Taylor-Couette stability analysis for a Doi-Edwards fluid

The stability of Taylor-Couette flow of entangled polymeric solutions to small axisymmetric stationary disturbances is analyzed using the Doi-Edwards constitutive equation in the small gap limit. A previous analysis of Karlsson, Sokolov, and Tanner for the general K-BKZ equation, of which the Doi-Edwards equation is a special case, reduces the problem to one of numerically evaluating seven viscoelastic functions of the shear rate in the gap. Of these seven, only three — two of which are related to the second normal stress difference, and one of them to shear thinning — significantly affect the flow stability. The negative second normal stress difference of the Doi-Edwards fluid stabilizes the flow at low values of the Weissenberg number ₁ , while shear thinning produces strong destabilization at moderate Weissenberg number. Here ₁ is the longest relaxation time. Non-monotonic effects of viscoelasticity on Taylor-Couette stability analogous to those predicted here have been observed in experiments of Giesekus. The extreme shear thinning of the Doi-Edwards fluid is also predicted to produce a large growth in the height of the Taylor cells, a phenomenon that has been seen experimentally by Beavers and Joseph.     

Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers

The problem of flow of a viscous fluid around a spherical drop has been examined for the limiting case of small and large Reynolds numbers in several investigations (see [1–3], for instance; there is a detailed review of various approximate solutions in [4]). For the intermediate range of Reynolds numbers (approximately 1Re100), where numerical integration of the complete Navier-Stokes equations is necessary, there are solutions of special cases of the problem —flow of air around a solid sphere [5–7], a gas bubble [8, 9], and water drops [10]. The present paper deals with flow around a spherical drop at intermediate Reynolds numbers up to Re=200 for arbitrary values of the ratio of dynamic viscosities =₁/₂ inside and outside the drop. It is shown that a return flow can arise behind the drop in flow without separation. In such conditions the circulatory flow inside the drop breaks up. An approximate formula for the drag coefficient of the drop is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 8–15, January–February, 1976.We thank L. A. Galin, G. I. Petrov, L. A. Chudov, and participants in the seminars led by them for useful discussions.     

Die freie Oberfläche einer Flüssigkeit über einer rotierenden Scheibe

Zusammenfassung Es wird eine modifizierte Form des Weissenberg-Effekts untersucht, wobei sich die viskoelastische Flüssigkeit in einem kreiszylindrischen Gefäß befindet, an dessen Boden eine Scheibe rotiert. Normalspannungsdifferenzen rufen in der Flüssigkeit eine Strömung hervor, die auf der Drehachse von unten nach oben gerichtet ist, und die freie Oberfläche wölbt sich nahe der Achse nach außen. Unter der Voraussetzung hinreichend langsamer Strömung wird eine Theorie zweiter Ordnung entwickelt. Sie führt auf elliptische Randwertaufgaben zweiter bzw. vierter Ordnung für das Geschwindigkeitsfeld der Primärströmung in Umfangsrichtung und für die Stromfunktion der Sekundärströmung in der Meridianebene. Ihnen werden äquivalente Variationsaufgaben zugeordnet und mit der Methode der Finiten Elemente numerisch gelöst. Die Gestalt der freien Oberfläche setzt sich bei geeigneter Normierung aus drei universellen Formfunktionen zusammen, die für verschiedene Füllhöhen berechnet werden. Im experimentellen Teil wird nachgewiesen, daß durch entsprechende Messungen der Auslenkung des Flüssigkeitsspiegels die unteren Grenzwerte der beiden Normalspannungskoeffizienten bestimmt werden können. Das Rheometer besitzt den Vorzug, daß die Oberflächenspannung der Flüssigkeit die Meßgröße nur unwesentlich beeinflußt.

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Some kind of Weissenberg effect is considered where the viscoelastic fluid, being within a cylindrical vessel, is set in motion by a rotating disc near the tank bottom. Because of normal-stress differences within the fluid a secondary flow arises which is directed upwards near the axis of symmetry, and thus the free surface is deformed. Under the assumption of sufficiently slow flow a second-order theory is developed. It leads to second-order and fourth-order elliptic boundary value problems for the velocity field in azimuthal direction and for the stream function of the secondary flow, respectively. Equivalent variational problems are formulated and solved by the method of finite elements. When normalized appropriately, the shape of the free surface consists of three shape functions, which are independent of any material constants. It is shown by corresponding experiments, that the zero-shear-rate normal-stress coefficients can be determined by measuring the displacement of the free surface. In this rheometer, the surface tension of the fluid causes only insignificant influence on the quantity to be measured.

