Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V/V, volume average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Perturbation analysis for periodic heat transfer in radiating fins

A perturbation analysis is presented for periodic heat transfer in radiating fins of uniform thickness. The base temperature is assumed to oscillate around a mean value. The perturbation expansion is carried out in terms of dimensionless amplitude of the base temperature oscillation. The zero-order problem which is nonlinear, and corresponds to the steady state fin behaviour, is solved by quasilinearization. A method of complex combination is used to reduce both the first and the second order problems to two, coupled linear boundary value problems which are subsequently solved by a noniterative numerical scheme. The second-order term is composed of an oscillatory component with twice the frequency of base temperature oscillation and a time-independent term which causes a net change in the steady state values of temperature and heat transfer rate. Within the range of parameters used, the net effect is to decrease the mean temperature and increase the mean heat transfer rate. This is in constrast to the linear case of convecting fins where the mean values are unaffected by base temperature oscillations. Detailed numerical results are presented illustrating the effects of fin parameter N and dimensionless frequency B on temperature distribution, heat transfer rate, and time-average fin efficiency. The time-average fin efficiency is found to reduce significantly at low N and high B.

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Störungsanalyse für periodische Wärmeübertragung an Strahlungsrippen

Zusammenfassung Eine Störungsanalyse wird für periodische Wärmeübertragung in Strahlungsrippen gleicher Dicke vorgelegt. Die Fußtemperatur wird als um einen Mittelwert schwingend angenommen. Die Störungsentwicklung wird in Termen einer dimensionslosen Amplitude e dieser Schwingung angesetzt. Das Problem nullter Ordnung, das nichtlinear ist und dem stationären Verhalten der Rippe entspricht, wird durch Quasilinearisierung gelöst. Eine Methode der komplexen Kombination wird angewandt, um die Probleme erster und zweiter Ordnung auf zwei gekoppelte Grenzwertprobleme zu reduzieren, die nacheinander nach einem nichtiterativen Schema gelöst werden. Der Term zweiter Ordnung besteht aus einer Schwingungskomponente mit der doppelten Frequenz der Schwingung der Fußtemperatur und einem zeitunabhängigen Term, der eine Nettoänderung der stationären Werte der Temperatur und der Wärmeübertragung verursacht. Im verwendeten Bereich der Parameter tritt eine Abnahme der mittleren Temperatur und eine Zunahme der mittleren Wärmeübertragung auf. Das steht im Gegensatz zum linearen Fall der Konvektionsrippe, bei dem die Mittelwerte durch Schwingungen der Fußtemperatur nicht beeinflußt werden. Detaillierte numerische Ergebnisse zeigen die Einflüsse des Rippenparameters N und der dimensionslosen Frequenz B auf Temperatur Verteilung, Wärmeübertragung und zeitliches Mittel des Rippengütegrades. Dieses zeitliche Mittel nimmt merklich ab bei kleinem N und hohem B.

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Nomenclature b fin thickness - B dimensionless frequency, L²/ - E emissivity - f₀, f₁ functions of X - g₀, g₁, g₂ functions of X - h₀, h₁, h₂ functions of X - k thermal conductivity - L fin Length - N fin parameter, 2EL²T_bm/bk - q heat transfer rate - Q dimensionless heat transfer rate, qL/kbT_bm - t time - T temperature - T_b fin base temperature - T_S effective sink temperature - T_bm mean fin base temperature - x axial distance - X dimensionless axial distance, x/L - dimensionless amplitude of base temperature (s. Eq.2) - thermal diffusivity - instantaneous fin efficiency - time-average fin efficiency - _ss steady state fin efficiency - dimensionless temperature, T/T_bm - ₀ zero-order approximation - ₁ first-order approximation - ₂ second-order approximation - _2s steady component of ₂ - , ₁, ₂ constants - complex function of X - ₁ real part of - ₂ imaginary part of - complex function of X - ₁ real part of Y - ₂ imaginary part of - dimensionless time, t/L² - frequency of base temperature oscillation     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Face-Centered and Volume-Centered Discrete Analogs of the Exterior Differential Equations Governing Porous Medium Flow I: Theory

This paper presents a physics-oriented approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcys law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant translation rules. In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the block-centered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.     

