Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

A solution for the problem of stokes flow past a porous sphere

The problem of a general non-axisymmetric Stokes flow of a viscous fluid past a porous sphere is considered. The expressions for the velocity and pressure, both inside and outside the sphere are given, when the flow outside satisfies the Stokes equations and the flow inside the sphere is governed by Darcy's law. The expressions for drag and torque are given. It is found that the drag is greater or smaller than the drag in the rigid case, depending on whether the undisturbed velocity is a pure biharmonic or a harmonic respectively. The torque is same as in the rigid case.     

Slow motion of a micropolar fluid through a porous sphere bounded by a solid sphere

The paper examines the slow motion of a micropolar fluid produced by the relative motion of a solid sphere to an inside porous sphere. The result extends the Cunningham’s problem to micropolar fluid when the inner sphere is porous with prescribed radial suction/injection velocity at the surface of the sphere. The result can also be taken as an extension of the work of Ramkissoon and Majumdar when the fluid is bounded at a radiusr=b (b>a) but the solid sphere is replaced by a porous sphere. The force experienced by the inner sphere has been calculated and particular cases of interest have been deduced.     

Slow magnetohydrodynamic flow past a circular cylinder

Oseen’ approximations are used to study the slow motion of a viscous, incompressible, electrically conducting fluid past a circular cylinder in the presence of a uniform aligned magnetic field. Using series truncation method, the analytical solutions for the first three terms in the Fourier sine series expansion of the stream function are obtained. Numerical values of the tangential drag for different values of magnetic interaction parameter and viscous Reynolds number are calculated.     

Flow past a porous sphere at small Reynolds number

low of an incompressible viscous fluid past a porous sphere has been discussed. The flow has been divided in three regions. The Region-I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear fluid. In Regions II and III clear fluid flows and Stokes and Oseen solutions are respectively valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Stream function of Region II is matched with that of Region I at the surface of the sphere by the conditions suggested by Ochao-Tapia and Whitaker and it is matched with that of Oseen’s solutions far away from the sphere. It is found that the drag on the sphere reduces significantly when it is porous and it decreases with the increase of permeability of the medium.Received: February 7, 2002; revised: April 8, 2003 / June 9, 2004     

Stokes flow past a slightly deformed fluid sphere

Summary The Stokes flow past a fluid spheroid whose shape deviates slightly from that of a sphere, is examined. To the first order in the small parameter characterizing the deformation, an exact solution is obtained. As an application, the case of a fluid oblate spheroid is considered and the drag experienced by it is evaluated. Special well-known cases are then deduced.

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Zusammenfassung Die Stokes-Strömung um einen Flüssigkeitstropfen, der nur leicht von einer perfekten Kugel abweicht, wird untersucht. Eine Lösung wird gefunden, die exakt ist bis zur ersten Ordnung im Deformations-Parameter. Als Beispiel wird der Strömungswiderstand eines abgeplatteten Flüssigkeits-Sphäroids berechnet. Die Methode liefert für bekannte Spezialfälle korrekte Lösungen.

     

Slow viscous flow past a cavity with infinite depth

Two-dimensional slow viscous flow on infinite half-plane past a perpendicular infinite cavity is considered on the basis of the Stokes approximation. Using complex representation of the two-dimensional Stokes flow, the problem is reduced to solving a set of Fredholm integral equations of the second kind. The streamlines and the pressure and vorticity distribution on the wall are numerically determined.     

Plane potential flow past a cylinder with porous surface

In the paper two models for the potential flow past a cylinder with porous or perforated surface are reduced to nonlinear Riemann-Hilbert problems for the complex velocity function and general existence and uniqueness theorems for the solution of the problems are derived. Further the problems are reduced to nonlinear singular integral equations of Hilbert type whose solvability is investigated by means of the Schauder and Banach fixed point theorem.     

Heat transfer from a porous sphere in low Reynolds number flow

This paper deals with the problem of the effect of permeability in the steady state heat transfer from a single porous sphere at low Reynolds number with the assumptionsPr=0(1) andR<1. The solution is sought by the method of matched asymptotic expansions. The Nusselt number,Nu, for isothermal sphere has been calculated for different values ofk, wherek being the permeability associated with the porosity of the sphere. We notice that the Nusselt number increases as the permeability increases in the range 0<R<0.35 with the Prandtl numberPr=0.7, the rate of increase is very small, while for the Reynolds numberR0.35 Nusselt number decreases ask increases and the rate of decrease is large.

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Zusammenfassung In dieser Arbeit wird der Einfluss der Durchlässigkeit von einer porösen Kugel auf den Wärmeübergang behandelt, bei kleiner Reynolds-ZahlR undPr=0(1). Die Lösung wird mit der Methode der angepassten asymptotischen Entwicklung gewonnen. Die Nusselt'sche ZahlNu für die isotherme Kugel wurde für verschiedene Werte des Durchlässigkeits-Koeffizientenk ausgerechnet. Man findet, dass die Nusselt'sche Zahl (schwach) zunimmt mit der Durchlässigkeit für 0<R<0.35 mitPr=0.7, wogegen fürR0.35Nu (stark) abnimmt mit wachsendemk.