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Symbole C _H [—] Verhältnis der FormfunktionenF ₂/F₁ - f [—] die Sekundärströmung treibende radiale Volumenkraft, dimensionslos - F ₀, F₁, F₂ [—] universelle Formfunktionen - Fr [—] Froude-Zahl - g [m s^–2] Erdbeschleunigung - h [—] Auslenkung der Oberfläche, aufr ₀ bezogen - H [—] dimensionslose Füllhöhe - K [—] Kennzahl der Kapillarität - r,z [m] Zylinderkoordinaten - r, z [—] dimensionslose Koordinaten - r ₀ [m] Radius des Meßbehälters - Re [—] Reynolds-Zahl - v _r, v, v_z [m s^–1] Geschwindigkeitskomponenten - We ₁, We₂ [—] Weissenberg-Zahlen - [Pa s] Nullviskosität der Flüssigkeit - [°C] Temperatur - [m] Kapillarlänge - v ₁, v₂ [Pa s²] untere Grenzwerte der Normalspannungskoeffizienten - [kg m^–3] Dichte der Flüssigkeit - [N m^–1] Oberflächenspannung - [—] Zylinderkoordinate - [—] Dissipationsfunktion der Sekundärströmung, dimensionslos - [—] Stromfunktion, dimensionslos - [—] örtliche Winkelgeschwindigkeit, dimensionslos - [s^–1] Winkelgeschwindigkeit der Scheibe     

Parametric excitation of waves at an interface

Experiments on the parametric excitation of waves at a fluid interface show a strong disagreement with theoretical results [1–3], since the latter do not take into account the influence of the second medium. This proves to be especially important at low frequencies. Thus, for a water-air interface with an excitation frequency = 60 sec^–1 the contribution amounts to 10%,and with = 30 sec^–1, even 20%. In this paper the stability of the interface of two viscous, incompressible fluids of finite depth in a variable gravity field is considered. The problem is put in the linear form by making an expansion with respect to the small viscosity and is solved by taking the Laplace transform with respect to time. A second-order integrodifferential equation with periodic coefficients is obtained for the deviation of the interface from the equilibrium position; its solution is sought by the method of averaging [4]. It is shown that the presence of the second fluid significantly raises the threshold of instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1977.     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

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Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

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Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

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.     

Computational evaluation of ship manoeuvring performance based on scale model tests

Summary A brief review of the most important existing mathematical models for predicting the manoeuvring performance of a ship at the design stage is presented. A model based on the derivation of the hydrodynamic coefficients from force measurements on scale models is used to develop a computer program for the evaluation of the ship performance in some standard manoeuvres such as turning circle and zig-zag manoeuvres.

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Sommario Viene presentata una breve descrizione delle metodologie attuali più seguite per la identificazione di un modello matematico atto alla previsione delle caratteristiche di manovrabilità di una nave. Utilizzando coefficienti idrodinamici ricavati da prove su modelli in scala si è sviluppato un codice di calcolo che consente di ottenere la risposta della nave in alcune manovre standard quali quelle di evoluzione e zig-zag.

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Symbols G Center of gravity - g Acceleration due to gravity - I _ZZ Moment of inertia aboutz-axis - i _EP Effective moment of inertia about propeller axis - L Length between perpendiculars - m Ship mass - N Hydrodynamic moment aboutz-axis - n Rate of revolutions of propeller - O Origin of shipbound coordinate system - Q Propeller torque - Q _E Engine torque - q _F Engine fuel rate - R _T Total hull resistance - r Rate of turn aboutz-axis (yaw rate) - U Along-track velocity of0 - u, v Components ofU alongx, y-axes - X, Y Hydrodynamic forces alongx, y-axes - x,y,z Shipbound coordinate axes - x _G ,y _G ,z _G Coordinate of center of gravity in the shipbound system - x _o,y ₀,z ₀ Coordinate of 0 in the earthbound system, Fig. 1 - Drift angle - Rudder angle - Characteristic time - Heading angle Presented at the II Convegno AIMETA di Meccanica Computazionale, Rome, June 2–5, 1987.     

Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

The dynamics of an analytic reversible vector field (X,) is studied in with one real parameter close to 0; X=0 is a fixed point. The differential D_x (0,0) generates an oscillatory dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a slow dynamics which changes from a hyperbolic type—eigenvalues are —to an elliptic type—eigenvalues are —as passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.     

Theory of weak turbulence of waves on a fluid surface

The nonlinear interaction of waves in a fluid of finite depth is discussed. Forbidden decay processes in the gravitational portion of the spectrum are eliminated from the Hamiltonian by means of a canonical transformation. This provides an opportunity to obtain a kinetic equation which takes into account scattering of capillary waves by gravitational waves, in addition to decays in the subsystem of gravitational waves. The distribution N_k P^1/2h^1/4k^–4 is obtained for capillary waves in shallow water with constant flow of energy P with respect to the spectrum in the space of the wave numbers k. The interaction of the gravitational and capillary turbulence spectra is discussed. An induced distribution of gravitational waves is found which results from their interaction with capillary waves. It is an increasing function of the wave numbers q in the region bounded by the capillary constant k_o, N_q q^9/4 (q < k_o). The coupling of spectra in the gravitational and capillary regions and the conversion from slightly turbulent distributions to universal distributions are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 97–106, November–December, 1974.     

Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet

In Sec. 1 the stability of small-amplitude steady-state periodic solutions of Eq. (0.1) in the neighborhood of k=k_n are investigated. The results of the investigations are consistent with those of [1]. In Sec. 2 the stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when =1 steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3 the properties of certain exact solutions — periodic waves and solitons — are studied in relation to the nature of the singular points of the dynamical system derived from (0.1). In Sec. 4 the evolution of rapidly decreasing Cauchy data is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1989.     

Messung des ersten Normalspannungskoeffizienten von Kunststoffschmelzen im Bogenspalt

Zusammenfassung Bei einer stationären Schichtenströmung in einem Bogenspalt (azimutale Druckströmung im Ringspalt) bildet sich zwischen Innen- und Außenwand eine Druckdifferenz aus, deren Größe ein Maß für den 1. Normalspannungskoeffizienten der elastischen Flüssigkeit im Spalt ist. Die Strömung läßt sich zur Messung des 1. Normalspannungskoeffizienten verwenden. Der Schergeschwindigkeitsbereich der Messung liegt, wie bei der Kapillarrheometrie zur Bestimmung der Viskosität, zwischen 1 und 1000 s^–1. Die Auswertung der Messungen ist wegen des inhomogenen Scherfeldes relativ kompliziert. In der Arbeit wird ein besonders wirkungsvolles numerisches Auswerteverfahren hergeleitet und auf bestehende Messungen angewendet. Eine Besonderheit des Auswerteverfahrens ist die Freiheit der Wahl des Approximationsansatzes für die Viskositätskurve, während analytische Verfahren meist an einen bestimmten Ansatz gebunden sind. Außerdem braucht, im Gegensatz zu anderen derartigen Verfahren, die Position des schubspannungsfreien Stromfadensr ₀ nicht bestimmt zu werden.

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Summary The stress in steady viscometric flow of molten polymers is determined by the viscosity and by the two normal stress coefficients ₁ and ₂. The paper describes a method of measuring ₁ by means of steady circumferential shear flow in an annulus. The cylinders are stationary and the fluid flows due to a circumferential pressure gradient. The radial normal stresses at the outer and at the inner wall are different from each other. The pressure-differencep is a measure for the 1. normal stress coefficient of the viscoelastic fluid. Due to the inhomogeneous shear field, the evaluation of ₁ fromp measurements is quite complicated. A powerful numerical method of evaluation has been developed and applied to existing data. The method is not restricted to a special empirical formula for the flow curve (as an analytical method would be) and does not require the knowledge of the positionr ₀ of the stress-free stream line.