Linear shear flow past a porous particle

Linear shear flow past a porous spherical particle is studied using a generalized boundary condition proposed by Jones. The torque on a porous sphere rotating in a quiescent fluid is calculated. Streamlines patterns are illustrated for the case of a particle freely suspended in a simple shear flow. These patterns are shown to differ significantly from those associated with an impermeable rigid sphere. Finally, an expression for the effective viscosity of a dilute suspension of porous spherical particles is obtained.Nomenclature A, B dimensionless flow parameter - a radius of the porous sphere - C, E, F constants of integration - d shear strength - d constant rate of deformation of ambient field - e rate of strain tensor - f, g functions of distance - k permeability of the porous medium - n unit normal vector - p pressure - p unit vector - Q coefficient of spherical harmonic - q filter velocity within the porous medium - r polar spherical coordinate - S _p surface of porous particle - S, T, T* coefficients of spherical harmonics - T torque exerted on the particle - u fluid velocity vector - x cartesian coordinates - dimensionless constant - , polar spherical coordinates - dimensionless flow parameter - viscosity of the fluid - stress tensor - rotational velocity of the particle - rotational velocity of the ambient field.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

Effect of Streamwise Streaky Structures on Turbulization of a Circular Jet

Experimental investigations of the influence of streamwise streaky structures on turbulization of a circular laminar jet are described. The qualitative characteristics of jet evolution are studied by smoke visualization of the flow pattern in the jet and by filming the transverse and longitudinal sections of the jet illuminated by the laser sheet with image stroboscopy. It is shown that the streaky structures can be generated directly at the nozzle exit, and their interaction with the Kelvin–Helmholtz ring vortices leads to emergence of azimuthal beams ( structures) by a mechanism similar to threedimensional distortion of the twodimensional Tollmien–Schlichting wave at the nonlinear stage of the classical transition in nearwall flows. The effect of the jetexhaustion velocity and acoustic action on jet turbulization is considered.     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Stationary combustion modes with allowance for heat losses

An attempt is made to investigate the number of possible stationary combustion modes in a continuous-flow semi-infinite pipe with allowance for heat losses through the walls. Cases of a zero-order reaction in the reaction mixture or similarity of the concentration and temperature fields are considered. The equations are averaged with respect to the transverse coordinate . Within the framework of these approximations it is found that the number of stationary combustion modes is determined by the roots _n of some function. The roots _2k correspond only to trivial unstable solutions. The roots _2k–1 correspond to modes possible within broad regions of variation of the parameters characterizing the temperature of the mixture, the mixture feed rate, and the rate of heat removal. These regions intersect, forming zones where several stationary modes coexist. In these zones, apart from monotonic solutions there may also be solutions that initially make several oscillations. It is shown that the latter are obviously unstable and, in the last analysis, lead to one of the monotonic modes. The common case of not more than three roots is examined in detail.If the heat release function can change sign, then a similar picture is also observed in the absence of heat losses through the walls (the roots _2k–1 and _2k may change roles). In this case it is no longer necessary to average the equations with respect to , since there will not be any corresponding derivatives.     

Mixed convection flow in narrow vertical ducts

The mixed convection flow in a vertical duct is analysed under the assumption that , the ratio of the duct width to the length over which the wall is heated, is small. It is assumed that a fully developed Poiseuille flow has already been set up in the duct before heat from the wall causes this to be changed by the action of the buoyancy forces, as measured by a buoyancy parameter . An analytical solution is derived for the case when the Reynolds numberRe, based on the duct width, is of 0 (1). This is extended to the case whenRe is 0 (^–1) by numerical integrations of the governing equations for a range of values of representing both aiding and opposing flows. The limiting cases, || 1 andR=Re of 0 (1), andR and both large, with of 0 (R ^1/3) are considered further. Finally, the free convection limit, large with R of 0 (1), is discussed.

