Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

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Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

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A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.     

Experiments on the flow past long circular cylinders in a shear flow

This paper describes an experimental investigation of the flow past circular cylinders, with the mean flow perpendicular to the cylinder axis, at conditions that yield a strong three-dimensional behaviour. The experiments were carried out in the subcritical regime. Long cylinders with end plates were subjected to shear flow with a linear velocity profile in the spanwise direction generated by means of a curved gauze. It was concluded that spanwise cellular structures of vortex shedding emerged in the wake, more clearly for some boundary conditions than others. These structures are characterised by a portion of spanwise length, a cell, having constant shedding frequency over a time average, which implies that there were no vortex dislocations inside that cell during that time. These features were studied using flow visualisation and hot-film anemometry. Spectra of the local shedding frequency are shown, revealing the effect of the shear parameter (=0.02 and 0.04) and aspect ratio L/D (=20.6 and 8) on the stability and geometry of the cells at several Reynolds numbers in the range of 3.13×10³Re_m1.25×10⁴.     

Stokes flow past a porous sphere with an impermeable core

Stokes flow of a viscous, incompressible fluid past a porous sphere with an impermeable core using Darcy law for the flow in the porous region is discussed. The formulae for drag and torque are found by deriving the corresponding Faxen's laws. It is found that torque is always less than that on a solid sphere and it does not depend on the radius of the impermeable core. Some illustrative examples are discussed.     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

Arbitrary oscillatory Stokes flow past a porous sphere using Brinkman model

The present paper deals with the hydrodynamics of a porous sphere placed in an arbitrary oscillatory Stokes flow. Unsteady Stokes equation is used for the flow outside the porous sphere and Brinkman equation is used for the flow inside the porous sphere. Corresponding Faxén’s law for drag and torque is derived and compared with few existing results in some special cases. Examples like uniform flow, oscillatory shear flow and oscillating Stokeslet are discussed. Also, translational oscillation of a weakly permeable sphere is discussed.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Magnetogasdynamic flow past an infinite porous flat plate in slip flow regime

This paper presents an exact solution for the flow of a rarefied ionized gas over an infinite porous plate in the presence of a transverse magnetic field, by using the well known continuum approach. An attempt is made to bring out the salient features of the interaction between the applied magnetic field and the flow of a rarefied conducting gas. The analysis reveals that the skin friction, and the heat transfer into the plate are reduced due to gas rarefaction.     

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.     

Forces on a spherical particle in shear flow at intermediate Reynolds numbers

When particles are submerged in a shear flow, there are lateral (lift) forces on the particles, and these lateral forces affect the dispersion of the particles very much. Recent literature survey indicates that there are large discrepancies among the results from the previous numerical investigations on this subject. A small computational domain ranging between 20–30 sphere radii was used in all the previous numerical investigations. However, the result from the present study reveals that the value of lift coefficient strongly depends on the size of computational domain. To provide correct numerical data and physical interpretation for the forces on a spherical particle in linear shear flow, accurate numerical computations were performed for 5≤Re≤200 using a computational domain of 101 sphere radii.     

Numerical study on viscous shear flow past a circular cylinder using integro-differential formulation

Numerical solutions of the unsteady Navier-Stokes equations are presented for the two dimensional flow about a cylinder immersed in a simple shear flow by using the integrodifferential formulation. Selected numerical results for Reynolds numbers 40 and 80 shear parametersK=0.1 and 0.2 are presented and compared with finite-difference results. Plots of streamlines and equi-vorticity lines and time histories of lift, drag, stagnation point and separation angles are given. It was found that two separation bubbles exist periodically behind the lowvelocity side of the cylinder; a main bubble and a secondary one, while there is only one bubble behind the high-velocity side. Numerical results showed that the lift force acts in the direction towards the low-velocity side of the free stream and its magnitude varies periodically with the nondimensional time. In addition, the magnitude of the lift force is approximately proportional to the shear parameterK.

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Numerische Studie einer viskosen Scherströmung hinter einem Zylinder unter Verwendung eines Integral-Differential-Verfahrens

Zusammenfassung Es werden numerische Lösungen der instationären Navier-Stokes Gleichungen für eine zwei-dimensionale Strömung um einen Zylinder mitgeteilt. Der Zylinder ist einer einfachen Scherströmung ausgesetzt. Zur Lösung wird ein integral-Differential-Verfahren verwendet. Es werden ausgewählte Ergebnisse für Reynolds-Zahlen zwischen 40 und 80 sowie für die ScherparameterK= 0,1 und 0,2 dargestellt. Die Ergebnisse werden mit Resultaten aus einer finiten Differenzenmethode verglichen. Weiterhin werden die Stromlinien, die Linien gleicher Geschwindigkeit und der zeitliche Verlauf von Auftriebskraft und Widerstandskraft sowie die Bewegung des Staupunktes und des Ablösepunktes graphisch dargestellt. Es stellte sich heraus, daß zwei Ablöseblasen periodisch hinter dem Zylinder im Gebiet kleiner Geschwindigkeit auftreten. Dabei handelt es sich um eine primäre und eine sekundäre Ablöseblase. Im Hochgeschwindigkeitsbereich existiert nur eine Ablöseblase. Die numerischen Ergebnisse zeigten, daß die Auftriebskräfte in Richtung des Bereiches Niedergeschwindigkeit wirken, wobei deren Größe periodisch mit der Zeit schwankt. Die Größe der Auftriebskraft ist näherungsweise proportional zum ScherparameterK.

