Laminar flow of a non-Newtonian fluid in channels with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing flow of a non-Newtonian fluid confined in two large plates separated by a small distance of h, with the upper plate stationary while the lower plate is uniformly porous and moving in the x-direction with constant velocity, is derived by accounting for the order of magnitude of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The one-dimensional governing equation for a power-law fluid flow confined between parallel plates, with the upper plate is stationary and the bottom plate subjected to sudden acceleration with a constant velocity in the x-direction and uniformly porous, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient velocity and thus the overall transient velocity distribution. The effects of mass suction/injection at the porous bottom plate on the flow of non-Newtonian fluids are examined for various values of time and power-law index. The results obtained from the present analysis are compared with the data available in the literature.     

A singular perturbation problem of non-newtonian flow between porous disks

The non-Newtonian flow between parallel porous stationary disks due to uniform suction at the disks is considered for both small and large suction Reynolds numbers. In the case of small suction Reynolds number the Navier-Stokes equations have been solved by a regular perturbation technique. The solution obtained is valid for both suction and injection Reynolds numbers. The velocity, pressure and shear distributions have been obtained and are compared with those of the Newtonian flow. We find, in the case of injection, that the combined effect of cross viscosity and visco-elastic co-efficients is to increase the maximum velocity at the centre of the channel and to decrease the magnitude of the velocity gradient at the disks. Whereas in the case of suction, velocity profile is flatter with higher magnitude for the velocity gradients at the disks.In the case of large suction, the Navier-Stokes equations have been solved by the method of matched asymptotic expansions. We find that the effect of large suction at the disks is to flatten the velocity profiles considerably and thereby to push the boundary layer towards the disks. The combined effect of visco-elastic and cross-viscosity terms is to decrease the radial velocity and to increase the axial velocity distributions.     

Unsteady MHD flow of a non-newtonian fluid due to eccentric rotations of a porous disk and a fluid at infinity

Summary An exact solution of the unsteady flow of a second-order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is investigated. It is once again shown that for uniform suction or uniform injection at the disk an asymptotic profile exists for the velocity distribution. The effects of the magnetic field, the material parameters of the second-order fluid, suction and injection on the velocity distribution are studied. Further, from the solution of a rigid disk, it is found that for parameter >.01, a non-Newtonian effect is present in the velocity field. However, for <.01 the velocity field becomes a Newtonian one.     

The effects of side walls on axial flow in rectangular ducts with suction and injection

Summary.  The effects of the side walls on the flows in ducts with suction and injection are examined. Three illustrative examples are considered. The first example considers the effect of the side walls on the flow over a porous plate. It is shown that the presence of the side walls provides a solution for both injection and suction, although, in the absence of the side walls, a solution exists only in the case of suction. The second example considers the flow between two porous plates and the third example the flow in a rectangular duct with two porous walls. Analytical solutions are obtained for the velocity, the volume flux across a plane normal to the flow and the vorticity. In order to show the effects of the side walls for the flow on a rectangular duct, a comparison of these quantities with those in the flow between two parallel porous plates is established. These three examples show that there are pronounced effects of the side walls on the flows in ducts with suction and injection. Received January 10, 2002; revised September 27, 2002 Published online: May 8, 2003 The author is grateful to Prof. D. Poulikakos for his suggestions and to a referee for careful corrections of an earlier version of this paper.     

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.     

Numerical solution of free convection micropolar fluid flow between two parallel porous vertical plates

The fully developed free convection micropolar fluid flow between two vertical porous parallel plates is studied in the presence of temperature dependent heat sources including the effect of frictional heating. The basic equations are solved using quasi-linearization finite difference technique with an error of order 0.5 × 10₋₆. The velocity, microrotation and temperature are displayed in graphs whereas the skin friction, couple stress and Nusselt numbers at the plates are shown in tables. It is noted that the couple stress on either plates increases numerically with increase in micropolar parameter. Also the Nusselt number follows the same pattern for a negative suction velocity.     

Unsteady Flow of Radiating and Chemically Reacting MHD Micropolar Fluid in Slip-Flow Regime with Heat Generation

An analytical study of the problem of unsteady free convection with thermal radiation and heat generation on MHD micropolar fluid flow through a porous medium bounded by a semi-infinite vertical plate in a slip-flow regime has been presented. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted for in the mass diffusion equation. A uniform magnetic field acts perpendicular on the porous surface absorbing micropolar fluid with a suction velocity varying with time. A perturbation technique is applied to obtain the expressions for the velocity, microrotation, temperature, and concentration distributions. Expressions for the skin-friction, Nusselt number, and Sherwood number are also obtained. The results are discussed graphically for different values of the parameters entered into the equations of the problem.     