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Mischkonvektion in engen senkrechten Rohren

Zusammenfassung Mischkonvektion in einem senkrechten Rohr wird unter der Voraussetzung untersucht, daß das Verhältnis der Rohrbreite zur Länge, über welche die Wand beheizt wird, klein ist. Es wird angenommen, daß sich bereits eine voll entwickelte Poiseuille-Strömung in dem Rohr eingestellt hat, bevor Antriebskräfte, gemessen mit dem Auftriebsparameter , aufgrund der Wandbeheizung die Strömung verändern. Es wird eine analytische Lösung für den Fall erhalten, daß die mit der Rohrbreite als charakteristische Länge gebildete Reynolds-ZahlRe konstant ist. Dies wird mittels einer numerischen Integration der wichtigsten Gleichungen auf den FallRe =f (^–1) sowohl für Gleich- als auch für Gegenstrom ausgedehnt. Weiterhin werden die beiden Grenzfälle betrachtet, wenn || 1 undR=Re konstant ist, sowieR und beide groß mit proportionalR ^1/3. Schließlich wird der Grenzfall der freien Konvektion, großes mit konstantem R, diskutiert.

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Nomenclature g acceleration due to gravity - Gr Grashof number - G modified Grashof number - h duct width - l length of the heated section of the duct wall - p pressure - Pr Prandtl number - Q flow rate through the duct - Q ₀ heat transfer on the wally=0 - Q ₁ heat transfer on the wally=1 - Re Reynolds number - R modified Reynolds number - T temperature of the fluid - T ₀ ambient temperature - T applied temperature difference - u, velocity component in thex-direction - v, velocity component in they-direction - x, co-ordinate measuring distance along the duct - y, co-ordinate measuring distance across the duct - buoyancy parameter - ₀ modified buoyancy parameter, ₀=R ^–1/3 - coefficient of thermal expansion - ratio of duct width to heated length, =h/l - (non-dimensional) temperature - _w applied temperature on the wally=0 - kinematic viscosity - density of the fluid - ₀ shear stress on the wally=0 - ₁ shear stress on the wally=1 - stream function     

Calculation of supersonic equilibrium-dissociating air-xenon mixture flow past a blunt body

The flow of an equilibrium-reacting multicomponent three-element air-xenon mixture is numerically investigated. The effect of multicomponent diffusion on the convective heat transfer to the body surface is examined. The dependence of the convective heat transfer to the body surface and the total shock-layer spectral radiation flux P_m on the xenon concentration is obtained. A comparison of the calculated data for P_m and the experimental data of [2] gives good agreement. A simple approximation for the convective heat flux at the stagnation point as a function of xenon concentration is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–164, July–August, 1991.In conclusion the authors wish to thank I. A. Sokolova for supplying data on the resistance coefficients of the various mixtures and S. A. Yunitskii for discussing the numerical method.     

Natural convection heat transfer through an enclosed horizontal layer of supercritical carbon dioxide

In natural convection heat transfer through a thin horizontal layer of carbon dioxide, maxima in the equivalent thermal conductivities are obtained in the vicinity of the respective pseudocritical temperatures at pressures of 75.8, 89.6 and 103.4 bar. The maxima are the more pronounced, the closer the critical point is approached.Comparison of experimental results with Nusselt equations shows good agreement except for the immediate vicinity of the pseudocritical temperature.In visual observations a distinct change in flow structure appears in the immediate vicinity of the pseudocritical temperature. A steady state polygon pattern and a boiling-like action could not be observed in this geometry.

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Zusammenfassung Beim Wärmetransport durch freie Konvektion in einer dünnen waagerechten Schicht von Kohlendioxid ergaben sich Maxima der scheinbaren Wärmeleitfähigkeit in der Nähe der pseudokritischen Temperaturen bei Drükken von 75,8, 89,6 und 103,4 bar. Die Maxima sind um so ausgeprägter, je mehr man sich dem kritischen Punkt nähert.Ein Vergleich der Versuchsergebnisse mit Nusseltbeziehungen ergibt gute Übereinstimmung außer in unmittelbarer Umgebung der pseudokritischen Temperatur. Direkte Beobachtungen der Konvektionsmuster zeigen in unmittelbarer Umgebung der pseudokritischen Temperatur eine deutliche Strukturänderung. Ein stationäres Zellmuster und siedeähnliche Vorgänge konnten in dieser Anordnung nicht beobachtet werden.

