The transverse force on a spinning porous sphere at small Reynolds number

This paper deals with the problem of the flow about a spinning porous sphere moving in an incompressible viscous fluid. The effect of permeability on the flow has been studied analytically at low Reynolds number by the method of matched asymptotic expansions. It turns out that the permeability on the drag force up to 0(R) reduces the effective radius by a factor (2+K/a ²)^–1,K being the permeability associated with the porosity of the reduced radius of the sphere. Further, it is found that the permeability affects the lift forceF _L , which is orthogonal to its direction of motion. The ratio of the effective angular velocity and actual angular velocity decreases forK>0.25. ForK=0.25 there is no spinning effect on the sphere.

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Zusammenfassung Diese Arbeit behandelt die Strömung um eine rotierende poröse Kugel in einer zähen inkompressiblen Flüssigkeit. Der Einfluss der Durchlässigkeit wurde für kleine Reynolds-Zahlen mit Hilfe von angepassten asymptotischen Entwicklungen bestimmt. Es wurde gefunden, dass die Durchlässigkeit den Widerstand O(R) vermindert, und die Auftriebskraft senkrecht zur Bewegungsrichtung beeinflusst.

     

Flow past a porous approximate spherical shell

In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.     

Stokes flow inside a porous spherical shell: Stress jump boundary condition

An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkmans equation for the flow in the porous region is discussed. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used. The drag and torque are found by deriving the corresponding Faxens laws. It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient. Critical permeability is found for which drag and torque change their behavior. As a limiting case the corresponding Faxens laws for the rigid spherical shell with internal singularities has been obtained.Received: December 17, 2002; revised: February 3, 2004     

Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid

The Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall at an arbitrary position from the wall is presented using a combined analytical-numerical method. A linear slip, Basset type, boundary condition on the surface of the sphere has been used. To solve the Stokes equations for the fluid velocity field and the microrotation vector, a general solution is constructed from fundamental solutions in both cylindrical, and spherical coordinate systems. Boundary conditions are satisfied first at the plane wall by the Fourier transforms and then on the sphere surface by the collocation method. The drag acting on the sphere is evaluated with good convergence. Numerical results for the hydrodynamic drag force and wall effect with respect to the micropolarity, slip parameters and the separation distance parameter between the sphere and the wall are presented both in tabular and graphical forms. Comparisons are made between the classical fluid and micropolar fluid.      

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.

     

Global well-posedness of incompressible flow in porous media with critical diffusion in Besov spaces

In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces with 1?p?∞ by the method of modulus of continuity and Fourier localization technique.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Solution of porous medium type systems by linear approximation schemes

Summary The convergence of semi-discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type. The enthalpy formulation and variational technique are used. The semi-discretization used reduces the original parabolic P.D.E. to linear elliptic P.D.E. The algebraic correction arising from nonlinearities is treated by Newton-like iterations in finite steps. Some numerical experiments are discussed and compared with the analytical solutions.Supported by the Alexander von Humboldt-Foundation in 1989, and by SFB 123, University Heidelberg     

Fluctuating flow at small Reynolds number

Zusammenfassung Für den Widerstand von Kugeln und Kreiszylindern in periodisch schwankendem Parallelstrom werden Formeln angegeben, gültig für kleine Reynolds-Zahlen.     

Fluctuating heat transfer from a sphere at small Reynolds numbers

Zusammenfassung Es wird die Wärmeleitung von einer Kugel an einen oszillierenden Flüssigkeitsstrom untersucht. Das Problem lässt sich lösen für kleine Reynoldszahlen, wobei sowohl für die Temperaturverteilung als auch für die Wärmeübertragung eine Entwicklung nach Potenzen der Pécletzahl angegeben werden kann.     

On the uniqueness of the solution of a nonlinear filtration problem through a porous medium

The purpose of this work is to study a fluid flow through a porous medium governed by a nonlinear Darcy's law. We also impose a nonlinear semi-permeability condition on some part of the boundary of this medium. The main results are the continuity of the free boundary and the uniqueness of the solution. Received May 5, 1996 / In a revised form November 16, 1996 / Accepted December 17, 1996     

Peristatcally induced MHD slip flow in a porous medium due to a surface acoustic wavy wall

The influences of Hall current and slip condition on the MHD flow induced by sinusoidal peristaltic wavy wall in two dimensional viscous fluid through a porous medium for moderately large Reynolds number is considered on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for the outer and inner flows for various values of the Reynolds number, slip parameter, Hall and magnetic parameters are drawn. The inner and outer solutions are matched by the matching process. An interesting application of the present results to mechanical engineering may be the possibility of the fluid transportation without an external pressure.     

Weak solutions to stochastic porous media equations

A stochastic version of the porous medium equation is studied. The corresponding Kolmogorov equation is solved in a space where is an invariant measure. Then a weak solution, that is a solution in the sense of the corresponding martingale problem, is constructed.     

Slow flow past a heterogeneous porous sphere

Stokes’ flow past a heterogeneous porous sphere has been studied, adopting the boundary conditions modified by Jones (1973) for curved surfaces at the interface of the free fluid region and porous material. The porous sphere is made up ofn + 1 concentric spheres of different permeability. The results for drag experienced by the sphere has been discussed and the following cases of interest are deduced:

1. WhenK ₁=K ₂=...=K _n+1=K.
2. WhenK _i is very small for eachi.

     

The security number of lexicographic products

A subset S of vertices of a graph G is a secure set if |N [X] ∩ S| ≥ |N [X] ? S| holds for any subset X of S, where N [X] denotes the closed neighborhood of X. The minimum cardinality s(G) of a secure set in G is called the security number of G. We investigate the security number of lexicographic product graphs by defining a new concept of tightly-securable graphs. In particular we derive several exact results for different families of graphs which yield some general results.     

Numerical solution of steady-state porous flow free boundary problems

Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.