically. It is observed that the velocity and temperature decrease as the viscosity ratio increases, while they increase with increases in frequency parameter. The effect of increasing the thermal conductivity ratio also suppresses the temperature in both fluid regions. The effect of periodic frequency on the flow is depicted in tabular form. It is predicted that both the velocity and temperature profiles decrease as the periodic frequency increases.     

Flow in a viscous jet escaping through a supersonic nozzle into a semi-infinite ambient space

The problem of an axisymmetric gas flow in a supersonic nozzle and in the jet escaping from the nozzle to a quiescent gas is solved within the framework of Navier-Stokes equations. The calculated pressure distribution is compared with that measured in the jet by a Pitot tube. The influence of the jet pressure ratio, Reynolds number, and half-angle of the supersonic part of the nozzle on nozzle flow and jet flow parameters is studied. It is shown that the distributions of gas-dynamic parameters at the nozzle exit are nonuniform, which affects the jet flow. The flow pattern for an overexpanded jet shows that jet formation begins inside the nozzle because of boundary-layer displacement from the nozzle walls. This result cannot be obtained with the inviscid formulation of the problem.     

Equations for transient behaviour of parallel-flow multichannel heat exchangers

A set of partial differential equations is formulated which describes the transient behaviour of parallel-flow multichannel heat exchangers. In the model, the heat capacitance of walls is neglected and heat is assumed to be transferred only in the direction perpendicular to the direction of flow of fluids. The set is solved by the MacCormack predictor-corrector method for a sample four-channel heat exchanger. Three cases of transient behaviour of the exchanger are investigated: uniform initial temperatures in all channels, a step change in flow rate in one channel, and a step change in inlet temperature of one fluid. In the last two cases, the exchanger is initially at steady state.     

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Transient-state flow of a conducting liquid in an MHD generator at constant flow rate in the presence of side walls

It is usual in studies of transient [nonsteady] flow for a viscous incompressible conducting fluid in an MHD channel to take the distance between the side walls as infinite, which allows the initial equations to be simplified, these reducing to a single equation for the velocity if the magnetic Reynolds number is small [1–3]. A real system has a finiteratio of the sides, so it is desirable to establish the effects of the side walls.