A filtration problem in a porous medium with variable permeability

Sunto Un problema di frontiera libera (moto di un fluido attraverso una diga in terra non omogenea) è ricondotto allo studio di una disequazione variazionale. Si danno condizioni sul coefficiente di permeabilità sufficienti ad assicurare che la parte bagnata della diga è un sottografico.

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Entrata in Redazione il 30 giugno 1976.

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This paper is partially supported by National Science Foundation Grant MPS 72-04959 A02, and by C.N.R. of Italy through L.A.N. of Pavia.     

An exact solution for flow in a porous pipe

Summary An exact solution of the Navier-Stokes equations for flow in a porous pipe is presented. This solution allows the suction or injection at the wall to vary with axial distance and will provide new insight into flows through porous pipes.

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Resumé Une solution exacte d'équation de Navier-Stokes est présentée pour l'écoulement d'un liquide visqueux dans un tube perméable. Ce liquide est aspiré ou injecté avec une vélocité variable et la solution donne une nouvelle optique quant aux tubes poreuses.

     

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.     

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.     

Nanofluid flow and heat transfer in a Brinkman porous channel with variable porosity

The problem of forced convection in a channel filled with a nanofluidsaturated porous medium is investigated, numerically. A finite difference Computational Fluid Dynamics (CFD) model with structured uniform grid system is employed to solve the momentum and energy equations. In modeling flow in the channel, the effects of flow inertia, variable porosity and Brinkman friction are taken into account. Studies are carried out for different nanoparticles with different volume fractions in the range 0%-4% and different nanoparticle diameters. Comparison made between our numerical and semi analytical Differential Transform Method (DTM) results with those in previous published research is found to be appropriate. Results show that increasing either nanoparticls volume fraction or pressure gradient parameter improves heat transfer. Further, for large quantities of nanoparticle concentration and pressure gradient, the channeling phenomenon is intensified.     

Steady flow of a viscous liquid in a porous elliptic tube

Flow of a viscous-liquid in a porous tube of elliptic cross-section is studied using the generalized momentum equation. As a particular case, flow of the liquid in a tube of a circular cross-section is obtained. It is observed that the classical Darcian effect is realized only in a core very near to the axis of the tube while the non-Darcian phenomenon is felt predominantly near the boundary of the tube.     

Exact solution for flow in a porous pipe with unsteady wall suction and/or injection

This paper presents an extension of the exact solution of the steady laminar axisymmetric flow in a straight pipe of circular cross section with porous wall, given by R.M. Terrill, to the case of unsteady wall injection and/or suction. The cases of the pulsating parabolic profile and of the developed pulsating flow are investigated as examples. The pulsating flow in porous ducts has many applications in biomedical engineering and in other engineering areas.     

On the flow of a dusty viscous liquid through a circular pipe

Analytical solution of the flow problem of a dusty viscous liquid through a circular pipe in case of axial symmetry is obtained when pressure gradient varies harmonically with time. It is found that the effect of the fine dust is to make the velocity of sedimentation zero and when dust is sufficiently coarse, the effect of the dust is equivalent to an extra frictional force proportional to the fluid velocity.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

The slow stationary flow of a viscous liquid about porous spheroids

The slow stationary flow of a viscous liquid about porous spheroids is examined. The solution describing the Stokes stream function is found to be a one-parameter family of surfaces and for special values of the parameter we recover the flow past impermeable spheroids. The ranges of the parameter are determined and the case of the porous sphere as well as that of a porous disc are deduced as limiting case.     

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.     

The non-steady flow in a magnetohydrodynamic shock-wave generator of finite length

Zusammenfassung Durch geeignete Transformationen der diese Vorgänge beschreibenden Grundgleichungen in differentieller Form, Impulssatz, Kontinuitätsgleichung und Energiesatz, können sogenannte Verträglichkeitsbedingungen erhalten werden. Diese gelten entlang den Charakteristiken beziehungsweise den Lebenslinien. Letzteres System von gewöhnlichen, nichtlinearen Differentialgleichungen darf linearisiert werden, sofern die elektromagnetischen Wechselwirkungen mit der Strömung schwach sind. Elementare Integrationen führen dann zu relativ übersichtlichen und einfachen Resultaten. Insbesondere sind die unbekannten Funktionen an drei ausgezeichneten Stellen des -t-Feldes explizite zu bestimmen und die Funktionswerte im übrigen Strömungsfeld lassen sich leicht graphisch ermitteln. Die stromaufwärts reflektierten Druckschwankungen, verursacht durch die elektromagnetische Wechselwirkung im Generator, können nach einer von den Betriebsbedingungen abhängigen Anlaufstrecke zu einem Stoss auflaufen. Die exakte Bestimmung dieser Anlaufstrecke ist im Rahmen der Linearisierung auch möglich. Es ist erstaunlich, wie rasch, gemessen an der Zeit, die die Stosswelle zum Durchlaufen des Generators braucht, die Strömung im Generator stationär wird.     

Multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall

In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed.     

An equivalent pipe network model for free surface flow in porous media

Free surface flow analysis in porous media is challenging in many practical applications with strong non-linearity. An equivalent pipe network model is proposed for the simulation and evaluation of free surface flow in porous media. On the basis of representative elementary volume with homogeneous pore-scale patterns, the pore space of the homogeneous isotropic porous media is conceptualized as a collection of capillary tubes. According to Hagen-Poiseulle's law and flux equivalence principle, equivalent hydraulic parameters and unified governing formulations for the pipe network model are deduced. The two-dimensional free surface flow problem is reduced to a one-dimensional problem of pipe networks and a one-dimensional procedure based on the finite element method is then developed by introducing a continuous penalized Heaviside function. The proposed equivalent pipe network model is verified with results from numerical solutions and laboratory-measured data available in the literature, and good agreements are obtained. The proposed equivalent pipe network model is shown to be effective in analyzing the free surface flow in porous media. The numerical results also indicate that the proposed equivalent pipe network model has weak sensitivity of the mesh size and penalty parameters.     

Laminar flow through parallel and uniformly porous walls of different permeability

Résumé Le problème de l'écoulement constant d'un liquide visqueux laminaire incompressible à travers un conduit à deux dimensions, le liquide étant aspiré ou injecté à travers ses parois uniformément poreuses, est généralisé dans le cas où les vitesses le long des parois sont différentes. L'écoulement qui s'ensuit satisfait une équation différentielle non linéaire ordinaire d'un nombre Reynolds d'absorbtion, et cette équation est résolue par la technique de perturbation. Cette équation différentielle est également résolue par le calcul, et l'on vérifie l'exactitude de la solution de série. Des expressions générales pour les distributions de vitesse, les coefficients de pression et la friction de surfaces sont données au moyen du nombre Reynolds. Le graphique de friction des surfaces indique que l'écoulement à travers un conduit ayant des parois poreuses de perméabilité inégale est régulier et stable comme dans le cas d'un écoulement symétrique.     

Impulsive displacement of liquid in a pipe

The solution of a viscous liquid, originally at rest in a pipe, which is set impulsively into motion by a pressure gradient is well known. Analyzing this solution shows that the displacement time – the time it takes to completely displace the liquid originally at rest in the pipe – falls primarily within an inviscid window. Assuming this result remains essentially unchanged when the displaced and displacing liquids are different we apply an inviscid analysis to determine the displacement time as a function of the density ratio of the liquids and the pressure difference. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)