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Nomenclature A area of the heating or cooling plate - C constant in the correlation - g acceleration of gravity - h heat transfer coefficient - k thermal conductivity of fluid in the gap - k _e equivalent thermal conductivity - m, n exponents of dimensionless numbers - q heat flux - T _C,PC absolute temperature; critical C, pseudocritical PC - Gr Grashof numberg ( _h– _c) ³/ ² - Nu Nusselt numberh/k - Pr Prandtl number/ - thermal diffusivity - coefficient of volume expansion - width of gap - _c,h temperature of cooling (c)-, heating (h)-plate - _m arithmetic mean temperature ( _c+ _h)/2 - kinematic viscosity - _c,h fluid density at the temperature of the cooling (c)- or heating (h)-plate - heat flow rate through the gap     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Heat transfer for laminar flow in internally finned pipes with different fin heights and uniform wall temperature

An analysis is presented for fully developed laminar convective heat transfer in a pipe provided with internal longitudinal fins, and with uniform outside wall temperature. The fins are arranged in two groups of different heights. The governing equations have been solved numerically to obtain the velocity and temperature distributions. The results obtained for different pipe-fins geometries show that the fin heights affect greatly flow and heat transfer characteristics. Reducing the height of one fin group decreases the friction coefficient significantly. At the same time Nusselt number decreases inappreciably so that such reduction is justified. Thus, the use of different fin heights in internally finned pipes enables the enhancement of heat transfer at reasonably low friction coefficient.Nomenclature A_f dimensionless flow area of the finned pipe, Eq. (8) - a_f flow area of the finned pipe - C_p specific heat at constant pressure - f coefficient of friction, Eq. (12) - H₁, H₂ dimensionless fin height h₁/r_o h₂/r_o - h₁, h₂ fin heights - average heat transfer coefficient at solid-fluid interface - KR fin conductance parameter, k_s/k_f - k_f thermal conductivity of fluid - k_s thermal conductivity of fin - l pipe length - mass flow rate - N number of fins - Nu Nusselt number, Eqs. (15) and (16) - P pressure - Q total heat transfer rate at solid fluid interface - Q_f1, Q_f2 heat transfer rate at fin surface - q_w average heat flux at pipe-wall, Q/(2 r_ol) - R dimensionless radial coordinate r/r_o - Re Reynolds Number, Eq. (13) - r radial coordinate - r_o radius of pipe - r₁, r₂ radii of fin tips - T temperature - T_b bulk temperature - U dimensionless velocity, Eq. (2) - U_b dimensionless bulk velocity - u_z axial velocity - z axial coordinate - angle between the flanks of two adjacent fins - half the angle subtended by a fin - angle between the center-lines of two adjacent fins - angular coordinate - dynamic viscosity - density - dimensionless temperature, Eq. (6) - _b dimensionless bulk temperature     

Spectral and correlation characteristics of the turbulent boundary layer on an extended flexible cylinder

In marine geophysical seismological prospecting extensive use is made of towed receiving systems consisting of extended flexible cylinders containing acoustic sensors over which the water flows in the longitudinal direction. The boundary layer pressure fluctuations on the cylinder surface are picked up by the sensors as hydrodynamic noise. This paper is concerned with the study of the energy and spacetime characteristics of the pressure fluctuations in the turbulent boundary layer on an extended flexible cylinder in a longitudinal flow. The pressure fluctuations on the cylinder surface have been investigated experimentally for Re_X=(2–4)·10⁷, a dimensionless diameter of the pressure fluctuation sensors d _p ⁺ =d_pu^*/=500, where d_p is the sensor diameter, u^* the dynamic viscosity, and the viscosity coefficient, and frequencies 0.02311.259 (=^*/U). The spectral and correlation characteristics of the pressure fluctuations on the surface of the flexible cylinder are found to differ from the corresponding characteristics for a rigid cylinder [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i aza, No, 5, pp. 49–54, September–October, 1989.     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.