Scaling mixing during miscible displacement in heterogeneous porous media

This paper discusses scaling of mixing during miscible flow in heterogeneous porous media. In large field systems dispersivity appears to depend on system length due to heterogeneities. Three types of scaling are discussed to investigate the heterogeneous effects. Dimensional analysis of mixing during flow through geometerically scaled heterogeneous models is illustrated using measured dispersion. Fractal analysis of mixing in statistically scaled heterogeneous porous media is discussed. Analog scaling of pressure transients in heterogeneous porous media is suggested as an in-situ method of estimating dispersion.Notation L Length - M mass - t time, (1) indicates dimensionless - a dispersivity (L) - V local velocity (L/t) - c concentration (l). - v velocity (L/t) - C₁ fluid compressibility (Lt²/M) - v time averaged velocity (LJt) - D dispersion VA) - W width (L) - D fractional dimension (1) - x coordinate (L) - d Euclidean dimension (1) - Y Y=In \-k (l) - \-d average particle size (L) - y coordinate (L) - g acceleration due to gravity (L/t2) - _c fractal cutoff (L) - \-k average permeability (L2) - viscosity (LM/t) - L length (L) - porosity (1) - L correlation scale (1/L) - density (N/L³) - N Number of sites (l) - ² variance (dimension depends on variable) - p pressure (W/t²L) - spectral exponent (l) - [R] randomnumber (1) - r radius (L) - t time (t)     

Numerical simulation of shear-thinning flow problems in mixing vessels

In many applications of chemical and process engineering numerous important flow situations appear, in which the shear-thinning properties of the fluid dominate the normal stress effects and time-dependent elastic effects. A mixed finite element method for steady incompressible flow is presented taking as a basis the model of a generalized Newtonian fluid. An augmented Lagrangian functional is constructed corresponding to the equations of motion and the continuity constraint together with appropriate boundary conditions. The treatment of the resulting nonlinear system of equations by the Newton-Raphson scheme is made for general models of the viscosity function. The linear subproblems are solved by Uzawa's algorithm. The method is applied to the numerical simulation of various mixing problems in cylindrical unbaffled vessels. The computations were performed for a real polymer liquid (CMC in water), where the viscosity data were found experimentally and were fitted to a suitable mathematical model. Several numerical results are discussed and as far as possible compared with experimental data.This work was supported by Deutsche Forschungsgemeinschaft.     

Nonlinear dynamics of multicomponent three-dimensional systems of noncanonical form

Kiev Polytechnic Institute. Translated from Prikladnaya Mekhanika, Vol. 24, No. 7, pp. 95–100, July, 1988.     

Two-dimensional problems of displacement in the case of nonlinear filtration laws

A method is proposed for calculating the multidimensional displacement of oil possessing anomalous properties from productive strata by water. Numerical solutions are obtained on the basis of the most generally accepted models of non-Newtonian oil within the framework of the theory of two-phase filtration of immiscible liquids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–75, September–October, 1976.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Slip boundary conditions on a catalytic surface in a multicomponent gas flow

The boundary conditions for the velocity slip and temperature and concentration jumps on the surface of a body in a rarefied multicomponent gas flow are obtained. The mathematical treatment is given in detail because of the need to examine critically some previous results which disagree with each other in spite of the fact that the initial premises and the methods of solution were the same. The results of this study, which are given in a convenient form, represent the boundary conditions for both the simplified and the complete Navier-Stokes equations in problems of hypersonic rarefied gas flow past bodies with a catalytically active surface.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–168, January–February, 1996.     

试论支座处的线位移对结点线位移数的影响

针对用换铰法确定位移法中独立的未知结点线位移时遇到的问题,提出了换铰时对定向支座的处理方法,论证了链杆支座、定向支座中的支杆相对于被支承的杆件轴线处于不同方位时支座处的线位移对位移法中独立的未知结点线位移的影响,总结出了用换铰法确定位移法中独立的未知结点线位移的简单计算公式.     

Variation in stress-intensity factor and back-surface displacement for surface cracks

Knowledge of the magnitude and variation in the stress-intensity factor (SIF) around the perimeter of a surface crack is essential for an accurate analysis of a flawed component. SIFs for surface flaws of various semi-elliptical geometries were analytically determined. Three-dimensional linear-elastic finite-element analysis was performed to determine the maximum SIF for bending and tension for each of 12 crack geometries which represent deep surface flaws in finite-thickness plates. Experimental verification of one of the crack geometries was performed. Interferometry techniques were used to determine the actual variation in SIF along the curve crack front due to bending. In addition to the SIF calculations, physical characteristics are noted as observed in the analytical and experimental evaluations.     