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a Pa s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten, s. Gl. [8] - AR _i — Koeffizient des Druckgefälles in-Richtung (Programm PFEIL) - AU _i — Koeffizient für Integration nach Simpson-Regel (Programm PFEIL) - b s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten - B _i — Koeffizient auf der rechten Seite des linearen Gleichungssystems (Programm PFEIL) - c — Exponent des Ansatzes des 1. Normalspannungskoeffizienten - CL _i CM _i CR _i — Koeffizienten der dimensionslosen Geschwindigkeit in dem linearen Gleichungssystem (Programm PFEIL) - F ₁,F ₂,F ₃ — Ableitungen der Summe der Fehlerquadrate nacha, b undc - G _k — Gewichtsfaktor - h m Spaltweite,r _a –r _i - H — dimensionslose Spaltweite, (r _a –r _i )/r _a - l m Länge des Bogenspaltes, 0,75(r _a +r _i ) - m — Exponent des Potenzansatzes, s. Gl. [13] - n — Dämpfungskonstante - N ₁ Pa 1. Normalspannungsdifferenz, _rr – - N ₂ Pa 2. Normalspannungsdifferenz - p Pa Druck - p Pa Druckgradient in-Richtung - P — dimensionsloser Druckgradient in-Richtung, s. Gl. [14] - p, p _k Pa Normalspannungsdifferenz zwischen Innen- und Außenwand im Bogenspalt, (– p + _rr ) _a – (–p + _rr ) _i - Q — Summe der Fehlerquadrate - r, R= r/r _a m, — Radiusvektor (Koordinate in Gradientenrichtung) - r ₀,R ₀=r ₀/r _a m, — Radius des neutralen Fadens - R — dimensionslose radiale Schrittweite - T, °C Temperatur bzw. Bezugstemperatur - v ms^–1 Geschwindigkeitskomponente in-Richtung - V ,V _,i — dimensionslose Geschwindigkeitskomponente in-Richtung - V _a ,V _k — dimensionslose Geschwindigkeit an der Außen- bzw. Innenwand - v _r ,v _z ms^–1 Geschwindigkeitskomponenten inr-undz-Richtung - ms ^–1 mittlere Geschwindigkeit in-Richtung - z m Koordinate in der indifferenten Richtung - K^–1 Temperaturkoeffizient der Viskosität - s^–1 Schergeschwindigkeit - s^–1 kritische Schergeschwindigkeit der Viskositätskurve, s. Gl. [13] - s^–1 Bezugsschergeschwindigkeit, - — dimensionslose Schergeschwindigkeit - — dimensionslose kritische Schergeschwindigkeit, - Pa s Viskosität - ₀ Pa s Nullviskosität - Pa s Bezugsviskosität, - — Radienverhältnis,r _i /r _a - ₁ Pa s ² 1. Normalspannungskoeffizient - Pa s² mittlerer 1. Normalspannungskoeffizient - ₂ Pa s² 2. Normalspannungskoeffizient - — Koordinate in Strömungsrichtung - Pa Spannung - a an der Außenwand - i, an der Innenwand - i laufender Index inr-Richtung - k Nummer des Meßpunktes - n Anzahl der Meßpunkte - n _i nord für Programm PFEIL - s _i süd für Programm PFEIL Mit 9 Abbildungen und 2 Tabellen     

The moving, plane heat source in an elastic medium

Summary This note presents an exact solution for the stress and displacement field in an unbounded and transversely constrained elastic medium resulting from the motion of a plane heat source travelling through the medium at constant speed in the direction normal to the source plane.Nomenclature mass density - diffusivity - thermal conductivity - Q heat emitted by plane heat source per unit time per unit area - speed of propagation of plane heat source - shear modulus - Poisson's ratio - T temperature - _x, _y, _z normal stress components - u _x, u_y, u_z displacement components - c speed of irrotational waves - t time - x, y, z Cartesian coordinates - =x–vt moving coordinate     

