Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Generalized Rouse model for a dilute solution of clustered polymers

A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG^1/2). In our case the storage modulus becomes proportional to ^3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters.     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

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.     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Determination of molar mass polydispersity index from dynamic rheological measurements

The polydispersity of the molar mass of a material influences the behaviour of its dynamic rheological propertiesG () andG (). This is exemplified by the deviation from unity of the indexI = [2 – (d logG/d log)]/[1 – (d logG/d log)] in the terminal zone, i.e. at low frequencies. For a normal logarithmic distribution of species, a quantitative correlation may be established between the rheological properties and the parameter that is characteristic of the polydispersity for these species. This correlation can, in certain cases, be drawn from measurements in just the terminal zone or it may require measurements both in the terminal zone and at the beginning of the plateau zone. In each case, the parameter of dispersion can be determined by simple graphical or numerical methods. Finally, an example of the application of these methods to entangled liquid polymers is presented.     

Stabilization of the interface instability of moving magnetizable fluids

A study is made of the stabilization of the interface between two moving magnetizable fluids by means of an external magnetic field H={H₀ cos t, H₀ sin t, 0} of circular polarization. The fluids are assumed to be ideal, incompressible, nonconducting, and electrically neutral. An equation of motion is derived for the perturbed interface. The Borg criterion is used to obtain sufficient conditions of stability of small perturbations of the interface; these conditions relate the amplitude H₀ and the frequency of the external magnetic field to the characteristic parameters of the problem and the wave vector k. The dependence of H₀ and on the modulus of the wave vector is investigated. The obtained results are compared with the results of [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 118–122, July–August, 1980.     

Relations between steady flow and oscillatory shear measurements

Summary Results are given of a comparison between dynamic oscillatory and steady shear flow measurements with some polymer melts. Comparison of the steady shear flow viscosity,, with the absolute value of the dynamic viscosity, ¦¦, at equal values of the shear rate,q, and the circular frequency,, has shown the relation thatCox andHerz had found empirically to be substantially correct.Further, the coefficients of the normal stress differences obtained by streaming birefringence techniques have been compared with 2G () · ^{– 2} in the same range of shear rates as covered by the viscosity measurements (G is the real part of the dynamic shear modulus). Two polystyrenes with narrow molecular weight distribution showed the same shift factor along the orq axis for the normal stress coefficients with respect to 2G () · ^{– 2} and the steady shear flow viscosities with respect to the real part of the dynamic viscosity,. For two polyethylenes the results are not so conclusive owing to the smallness of the shift factor found. An empirical equation is proposed predicting the main normal stress difference from dynamic measurements only.

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Zusammenfassung Die Ergebnisse von Messungen unter erzwungenen Schwingungen und stationärer Scherströmung an einigen Polymerschmelzen werden miteinander verglichen. Der Vergleich der stationären Viskosität mit der absoluten dynamischen Viskosität ¦¦ bei gleichen Werten des Strömungsgradientenq und der Kreisfrequenz zeigt die Gültigkeit der empirischen Beziehung vonCox undHerz.Weiter wurden die Koeffizienten der Normalspannungsdifferenzen, welche durch Messung der Strömungsdoppelbrechung erhalten wurden, mit 2G() · ^–2 verglichen, und zwar wiederum bei gleichen Werten vonq und, wobeiG die Speicherkomponente des dynamischen Schubmoduls ist. Zwei Polystyrole mit enger Molekulargewichtsverteilung zeigen die gleiche Verschiebung entlang der-oderq-Achse für die Normalspannungskoeffizienten in bezug auf2G()· ^–2 und für die stationären Scherviskositäten in bezug auf den Realteil der dynamischen Viskosität. Für zwei Polyäthylene sind die Ergebnisse weniger signifikant, da die entsprechenden Verschiebungen zu klein waren. Eine empirische Beziehung zwischen den Hauptnormalspannungsdifferenzen und den dynamischen Meßwerten wird vorgeschlagen.

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Paper presented at the British Society of Rheology Conference, held at Shrivenham, from 9th–12th September, 1968.     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.     

Numerical calculation of creep compliance from dynamic data for linear viscoelastic materials

Summary Numerical formulae are given for calculation of creep compliance from the known course of the storage and loss compliance with frequency for linear viscoelastic materials. These formulae involve values of the storage compliance and/or loss compliance at frequencies which are equally spaced on a logarithmic frequency scale. The ratio between successive frequencies corresponds to a factor of two.A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tan, at the angular frequency, ₀, at which the calculation is performed. The lower this damping, the easier is the calculation of the creep compliance. This calculation involves either the value of the storage compliance at a frequency ₀ = 1/t, and the values of the loss compliance in a rather narrow frequency region around ₀; or the value of the storage compliance at frequency ₀, the value of the loss compliance at frequency ₀/2, and the derivative of the storage compliance with respect to the logarithm of frequency in a frequency region around ₀.

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Zusammenfassung Numerische Formeln werden gegeben, die die Berechnung der Kriechfunktion aus der dynamischen Nachgiebigkeit ermöglichen. In diesen Formeln treten Werte der Speicher- bzw. Verlustkomponente der dynamischen Nachgiebigkeit auf, die bei logarithmisch äquidistanten Frequenzen gemessen wurden. Das Verhältnis zweier aufeinanderfolgender Frequenzen entspricht stets einem Faktor 2.Für alle Formeln werden obere und untere Schranken für den relativen Fehler abgeleitet. Diese Schranken hängen vom Werte der Dämpfung (tan) ab, die bei der Kreisfrequenz ₀ auftritt, für die die Berechnung erfolgt. Die Berechnung der Kriechfunktion ist desto leichter, je niedriger der Wert der Dämpfung ist. Zu dieser Berechnung benötigt man entweder den Wert der Speicherkomponente der dynamischen Nachgiebigkeit bei der Kreisfrequenz ₀ = 1/t und die Werte der Verlustkomponente der dynamischen Nachgiebigkeit in einem ziemlich engen Frequenzintervall um ₀; oder den Wert der Speicherkomponente bei der Kreisfrequenz ₀, den Wert der Verlustkomponente bei der Kreisfrequenz ₀/2 und den Wert der logarithmischen Frequenzableitung der Speicherkomponente in einem Frequenzintervall um ₀.