A noninvasive method for the measurement of flow-induced surface displacement of a compliant surface

A noninvasive optical method is described which allows the measurement of the vertical component of the instantaneous displacement of a surface at one or more points. The method has been used to study the motion of a passive compliant layer responding to the random forcing of a fully developed turbulent boundary layer. However, in principle, the measurement technique described here can be used equally well with any surface capable of scattering light and to which optical access can be gained. The technique relies on the use of electro-optic position-sensitive detectors; this type of transducer produces changes in current which are linearly proportional to the displacement of a spot of light imaged onto the active area of the detector. The system can resolve displacements as small as 2 m for a point 1.8 mm in diameter; the final output signal of the system is found to be linear for displacements up to 200 m, and the overall frequency response is from DC to greater than 1 kHz. As an example of the use of the system, results detailing measurements obtained at both one and two points simultaneously are presented.List of symbols C _t elastic transverse wave speed = (G/)^1/2 - d ⁺ spot diameter normalized by viscous length scale - G frequency average of G() - G() shear storage modulus - G() shear loss modulus - l. viscous length scale = v/u _* - N total number of sampled data values - r separation vector for 2-point measurements = (, ) - rms root-mean-square value - R momentum thickness Reynolds number = U _t8/v - t time - u (y) mean streamwise component of velocity in boundary layer - u _* friction velocity = (t _w/)^1/2 - U free-stream velocity - x, y, z longitudinal, normal and spanwise directions - y _o undisturbed surface position - vertical component of compliant surface displacement - ₉₉ boundary layer thickness for which u(y) = 0.99 U _t8 - _l viscous sublayer thickness 5 l _* - frequency average of G()/ - boundary layer momentum thicknes = - fluid dynamic viscosity - v fluid kinematic viscosity = / - , longitudinal, spanwise components of separation vector r - fluid density - time delay - _w wall shear stress     

Stability of two-layer systems with respect to convective mixing

The occurrence and development of convection in a two-layer system heated below has been investigated [1–5] under the assumption that the interface of the fluids is horizontal and is not subject to deformations. However, this assumption may not be satisfied if the surface tension on the interface is small and the fluids have either nearly equal densities or the heavier fluid is situated at the top. In the present paper, an attempt is made to study the convection regimes in a two-layer system with deformation of the interface when there is heating from below or above. The simultaneous influence of the convective and Rayleigh-Taylor instability mechanisms is taken into account; the surface tension on the interface is assumed to be infinitesimally small, and thermocapillary effects are ignored. A two-fluid variant of the method of markers and cells [6–9] is used for the numerical solution of the convection equations. A diagram of the regimes is constructed. It is shown that depending on the values of the parameters the system either preserves its two-layer structure, or the development of the conveetive motion leads to the breakup of the interface and complete mixing of the fluids.     

The measurement of flow-induced surface displacement on a compliant surface by optical holographic interferometry

Flow-induced surface displacements that form on a singlelayer passive isotropic viscoelastic compliant surface as a result of the interaction with a turbulent boundary layer are measured by non-intrusive optical holographic interferometry in connection with an interactive fringe-processing system. The purpose for developing this method is to obtain the instantaneous topographic features of a whole field of the displacements of the compliant surface. Information about dimensions of the foot prints of the turbulence on the compliant surface are obtained in the form of line contours and isometric phase maps. These experimental data are essential in order to determine statistical measures of the random topography of the compliant surface. Furthermore, by coupling with the simultaneous measurements of the turbulence field, the physics of the alteration of the turbulent boundary layer by the undulating surface can then be better understood.List of symbols A light amplitude - A ₀ light amplitude of object beam - A _r light amplitude of reference beam - _c light speed - d _e diameter of laser beam - d _p diameter of pinhole - d _s spatial frequency - E exposure energy - f focal length - I _in light intensity at photographic plate - I ₀ light intensity of object beam - I _r light intensity of reference beam - k wave number, 2 / - L _in light amplitude records on the emulsion surface - L ₀ light ray of object beam - L _r light ray of reference beam - R Reynolds number based on momentum thickness, V / - t time - t _e exposure time - T ₀ uniform background light transmittance - T _re reconstructed light through hologram - T _t amplitude transmittance of hologram - V freestream flow speed - radian frequency, k c - x distance from the leading edge of the flat plate - z position - slope - optical path difference - phase - ₀ phase of object beam - _r phase of reference beam - wavelength - momentum thickness - ₀ angle between the object beam with respect to the normal of the photographic plate - _r angle between the reference beam with respect to the normal of the photographic plate - kinematic viscosity of water     

Component desorption effect on the material catalyticity in high-temperature multicomponent gases

The distinctive features of the formation of the catalyticity of materials with respect to atom recombination on the material surface are investigated for mixtures of different high-temperature gases under conditions of hypersonic atmospheric flight or bench setups. It is shown that in general the catalyticity constants (heterogenous recombination probabilities) of individual components determined experimentally in dissociated flows of “pure” gases are improperly used for calculating the heat fluxes to material surfaces in multicomponent gas flows, owing to differences in the occupation of the surface by atoms in pure gases and mixtures. This effect must be taken into account in interpreting the experimental data which so far have been the only source of information on material catalyticity in gas mixtures. Otherwise, the results of calculations of the heat transfer to hypersonic flight vehicles could turn out to be invalid. Examples of the possible effect of ignoring this factor on the calculated heat fluxes are presented.     