     

Self-sustained oscillations of shear flow past a slotted plate coupled with cavity resonance

Shear flow past a slotted plate configuration can give rise to highly coherent, self-sustained oscillations when coupling occurs with a resonant mode of an adjacent cavity. The distinctive feature of these oscillations is that the wavelength of the coherent instability along the plate is of the order of the plate length. This observation is in contrast to previous investigations of flow past perforated or slotted surfaces, where the instability scales on the diameter of the perforation or the gap length of a slot. The present oscillations occur even when the inflow boundary layer is turbulent and an inflectional form of the shear flow cannot develop along the cavity opening, due to the presence of the slotted plate. Instigation of a resonant mode of the cavity, in conjunction with an inherent instability of the shear flow along the plate, gives rise to ordered clusters of instantaneous vorticity and instantaneous velocity correlation. During the oscillation, ejection of flow occurs from the cavity to the region of the shear flow; this ejection is in accord with the convection of the large-scale cluster of vorticity along the slotted plate. This oscillation can be effectively detuned by adjusting the inflow velocity, such that the inherent instability of the shear flow past the slotted plate is no longer coincident with the resonant frequency of the cavity. Certain features of this self-sustained oscillation are directly analogous to recent findings of oscillations due to shear flow past a perforated plate bounded by a cavity, but in the absence of cavity resonance effects.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

The boundary-layer flow past a suddenly heated vertical surface in a saturated porous medium

The boundary-layer flow generated on an impermeable vertical surface in a saturated porous medium is considered in the case when wall heating at a rate proportional tox is switched on at timet=0, (x measures distance along the wall and is a constant). The similarity equations which hold in the limit of larget are discussed and are shown to have a solution only for >–1. The behaviour of the solution as –1 and as is obtained. Numerical solutions of the initial value problem are then obtained for a range of values of . A direct numerical integration is possible for 1, while an iterative procedure is required for <1, with the numerical scheme becoming unstable for =–0.5.

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Grenzschichtströmung an einer plötzlich aufgeheizten vertikalen Fläche, in einem gesättigten porösen Medium

Zusammenfassung Es wird die an einer undurchlässigen, vertikalen Fläche hervorgerufene Grenzschichtströmung im Falle eines Einschalten der Heizung beit=0 betrachtet. Die Stärke der Wandheizung is proportional zux , wobeix die Koordinate längs der Wand ist und eine Konstante. Die Ähnlichkeitsgleichungen werden für den Bereich von großen Zeitent besprochen und es wird gezeigt, daß eine Lösung nur für >–1 vorliegt. Es wird das Verhalten der Lösungen für –1 und erhalten. Numerische Lösungen für die Anfangsbedingungen des Problems werden für eine Reihe von -Werten errechnet. Eine direkte numerische Integration ist für 1 möglich, während für <1 eine Iteration erforderlich ist, wobei das numerische Verhalten für =–0.5 instabil wird.

     

Steady laminar flow past a heated horizontal plate embedded in a saturated porous medium

In this paper we study the boundary layer equations for steady laminar flow past a heated horizontal plate embedded in a saturated porous medium by adopting the formulation of Chandrasekhara [3], Kolar and Sastri [7]. The velocity distribution and temperature distribution are determined by using the implicit Crank-Nicolson-Predictor-Corrector method of finite difference scheme [7] and [1]. With the help of a compute the distributions are estimated at both (i+1/2)th and (i+1)th levels and they are presented in tabular form. The curves for these distributions are plotted. We calculate the shear stress and skin friction at the wall and observe that the skin-friction directly depends upon the dimensions of the plate and inversely depends upon the Reynolds numberRe. The heat flux and the Nusselt number are evaluated. Further we observe that the Nusselt number depends upon the length of the porous plate.

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Stetige laminare Strömung über eine in einem gesättigten porösen Medium eingebettete horizontale Platte

Zusammenfassung Diese Untersuchung befaßt sich mit den Grenzschichtgleichungen für eine stetige laminare Strömung über eine beheizte Platte, die in ein gesättigtes poröses Medium eingebettet ist, mittels des Formalismus von Chandrasekhara [3], Kolar und Sastri [7]. Die Geschwindigkeits- und Temperaturverteilung wurden unter Benutzung der impliziten Crank-Nicolson-Korrekturmethode des Finiten-Elemente-Schemas bestimmt. Die (i+1/2). und die (i+1). Ebene der Verteilungen wurden mit Computer-Hilfe berechnet und in Tabellenform dargestellt. Die Graphen der Temperatur- und Geschwindigkeitsverteilung wurden ausgeplottet. Die Schubspannungen und die Oberflächenreibung an der Wand wurden berechnet und es konnte festgestellt werden, daß die Oberflächenreibung direkt von der Größe der Platte abhängt und umgekehrt proportional der Reynolds-ZahlRe ist. Der Wärmestrom und die Nusselt-Zahl wurden bestimmt. Weiterhin konnte festgestellt werden, daß die Nusselt-Zahl von der Länge der porösen Platte abhängt.

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Nomenclature C _p specific heat of the convective fluid - D skin friction - k permeability of the porous medium - k _f thermal conductivity of the fluid - k _m the coefficient of thermal conductivity of the porous medium - k _s the conductivity of the solid matrix - N(x) Nusselt number - q(x) specific heat flux - Re local Reynolds number - T temperature - T ₀ temperature of the free stream - T _w temperature of the plate - u velocity in thex-direction - u ₀ velocity of the free stream - V velocity iny-direction - x coordinate axis along the plate - y coordinate axis normal to the plate Greek symbols thermal diffusivity - thickness of the velocity boundary layers in thex direction - thickness of the velocity boundary layer in they-direction - the porosity of the medium - dimensionless variable - kinematic viscosity of the fluid - density of the fluid - shear stress