Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection

The steady flow of an incompressible viscous non-Newtonian fluid above an infinite rotating porous disk in a porous medium is studied with heat transfer. A uniform injection or suction is applied through the surface of the disk. Numerical solutions of the non-linear differential equations which govern the hydrodynamics and energy transfer are obtained. The effect of the porosity of the medium, the characteristics of the non-Newtonian fluid and the suction or injection velocity on the velocity and temperature distributions is considered. The inclusion of the three effects, the porosity, the non-Newtonian characteristics, and the suction or injection velocity together has shown some interesting effects.     

Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection

The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate moves with a constant velocity and the two plates are kept at different but constant temperatures. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions are studied.     

Effect of Hall current on transient hydromagnetic Couette–Poiseuille flow of a viscoelastic fluid with heat transfer

The unsteady Couette–Poiseuille flow of an electrically conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal non-conducting porous plates is studied with heat transfer considering the Hall effect. A sudden uniform and constant pressure gradient, an external uniform magnetic field that is perpendicular to the plates and uniform suction and injection through the surface of the plates are applied. The two plates are kept at different but constant temperatures while the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained using finite difference approximations. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions is examined.     

The study of Heat and Mass Transfer in a Visco elastic fluid due to a continuous stretching surface using Homotopy Analysis Method

In this paper, an approximate analytical solution is derived for the flow velocity and temperature due to the laminar, two-dimensional flow of non-Newtonian incompressible visco elastic fluid due to a continuous stretching surface. The surface is stretched with a velocity proportional to the distance $x$ along the surface. The surface is assumed to have either power-law heat flux or power-law temperature distribution. The presence of source/sink and the effect of uniform suction and injection on the flow are considered for analysis. An approximate analytical solution has been obtained using Homotopy Analysis Method(HAM) for various values of visco elastic parameter, suction and injection rates. Optimal values of the convergence control parameters are computed for the flow variables. It was found that the computational time required for averaged residual error calculation is very very small compared to the computation time of exact squared residual errors. The effect of mass transfer parameter, visco elastic parameter, source/sink parameter and the power law index on flow variables such as velocity, temperature profiles, shear stress, heat and mass transfer rates are discussed.     

Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection

The unsteady flow of a viscous conducting fluid due to the rotation of an infinite, non-conducting, porous disk in the presence of an axial uniform steady magnetic field is studied without neglecting the Hall effect. The fluid is acted upon by a uniform injection or suction through the disk. The relevant equations are solved numerically with a special technique to resolve the conflict between the initial and boundary conditions. The solution shows that the inclusion of the injection or suction through the surface of the disk in addition to the Hall current gives some interesting results.     

Steady laminar flow with suction or injection and heat transfer through porous pipe

Following the method due to Bhatnagar (P. L.) [Jour. Ind. Inst. Sic., 1968, 1, 50, 1], we have discussed in this paper the problem of suction and injection and that of heat transfer for a viscous, incompressible fluid through a porous pipe of uniform circular cross-section, the wall of the pipe being maintained at constant temperature. The method utilises some important properties of differential equations and some transformations that enable the solution of the two-point boundary value and eigenvalue problems without using trial and error method. In fact, each integration provides us with a solution for a suction parameter and a Reynolds number without imposing the conditions of smallness on them. Investigations on non-Newtonian fluids and on other bounding geometries will be published elsewhere.     

Laminar non-Newtonian fluid flow in a porous annulus

The effect of suction and injection through the porous walls of an annulus on a two-dimensional steady laminar flow of a non-Newtonian fluid is investigated by solving directly the Navier-Stokes equations in cylindrical coordinates. A quasi-linearization method is used to solve the flow problem for both small and moderately large cross flow through the porous walls. With very approximate starting values of functions, only a few iterations are needed to obtain results with very high accuracy. For comparison, a perturbation method is also used to solve the same flow problem. For small values of cross flow through the porous wall, results obtained by both methods agree very well; while for the moderately large values of cross flow, the perturbation technique is impractical and may give an inaccurate result. The effects of the non-Newtonian fluid on the axial velocity, the pressure drop, and skin friction are also examined.     

Viscous flow and temperature distribution in rotating annulus with suction and injection

Part A.—Steady viscous incompressible flow in a rotating coaxial cylindrical annulus with suction and injection is studied. The unsteady flow is also considered. Part B.—An exact solution for temperature distribution at different constant wall temperatures is obtained. It is assumed that the rate of injection at the inner wall equals the rate of suction at the outer wall.     

Falkner-Skan equation for flow past a moving wedge with suction or injection

The characteristics of steady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid past a moving wedge with suction or injection are theoretically investigated. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of Falkner-Skan power-law parameter (m), suction/injection parameter (f0) and the ratio of free stream velocity to boundary velocity parameter (λ) are discussed in detail. The numerical results for velocity distribution and skin friction coefficient are given for several values of these parameters. Comparisons with the existing results obtained by other researchers under certain conditions are made. The critical values off ₀,m and λ are obtained numerically and their significance on the skin friction and velocity profiles is discussed. The numerical evidence would seem to indicate the onset of reverse flow as it has been found by Riley and Weidman in 1989 for the Falkner-Skan equation for flow past an impermeable stretching boundary.     