Automated analysis of holointerferograms for the determination of surface displacement

A method for using an image digitizing system with a minicomputer to automatically determine surface-displacement fields by evaluating holographic fringe patterns is presented. The problem of determining the sign of the displacement vector as well as the need to produce a monotonic phase change across the surface under observation is solved by utilizing the method of carrier fringes. A set of programs has been written so that the camera/computer system can view the pattern of deformation-modulated carrier fringes and, with a knowledge of the initial carrier-fringe pattern, draw a profile plot of the deformed surface. The procedure is exemplified with a study of a centrally loaded disk.Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14.     

Measurement of surface displacement normal to the line of sight

Two methods for measuring the displacement of a surface normal to the line of sight are described and experimental results presented. The displacement in either case may consist of a simple lateral or rotational shift of the body in total, or it may consist of small local displacements due to some type of mechanical stress. The displacement is displayed as a system of interference fringes over the image of the surface. The first method permits observation of the displacement fringes in real time; or, if a double exposure is recorded, the fringes can be observed directly on the doubly exposed photograph in white light. The second method requires spatial filtering of the image after recording a double exposure of the surface before and after displacement. However it permits selection of the component of displacement in any direction normal to the line of sight with a continuously variable sensitivity from the single (doubly exposed) transparency. Both methods are noncontacting, use only a single laser beam for illuminating the object and require no previously constructed grids or rulings.     

Inverse problems of multiphase displacement under conditions of interphase mass transfer

Inverse problems of determining the relative phase permeabilities and flow distribution function from the multiphase displacement data under conditions of interphase mass transfer are solved. A method of identifying the phase equilibrium model of a multicomponent system in a porous medium from the displacement data is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 78–87, July–August, 1988.     

Free surface problems in rheological fluid mechanics

Summary In this paper we review our work on the theory of domain perturbations of the rest state of a viscoelastic fluid and its applications to the science of rheometry. To explain the principle behind the domain perturbation analysis we begin the paper with a discussion of a model problem, free of rheological complications, in which the principles involved may be clearly demonstrated. In Chapter two we formulate the analysis for rheological problems and we present some new, previously unpublished, simplifications of the theory. The canonical forms for stress tensors which perturb the state of rest are given in Chapter three. Chapter four is devoted to the problem of steady rod climbing. There we discuss the main physical features of the motion, the phenomenon of the critical radius, the effects of surface tension and temperature, secondary motions, the applicability of theory and experiment for rheological measurements, and an interesting new normal stress anplifier. Unsteady problems of rod climbing are considered in Chapter five. We review recent results on the free surface induced by torsional oscillations of a rod, and we show how to use these results to find the form of the material functions in the canonical forms of the stress. The breathing instability of steady axisymmetric rod climbing (aHopf bifurcation) and the flower instability of time-periodic climb induced by the oscillating rod (a symmetry-breakingPoincaré bifurcation) are also described in Chapter five. In Chapter six we consider other free surface problems: the free surface on a fluid between oscillating planes, die swell and edge effects in rheometers.

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Zusammenfassung Es wird ein Überblick über die Untersuchungen der Verfasser zum Thema Gebietsstörungen des Ruhezustandes (domain perturbations of the rest state) einer viskoelastischen Flüssigkeit und ihre Anwendung auf die Rheometrie gegeben. Zur Erläuterung der Grundlagen der Gebietsstörungs-Analyse wird in Kap. 1 die Diskussion eines Modellproblems vorangestellt, das keinerlei rheologische Komplikationen enthält, so daß die betreffenden Prinzipien in einfacher Weise erläutert werden können. In Kap. 2 wird die Analyse auf rheologische Probleme angewendet, und es werden einige bisher unveröffentlichte Vereinfachungen der Theorie vorgestellt. In Kap. 3 werden die kanonischen Formen der die Störung des Ruhezustandes hervorrufenden Spannungstensoren angegeben. In Kap. 4 wird das Problem des Hochkletterns einer Flüssigkeit am stationär rotierenden Zylinder behandelt. Dabei werden die Hauptmerkmale der Bewegung, das Phänomen des kritischen Radius, die Wirkung von Oberflächenspannung und Temperatur, Sekundärbewegungen, die Anwendbarkeit von Theorie und Experiment auf rheologische Messungen, sowie ein interessanter neuer Normalspannungs-Verstärker diskutiert. In Kap. 5 wird das Problem des Hochkletterns bei instationärer Bewegung des Zylinders betrachtet. Es werden neuere Ergebnisse über die durch eine Torsionsschwingung erzeugte freie Oberfläche referiert, und es wird gezeigt, wie diese Ergebnisse zur Bestimmung der Stoff-Funktionen in der kanonischen Darstellung der Spannungen verwendet werden können. Die atmende Instabilität beim stationären Hochklettern (eine Verzweigungslösung vomHopfschen Typ) und die blütenförmige Instabilität beim zeitlich-periodischen Hochklettern infolge der Zylinder-Oszillation (eine die Symmetrie brechende Verzweigungslösung vomPoincaréschen Typ) werden hier ebenfalls beschrieben. In Kap. 6 schließlich werden weitere an freien Oberflächen auftretende Probleme behandelt: nämlich die freie Oberfläche einer Flüssigkeit zwischen oszillierenden Platten, die Strahlaufweitung und die Randeffekte in Rheometern.

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With 13 figures