     

Stationary modes of combustion in a fine-scale turbulent stream

The majority of models of the turbulent combustion of gases are based mainly on intuitive concepts concerning the processes occurring in the flame. The characteristics of a turbulent flame are estimated from considerations of dimensionality and similarity. A detailed review of works on turbulent combustion is given in [1]. Problems on the calculation of the combustion rate in a turbulent stream as a proper value of the equations of heat and mass transfer and of the corresponding boundary conditions have recently been raised. Here too one must rest on assumptions of a semiempirical nature, which in large measure is connected with the inadequate level of development of turbulence theory. In the present work the equation of propagation of the zone of chemical reactions in the stream is averaged statistically by analogy with studies of turbulent flows. Correct averaging is possible at scales of hydrodynamic disturbances smaller than the flame thickness (fine-scale turbulence). The temperature pulsations are related with the size of the heat flux using the theory of mixing lengths. The main influence is specific to effects arising during averaging of the heat release function. Two stationary modes, distinguished by the normal propagation velocity ₁, are isolated within the framework of the Cauchy problem with a given initial mixture temperature and zero heat flux in the burned gas. A heat conduction mode occurs with a stream velocity > ₁ and an induction mode with < ₁. An expression is found for ₁ which reflects the principal effects in the flame and which in the limit coincides with the equation of Zel'dovich and Frank-Kamenetskii for a laminar flame. In those cases when the distorting effect of the heat release function is small, the turbulence affects the combustion rate through mechanisms of intensification of transport processes.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 118–124, September–October, 1973.     

Stability of spherical Couette flow in thick layers when the inner sphere revolves

A natural generalization of cylindrical Couette flow is the flow of a viscous incompressible liquid between two concentric spheres rotating about the same axis with different angular velocities. As has often been noted, spherical Couette flow is, despite the apparent similarity, considerably more complex than cylindrical flow. It consists of differential rotation about the axis and one- or two-eddy circulation (depending on the ratio between the angular velocities of the two spheres = ₂/₁) in the meridional plane and depends significantly on the Reynolds number Re = ₁r ₁ ² and the relative thickness of the layer = (r₂-r₁)/r₁ (₁, ₂ and r₁, r₂ are the angular velocities and radii of the inner and outer spheres, respectively. The investigation of spherical Gouette flow and its stability has begun relatively recently (within the last 10 years) and has evidently been stimulated by applied problems associated with astro- and geophysics. Because of the great computational difficulties encountered in investigating such flow theoretically, experimental investigations have yielded more extensive and interesting results [1–8], although all the published results refer to the case of rotation of one inner sphere ( = 0).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–15, March–April, 1978.It remains to thank S. A. Shcherbakov for help in organizing automatic input of the signals to the BÉSM-6 computer.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [ + (cos ₁ t + cos ₂ t)] x + x³ = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the usual trigonometric functions – to study subharmonic resonances associated with orbits in 2m:1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (, , ₁, ₂) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov's overlap criterion to obtain an overview of the O() global behavior of equation (1) as a function of and ₂ with ₁, , and fixed.     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Harmonic solutions of micropolar elastodynamics

The equations of micropolar elastodynamics are considered for an unbounded continuum subjected to a body force and a body couple. These act harmonically with the same real frequency , but with individual arbitrary spatial distributions. Over a harmonic state, the displacement and microrotation are related to two radiation conditioned harmonic vectors, each acquiring three eigenvalue contributions, assuming a noncritical -frequency. Altogether, four distinct eigenvalues are admissible. If ²<2² ₀, ₀ being a frequency parameter of the continuum, two of these are real while two are purely imaginary. But if ²<2² ₀, then all admissible eigenvalues are real. Each eigenvalue contribution resolves into a series of Hankel and Bessel functions coupled to Hankel type transforms of: (i) spherical integrals which, in turn, can be expanded via spherical harmonics for the 3-dimensional problem, (ii) circular integrals for the 2-dimensional problem. Axisymmetric and spherically symmetric results are deduced in 3-dimensions. Asymptotic solutions are also established; they disclose long-range formation of radially attenuated spherical (or circular) waves propagating with, generally, anisotropic amplitudes but, invariably, isotropic eikonals.If, in the absence of a body couple, a body force acts radially in 3-dimensions with a spherically symmetric strength, then the elastic displacement behaves likewise while the microrotation vanishes identically. Another application is made to a 2-dimensional problem for a 1 × 3 source system of body force plus body couple without longitudinal variation but with magnitudes symmetric about a longitudinal axis.As approaches a certain critical frequency , dependent solely on the continuum, at least two eigenvalues approach the same value. The phenomenon is explored for a continuum consistent with ²<2² ₀ and under the hypothesis ²<2² ₀. All admissible eigenvalues are then real throughout an -neighbourhood of . Here, two associated eigenvalue contributions behave singularly. Nevertheless, their essential singularities cancel out within the relevant combination. Examination of a far-field suggests that critical frequency attainment sets off a slow instability in the 2-dimensional configuration. In the 3-dimensional configuration, however, it preserves stability and eliminates radial attenuation; an exact solution is formulated for this case.     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.