Unsteady flow and heat transfer in a thin film of Ostwald–de Waele liquid over a stretching surface

In this paper, the effects of viscous dissipation and the temperature-dependent thermal conductivity on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian Ostwald–de Waele fluid over a horizontal porous stretching surface is studied. Using a similarity transformation, the time-dependent boundary-layer equations are reduced to a set of non-linear ordinary differential equations. The resulting five parameter problem is solved by the Keller–Box method. The effects of the unsteady parameter on the film thickness are explored numerically for different values of the power-law index parameter and the injection parameter. Numerical results for the velocity, the temperature, the skin friction and the wall-temperature gradient are presented through graphs and tables for different values of the pertinent parameter. One of the important findings of the study is that the film thickness increases with an increase in the power-law index parameter (as well as the injection parameter). Quite the opposite is true with the unsteady parameter. Furthermore, the wall-temperature gradient decreases with an increase in the Eckert number or the variable thermal conductivity parameter. Furthermore, the surface temperature of a shear thinning fluid is larger compared to the Newtonian and shear thickening fluids. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena.     

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [ 1 ], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [ 2 ] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter     , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [ 1 ] for zero-suction compressible modes.
The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.     

MHD Couette flow of a viscous stratified fluid of large conductivity

The MHD Couette flow of a viscous stratified fluid of large electrical conductivity with suction and injection at the plane boundaries is studied when the plane boundaries are maintained at different temperatures. The Oseen type governing equations are formulated using the method suggested by Greenspan for stratified fluids. Introducing the similarity variables, the linearised equations are solved to obtain the velocity and temperature distributions. The results show that the behaviour of velocity and temperature in fluids of large conductivity is different from the behaviour of velocity and temperature for fluids of finite conductivity. The effect of the magnetic field on the load capacity is investigated for the case when the width of the channel is small.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

Torque on a sphere inside a rotating cylinder

Summary The couple on a sphere in the centre of a finite rotating circular cylinder is measured over a wide range of Reynolds numbers for both Newtonian and non-Newtonian fluids. Wall effects are calculated. Experimental results are compared with Collins' analysis. Secondary flow is made visible. For non-Newtonian fluids the apparatus determines accurately the zero shear rate viscosity.

---

Résumé Le couple sur une sphère dans un cylindre circulaire rotatoire est mesuré pour des divers nombres de Reynolds et pour des fluides Newtoniens et non-Newtoniens. L'effet des parois sur le couple est calculé. Les resultats obtenus sont comparés avec l'analyse de Collins. Pour les fluides non-Newtoniens l'appareil determine la viscosité de zéro cisaillement.

     

热幅射对抽吸或喷射三维Couette流温度分布的影响

在固定的底板上有横向正弦射流,而匀速运动的多孔介质顶板以常速率完全抽出的情况下,理论分析了热幅射对三维Couette流动温度分布的影响.在这种射流速度下,流动呈现三维流动.利用图形分析了Prandtl数、幅射参数和射流参数对传热速率的影响.Prandtl数对温度分布的影响比射流参数或幅射参数大得多.     

Unsteady hydromagnetic rotating flow of a conducting second grade fluid

The purpose of this work is to investigate the hydromagnetic oscillatory flow of a fluid bounded by a porous plate, when the entire system rotates about an axis normal to the plate. The fluid is assumed to be non-Newtonian (second grade), incompressible and electrically conducting. The magnetic field is applied transversely to the direction of the flow. Such a flow model has great significance not only of its theoretical interest, but also for applications to geophysics and engineering. The resulting initial value problem has been solved analytically for steady and unsteady cases. The analysis of the obtained results showed that the flow field is appreciably influenced by the material parameter of the second grade fluid, the applied magnetic field, the imposed frequency, rotation and suction and blowing parameters. It is observed in a second grade fluid that a steady asymptotic hydromagnetic solution exists for blowing and resonance which is different from the hydrodynamic situation.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparative study of some constitutive equations characterising non-Newtonian fluids

This paper compares, in a general way, the predictions of the constitutive equations given by Rivlin and Ericksen, Oldroyd, and Walters. Whether we consider the rotational problems in cylindrical co-ordinates or in spherical polar co-ordinates, the effect of the non-Newtonicity on the secondary flows is collected in a single parameterα which can be explicitly expressed in terms of the non-Newtonian parameters that occur in each of the above-mentioned constitutive equations. Thus, for a given value ofα, all the three fluids will have identical secondary flows. It is only through the study of appropriate normal stresses that a Rivlin-Ericksen fluid can be distinguished from the other two fluids which are indistinguishable as long as this non-Newtonian parameter has the same value.     

Couette flow of a non-homogeneous fluid

The two dimensional Couette flow of a non-homogeneous viscous fluid is studied. The plane boundaries of the channel are maintained at different temperatures. The upper plane moves with a uniform horizontal velocity and the lower plane is at rest. The fluid is subjected to suction and injection at the boundaries. Thesteady equations are solved by introducing similarity variables which are expanded in series of powers of a small stratification parameter. The non-linear theory predicts that the temperature depends on the distancex from the throat section, an observation which is not predicted by the linear theory. The non-linear effects on velocity and temperature are studied. The rate of heat transfer is discussed.