Free convection boundary layer flow with uniform suction or injection over a cone

Summary The effect of uniform suction or injection on free convection boundary layer over a cone was theoretically investigated. The non-linear ordinary differential equations were obtained by the difference-differential method after transforming it to an equivalent two-dimensional problem by Mangler's transformation. The solutions of the resulting equations can be expressed in the form of integral equations. Numerical calculations were performed solving the integral equations by the iterative numerical quadrature. The velocity profiles, temperature profiles, skin friction parameters and heat transfer parameters with constant wall temperature were computed for various values of suction/injection parameter and cone angle parameter.     

Transient Couette flow in a rotating non-Darcian porous medium parallel plate configuration: network simulation method solutions

The transient, viscous, incompressible, hydrodynamic Couette flow in a rotating porous medium channel is studied in this paper. The channel comprises a pair of infinitely long parallel plates which rotate with uniform angular velocity about an axis normal to the plates. The porous medium is simulated using a Darcy–Forchheimer drag force model which includes both bulk matrix porous drag (dominant at low Reynolds numbers) and second order inertial impedance (dominant at higher Reynolds numbers). The two-dimensional Navier–Stokes equations are reduced to a (z*, t*) coordinate system incorporating Coriolis terms, and appropriate initial and boundary conditions are prescribed. Separate porous drag body force terms are incorporated in both the primary and secondary flow momentum equations. Using a set of transformations, the model is rendered dimensionless and shown to be dictated by the Ekman number, Forchheimer number, Darcy number and Reynolds number in a (z, t) coordinate system. Numerical solutions are obtained for the transformed model using the Network Simulation Method. The influence of the hydrodynamic parameters are computed graphically and also the interaction of parameters on the velocity fields is discussed at length. Excellent agreement is found with earlier non-porous flow studies. The analysis has important applications in geophysics and also chemical engineering systems.     

Longitudinal and torsional oscillations of a rod in an Oldroyd-B fluid with suction or injection

Summary The axisymmetric flow of a homogeneous Oldroyd-B fluid due to the longitudinal and torsional oscillations of an infinite circular rod is studied. At the surface of the rod, suction or injection velocity is applied. The motion and the constitutive equations from a system of P.D.E.s solved numerically. Only in few cases can the numerical results be compared with some known analytical solutions. Numerical experiments show the effect of the non-Newtonian dimensionless parameters on the velocity and on the shear stresses.     

Free convective oscillatory flow of a polar fluid through a porous medium in the presence of oscillating suction and temperature

The aim of this work is to analyze a two-dimensional oscillatory free convective flow of an incompressible polar fluid through a porous medium bounded by an infinite vertical porous plate with oscillating suction and temperature at the wall. The governing equations are based on the volume averaging technique. Analytical expressions for the velocity, angular velocity, and temperature fields are obtained by using the regular perturbation technique. The analysis reveals a multiple boundary layer structure near the wall for the fields mentioned.     

Oscillatory flow through a porous medium by the presence of free convective flow

The purpose of this work is to study the effect of non-constant 2-dimensional free convective flow during the motion of a viscous incompressible fluid through a highly porous medium. The porous medium is bounded by a vertical plane surface of constant temperature. This surface absorbs the fluid with a constant velocity and the free stream velocity of the fluid vibrates about a mean constant value. Analytical expressions for the velocity of the fluid are given. The effects of Grashof number and the permeability parameter upon the velocity field are also shown in a graphic representation.     

Rheological effects of non-newtonian fluids on gravitational segregation mechanism in a porous medium

This paper investigates the rheological behaviour effects of non-Newtonian fluids on the evolution in time of the shape of a moving interface separating two immiscible and incompressible fluids during the gravitational segregation process in a porous medium. A rheological model of Bingham type was used to illustrate these effects. An approximate analytical solution has been developed for determining the shape of the moving interface in function of time, assuming that its initial shape, i.e. at $\text{t = 0}$, is known. The pressure and velocity distributions ahead and behind the interface have also been obtained. The conditions in which the separation of the two phases, oil and water, may be complete are discussed and implications of the presence of a threshold gradient in gravitational segregation mechanism in porous medium are shown.     

Slip at a uniformly porous boundary: effect on fluid flow and mass transfer

An approximate solution to the 2-D Navier-Stokes equations for steady, isothermal, incompressible, laminar flow in a channel bounded by one porous wall subject to uniform suction is derived. The solution is valid for small values of the Reynolds number based on the suction velocity and channel height. Solute transport is considered numerically by decoupling the equations representing momentum and mass transfer. The effect of fluid slip at the porous boundary on the axial and transverse components of fluid velocity, axial pressure drop and mass transfer is investigated.     

Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid

Summary The present work examines the flow of a third grade fluid and heat transfer analysis between two stationary porous plates. The governing non-linear flow problem is solved analytically using homotopy analysis method (HAM). After combining the solution for the velocity, the temperature profile is determined for the constant surface temperature case. Graphs for the velocity and temperature profiles are presented and discussed for various values of parameters entering the problem.     

Theoretical analysis on mixed convection boundary layer flow over a wedge with uniform suction or injection

Summary Forced and free mixed convection boundary layer flow over a wedge with uniform suction or injection is theoretically investigated. Nonsimilar partial differential equations are transformed into ordinary differential equations by means of difference-differential method. The solutions of the resulting equations are obtained in integral forms and are calculated by iterative numerical procedures. The results were given for velocity profiles, temperature profiles, friction and heat transfer parameters for various values of suction/injection parameter, pressure gradient parameter and buoyancy parameter.     

Laminar flow in an annulus between two concentric rotating porous spheres

Summary An analysis is presented for the steady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric porous spheres with injection/suction at their boundaries. The inner sphere rotates with constant angular velocity about its own fixed axis, while the outer sphere is stationary. A solution of the Navier-Stokes equations is obtained by employing a regular perturbation technique. The solution obtained is in the form of a power series expansion in terms of the rotational Reynolds number Re, and an injection/suction Reynolds number Re _w , and is valid for small values of these parameters. Results for the velocity distributions, streamlines, and viscous torques for various values of the flow parameters Re, Re _w , and radius ratios are presented. Viscous torques at the inner and outer spheres are compared with those obtained from the numerical solution of the Navier-Stokes equations, in order to find the range of Re and Re _w for which this solution is accurate.     

Forced and free mixed convection boundary layer flow with uniform suction or injection on a vertical flat plate

Summary A laminar forced and free mixed convection flow on a flat plate with uniform suction or injection was theoretically investigated. Nonsimilar partial differential equations are transformed into nonsimilar ordinary ones by means of difference-differential method. The solutions of the resulting equations are obtained with integral forms, and are calculated by the method of successive iteration. The velocity profiles, temperature profiles, friction coefficient and heat transfer coefficient are obtained for various values of suction/injection parameter and buoyancy parameter.     

Dynamics of moving interface in porous media for power law fluids with yield stress

In this paper the dynamic of moving interface in a porous medium for non-Newtonian fluids of power law with yield stress is investigated. The frontal advance theory, which describes the flow of two immiscible fluids separated by a moving interface, in which the displacing fluid is of power law with yield stress and the displaced one is Newtonian, has been used to illustrate the deviation from Newtonian behaviour in oil displacement mechanism. The limitations of this theory for non-Newtonian fluids have been shown and discussed. A criterion for determining the conditions under which the viscous fingering effect is eliminated and the interface movement may be in a regular manner has also been obtained. An approximate analytical solution for determining the interface position and its velocity at any time is presented.     

Coupled transport of multi-component solutes in porous media

Coupled convective-diffusive transport of multicomponent solutes in spatially-periodic models of porous media is considered. Species coupling at the micro- or interstitial scale results from a first-order irreversible surface reaction on the bed elements, composing the porous medium, and from the off-diagonal terms of the microscale matrix transport coefficients.The coarse-scale long-time solute matrix properties are calculated, namely, mean effective reactivity, velocity and dispersivity. These coefficients are analyzed in several important particular cases, pertaining to reactive and nonreactive constituents. The solution scheme is illustrated by an example of two reactive solute components with diffusional coupling, flowing in a bundle of tubes model porous medium. The effective matrix axial transport coefficients are analyzed for various values of the dimensionless Damkohler number, Da, associated with the surface-reaction constant. Analytical expressions for the effective axial transport properties are obtained in cases of extreme (small and large) values of the dimensionless Damkohler numbers.The microscale molecular diffusive coupling provides for each solute constituent two diffusive pathways to the reactive tube wall: one – via the direct diffusivity component, another – via the coupling diffusivity. The macroscopic manifestation of this microscale coupling is to give rise to coupling off-diagonal terms in the effective matrix transport coefficients: positive off-diagonal terms in the reactivity matrix and negative off-diagonal terms in velocity and dispersivity matrices. From a physical viewpoint microscale coupling brings about a more uniform solute distribution within the tube cross section, which reduces the effective axial transport.