     

Hydromagnetic effects on the three-dimensional flow past a porous plate

Hydromagnetic effects on the three-dimensional flow of an electrically conducting viscous incompressible fluid past a porous plate with periodic suction has been analysed. The uniform flow is subjected to a transversely applied magnetic field. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. Approximate solutions for the components of velocity field and the skin-frictions due to them are obtained and discussed with the help of a graph and tables.     

Boundary integral method for Stokes flow past a porous body

In this paper we obtain an indirect boundary integral method in order to prove existence and uniqueness of the classical solution to a boundary value problem for the Stokes–Brinkman-coupled system, which describes an unbounded Stokes flow past a porous body in terms of Brinkman's model. Therefore, one assumes that the flow inside the body is governed by the continuity and Brinkman equations. Some asymptotic results in both cases of large and, respectively, of low permeability are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

The flow due to a porous sphere embedded in a pure straining motion

Résumé Dans le présent travail on étudie les mouvements stationnaires lents d'un fluide visqueux en présence d'une sphère poreuse. Les mouvements considérés sont l'écoulement en présence d'une sphère poreuse dans une translation uniforme, dans une rotation uniforme, et dans un mouvement de déformation pure. On donne aussi une expression pour la viscosité de suspension des sphères poreuses. Elle montre que la viscosité de suspension des sphères poreuses est plus petite que celle des sphères rigides.     

Numerical simulation of flow past a sphere in vertical motion within a stratified fluid

In the present study we consider a viscous fluid, stratified by a diffusive saline agent and compute numerically the flow produced by a solid sphere moving vertically and uniformly. The governing equations describing this situation are solved on a variational grid. The results show the dependence of the boundary-layer separation point and the vanishing of vortices behind the sphere as the stratification increases at moderate Reynolds number flows. Details of the flow, density and pressure fields near the sphere are also shown. Important quantities for engineering use (drags, pressure and skin coefficients) are also computed and displayed in the Richardson vs. Reynolds number space. Comparison with experimental evidence shows and excellent agreement.     

MHD slow motion past a sphere

The slow motion of an incompressible, viscous electrically conducting fluid, in the presence of a uniform aligned magnetic field, past a sphere is studied. Solutions obtained by Chester, using Stokes’ approximations, and by Blerkom and Ludford, using Ossen’ approximations, are reviewed. Expressions for stream functions are obtained for MHD Stokes’ flow and Oseen’ flow respectively.     

Potential flow past a porous body with a core of different permeability

It is well known that a uniform flow past a non-permeable rigid body does not exert a total force upon the surface of the body, however this is not the case when the body is permeable. Power et. al. (1984, 1986) first solved the problem of uniform potential flow past a two-dimensional permeable circular cylinder, with constant permeability, and found that the exterior flow exerts a drag force upon the surface of the cylinder independent of its size and secondly the problem when the uniform potential flow past a porous sphere, with constant permeability, in this case the exterior flow exerts a drag force on the sphere which is linearly dependent on the radius of the sphere. Here we will present the solution of two problems, a uniform potential flow past a porous circular cylinder and past a porous sphere, for each case the porous body is composed of two materials with different permeabilities. In both cases the total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials, and in the limit when the two permeabilities are equal, the previous results, circular cylinder and sphere, with constant permeability, are recovered. Atlhough, the mathematics involved in the solution of the present problem is simple, due to the nice boundary geometry of the bodies, the final expression for the total force found in each case is quite interesting on the way it depends on the permeability relation, in particular, in the limiting cases of a porous body with solid or hollow core.     

Average steady flow toward a drain through a randomly heterogeneous porous formation

We consider the problem of steady pumping of water from a line drain on the surface of a wet ground. Unlike the classical formulation, which regards the conductivity parameter K as uniformly distributed in the domain, the problem here is solved within a stochastic framework in order to account for the irregular (random), and more realistic, spatial variability of K. Due to the linearity of the problem at stake, we focus on the derivation of the mean Green function G. This is computed by means of an asymptotic expansion.The fundamental result is an analytical (closed form) expression of G which generalizes the classical solution. Based on this, we develop an equivalent conductivity K^eq which enables one to tackle the problem similarly to the classical one. In particular, it is shown that the equivalent conductivity grows monotonically with the radial distance r from the drain, and it lies within the range K^eq(0) ≤ K^eq(r) ≤ K^eq(∞) < ∞.     

Fluctuating flow of a Maxwell fluid past a porous plate with variable suction

The problem dealing with the two-dimensional flow of an incompressible viscoelastic Maxwell fluid past an infinite porous plate is investigated. It is assumed that the suction velocity is normal to the plate and oscillates about a mean value. The external free-stream velocity varies periodically in time. The resulting differential equation subject to the relevant boundary and initial conditions is numerically solved by means of a numerical technique, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The effects of various values of the emerging parameters, e.g. the elasticity parameter, the oscillation amplitude and frequency of the external flow and the suction velocity, on the time series of velocity, especially on the boundary-layer structure near the plate, are discussed. The nature of the shear stress engendered due to the flow is also investigated.