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

Motion of discrete particles in a turbulent fluid

Summary Various approximations to Basset's equation for the motion of a particle in a viscous fluid have been applied to the complex phenomenon of dispersion in a turbulent fluid. The deviations of the particle motion from the fluid motion, as predicted by the various approximations, is explored, and the frequencies for which this deviation is large are described. The approximations are found to be invalid for such cases as sediment transport and motion of gas bubbles in liquids. For small, 7 micron, liquid or solid particles in air, however, all approximations are shown to be valid for turbulent frequencies below 812 cps.Nomenclature a parameter in equation (2.3) - b parameter in equation (2.3) - c parameter in equation (2.3) - d diameter of sphere - E _f energy spectrum of the fluid - E _p energy spectrum of the particle - F frequency of oscillation - f ₁ parameter defined by equation (2.10) - f ₂ parameter defined by equation (2.10) - g acceleration of gravity - N _S , Stokes number - s density ratio - t time - t ₀ initial time - u _f fluid velocity - u _p particle velocity - V velocity of sphere - phase angle - parameter in equation (2.8) - amplitude ratio - parameter in equation (2.8) - dynamic viscosity - kinematic viscosity - f density of the fluid - p density of the particle - parameter in equation (2.8) - parameter in equation (2.8) - circular frequency of the motion     

Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x,y, u,ux, uy)

This paper develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x ₀)=y ₀) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation u _xy =f(x, y, u, u _x , y _y ), u(x, y ₀)=(x), u(x ₀, y)=(y), where (x ₀)=(y ₀). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions (x) and (y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.

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Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

This paper presents a study on the deformation of anisotropic fibrous porous media subjected to moistening by water in the liquid phase. The deformation of the medium is studied by applying the concept of effective stress. Given the structure of the medium, the displacement of the solid matrix is not taken into account with respect to the displacement of the liquid phase. The transport equations are derived from the model proposed by Narasimhan. The transport coefficients and the relation between the variation in apparent density and effective stress are obtained by test measurements. A numerical model has been established and applied for studying drip moistening of mineral wool samples capable or incapable of deformation.Nomenclature D mass diffusion coefficient [L²t^–1] - e void fraction - g gravity acceleration [Lt^–2] - J mass transfer density [ML^–2t^–1] - K hydraulic conductivity [Lt^–1] - K _s hydraulic conductivity of the solid phase [Lt^–1] - K ^* hydraulic conductivity of the deformable porous medium [Lt^–1] - P pressure of moistening liquid [ML^–1 t^–2] - S degree of saturation - t time [t] - V speed [Lt^–1] - X horizontal coordinate [L] - Z vertical coordinate measured from the bottom of porous medium [L] - z z-coordinate [L] Greek Letters porosity - ₁ total hydric potential [L] - _g gas density [ML^–3] - ₁ liquid density [ML^–3] - ₀ apparent density [ML^–3] - _s density of the solid phase [ML^–3] - density of the moist porous medium [ML^–3] - external load [ML^–1t^–2] - effective stress [ML^–1t^–2] - bishop's parameter - matrix potential or capillary suction [L] Indices g gas - 1 moistening liquid - p direction perpendicular to fiber planes - s solid matrix - t direction parallel to fiber planes - v pore Exponent * movement of solid particles taken into account     

Stability of flows of the Hamel type in channels with permeable walls

The stability of nonparallel flows of a viscous incompressible fluid in an expanding channel with permeable walls is studied. The fluid is supplied to the channel through the walls with a constant velocity v₀ and through the entrance cross section, where a Hamel velocity profile is assigned. The resulting flow in the channel depends on the ratio of flow rates of the mixing streams. This flow was studied through the solution of the Navier—Stokes equations by the finite-difference method. It is shown that for strong enough injection of fluid through the permeable walls and at a distance from the initial cross section of the channel the flow approaches the vortical flow of an ideal fluid studied in [1]. The steady-state solutions obtained were studied for stability in a linear approximation using a modified Orr—Sommerfeld equation in which the nonparallel nature of the flow and of the channel walls were taken into account. Such an approach to the study of the stability of nonparallel flows was used in [2] for self-similar Berman flow in a channel and in [3] for non-self-similar flows obtained through a numerical solution of the Navier—Stokes equations. The critical parameters *, R*, and Cr* at the point of loss of stability are presented as functions of the Reynolds number R₀, characterizing the injection of fluid through the walls, and the parameter , characterizing the type of Hamel flow. A comparison is made with the results of [4] on the stability of Hamel flows with R₀ = 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–129, November–December, 1977.The author thanks G.I. Petrov for a discussion of the results of the work at a seminar at the Institute of Mechanics of Moscow State University.