Equilibrium Configurations for a Floating Drop

If a drop of fluid of density ₁ rests on the surface of a fluid of density ₂ below a fluid of density ₀, ₀ < ₁ < ₂, the surface of the drop is made up of a sessile drop and an inverted sessile drop which match an external capillary surface. Solutions of this problem are constructed by matching solutions of the axisymmetric capillary surface equation. For general values of the surface tensions at the common boundaries of the three fluids the surfaces need not be graphs and the profiles of these axisymmetric surfaces are parametrized by their tangent angles. The solutions are obtained by finding the value of the tangent angle for which the three surfaces match. In addition the asymptotic form of the solution is found for small drops.     

Analytic Approximation of the Homoclinic Orbits of the Lorenz System at σ = 10, b = 8/3 and ρ = 13.926...

We present an iterative technique to analytically approximate the homoclinic loops of the Lorenz system for = 10, b = 8/3 and = _H = 13.926.... First, the local structure of the homoclinic solution for t 0 ± and t ± is analyzed. Then, global approximants are used to match the local expansions. The matching procedure resembles the one used in Padé approximations. The accuracy of the approximation is improved iteratively, with each iteration providing estimates for the initial conditions of the homoclinic orbit, the value of _H, and three undetermined constants in the local expansions. Within three iterations the error in _H falls to the order of 0.1%. Comparisons with numerical integrations are made, and a discussion on ways to extend the technique to other types of homoclinic or heteroclinic orbits, and to improve its accuracy, is given.     

Problem of a jet flowing into the surface of a heavy fluid

This paper is a continuation of [1]. The problem of the nonuniqueness of the angle of incidence of a fine jet into water is considered and the mathematical formulation of the problem is improved. A diagram of the flow is shown in Fig. 1.; the jet is an inviscid, incompressible, weightless fluid of density ₁ flowing from a nozzle onto the surface of a still heavier fluid of density ₂. The problem is two-dimensional.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 82–89, March–April, 1973.     

Stability of a laminar boundary layer in an incompressible fluid with variable physical properties

The stability of plane-parallel flows of an incompressible fluid with variable kinematic viscosity in the presence of solid walls has been discussed in [1–5]. The stability of Couette flow is considered in [1]. The method of solution, which is the same as that used in [2], differs from the Tollmien-Schlichting method, since the expansion of the solutions in powers of R which is used assumes the smallness of this quantity. A general formulation of the problem of stability of a nonuniform fluid is presented in [3]. Di Prima and Dunn [4] used the Galerkin method to study the stability of the boundary layer relative to vortex-like disturbances in the case of variable kinematic viscosity. Since the development of this sort of disturbance depends only weakly on the form of the velocity profile in the boundary layer, a marked change of the viscosity had little effect on the critical Reynolds number. This same problem is considered in [5], The present author was not able to find in the literature any references to study of the stability of the laminar boundary layer in an incompressible fluid with variable kinematic viscosity relative to disturbances of the Tollmien-Schlichting type, with the exception of mention in [4] of an unpublished work of MacIntosh which showed the essential dependence of the critical Reynolds number on the viscosity gradient.In Section 1 an analysis is made of the effect of variability of the kinematic viscosity on the stability of the boundary layer relative to Tollmien-Schlichting waves under the condition of constant fluid density.Two approaches are possible to the study of the development of disturbances in a heterogeneous fluid. On the one hand, we can assume that the displacement of the fluid particles does not cause changes in the distribution of p(y) and _*(y), i.e., the velocity pulsations are not accompanied by pulsations of and _*. This will be the case if a particle which is characterized by the quantities ₁, ₁ entering a layer with the different values _{2 2}, instantaneously alters its properties so that its density becomes equal to ₂, and its kinematic viscosity becomes equal to ₂. On the other hand, we can consider that a fluid particle moving from layer 1 into layer 2 fully retains the properties which it had in layer 1. In this case the velocity pulsations naturally lead to pulsations of the quantities and _*.In actuality, the phenomenon develops along some intermediate scheme, since the particle alters its properties as it moves in a heterogeneous fluid. The degree of approximation of the process to the first or second scheme depends on the rate of these changes. The analyses below are based on the first scheme.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

On the Cρ-Smoothness of an Invariant Torus of a Countable System of Difference Equations Defined on an m-Dimensional Torus

Using the method of truncation, we establish sufficient conditions of differentiability of order 2 with respect to the angular variable for an invariant torus of a countable system of difference equations with deviation of the discrete argument.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Flow stability of a Grad-model liquid layer on an inclined plane

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force.It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector , the internal moment stresses are negligibly small, and in the general case the force stress tensor _kj is asymmetric. The model is characterized by the usual Newtonian viscosity , the Newtonian rolling viscosity _r, and the relaxation time = J/4 _r, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, is the density. It is assumed that the medium is incompressible, the coefficients , _r, J are constant [2].The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid.The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].     

Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties

The steady laminar boundary layer flow, with an external force, along a vertical isothermal plate is studied in this paper. The external force may be produced either by the motion of the plate or by a free stream. The fluid is water whose density-temperature relationship is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations with , k and variable across the boundary layer. Both upward and downward flow is considered. It was found that the variation of , k and with temperature has a strong influence on mixed convection characteristics.Nomenclature c_p water specific heat - f dimensionless stream function - g gravitational acceleration - Gr_x local Grashof number - k thermal conductivity - Nu_x local Nusselt number - Pr Prandtl number - Pr_a ambient Prandtl number - Re_x local Reynolds number - s salinity - T water temperature - T_a ambient water temperature - T_o plate temperature - u vertical velocity - u_a free stream velocity - u_o plate velocity - v horizontal velocity - x vertical coordinate - y horizontal coordinate - pseudo-similarity variable - nondimensional temperature - dynamic viscosity - _f film dynamic viscosity - _o dynamic viscosity at plate surface - kinematic viscosity - buoyancy parameter - water density - _a ambient water density - _f film water density - _o water density at plate surface - physical stream function     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Analytical results on two-phase rows in ducts

A deterministic stochastic approach is successfully applied to the investigation of some problems of the fluid-dynamics of two-phase systems. The method follows the guidelines of the theory of differential equations with random initial conditions.Dusty gas flows and bubble flows are considered in circumstances where the action of the particulate matter on the fluid flow field is negligible. In all the cases, collisions between particles of the disperse phase are neglected. As significant applications, the entrance flow of dust particles in a tube and the behaviour of a population of bubbles imbedded in a pipe-flow subject to an abrupt area, change are considered. The probable distributions of the particles are evaluated as functions of assigned statistical distributions of the objects at the initial time.

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Analytische Ergebnisse von Zweiphasen-Strömungen in Kanälen

Zusammenfassung Ein deterministisch-stochastischer Ansatz wird erfolgreich auf die Untersuchung einiger Strömungsprobleme von Zweiphasensystemen angewandt. Die Methode entspricht der Theorie von Differential-gleichungen mit zufälligen Anfangsproblemen.Staub-Gas-Strömungen und Blasen-Strömungen werden unter Bedingungen behandelt, bei denen der Einfluß des betreffenden Stoffes auf das Strömungsfeld vernachlässigt werden kann. In allen Fällen werden auch Kollisionen zwischen Partikeln der dispersen Phase vernachlässigt. Als kennzeichnende Anwendungen werden die Eintrittsströmungen von Staub in ein Rohr und das Verhalten eines Blasenschwarmes an einer sprunghaften Querschnittsveränderung eines Rohres behandelt. Die wahrscheinlichen Teilchenverteilungen werden als Funktionen bestimmter Anfangsverteilungen ermittelt.

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Nomenclature A normalization factor - A,b,c,kj,G,, dimensionless quantities defined in the text - h _a/_c - H _w/_a - J operator defined at page 2 - K Boltzmann constant - p dimensionless pressure - P probability density - p probability - r⁺ radius of the tube, reference length - R radial co-ordinate (dimensionless) - Re Reynolds number of the relative motion of the particles with respect to the fluid - t⁺ reference time, r⁺ /U⁺ - t time (dimensionless) - U⁺ reference velocity - U_a,U_r axial and radial velocity components, respectively, of the fluid flow - 183-01 state vector of a particle - V_a,V_r axial and radial velocity components respectively of a particle - Z dimensionless axial co-ordinate along the tube - viscosity - mass density - diameter of a particle, dimensionless with respect to r⁺ Indexes ()_a air - ()_c coal - ()_t at the time t - ()_w water - ()_o at the time t=o     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Generalization of the experimental data on extended crisis in heat transfer in the boiling of a liquid

A generalized formula is given for the critical heat flux, and it is shown that crises of this type are most characteristic of the boiling of organic liquids at high temperatures.Notation q_* critical heat flux - q heat flux - W mean flow speed of liquid in crisis section; - W_g mass flow rate - r latent heat of evaporation - coefficient of surface tension - -@#@ density of dry saturated vapor - density of liquid on saturation line - i enthalpy of liquid on saturation line - i mean enthalpy of liquid in crisis cross section - cf coefficient of friction - g acceleration due to gravity - P static pressure in crisis cross section - T saturation temperature - T_* temperature of surface of tube - mean density of liquid in crisis cross section I am indebted to I. N. Svorkova for assistance.I am also indebted to S. S. Kutateladze and A. I. Leont'ev for discussions and valuable comments.     

Stability of couette flow between free cylinders with allowance for self-gravitation

The free two-dimensional motion of a gravitating system consisting of two unfixed rotating cylinders and the incompressible fluid between them is considered. The undisturbed state of the system is determined by the viscous Couette flow between the coaxial cylinders. The stability of the central position of the cylinders is investigated within the framework of the equations of an inviscid fluid in a rotating coordinate system in which the flow is assumed to be potential. The critical values of the parameters and the spectral characteristics of the system are found. The qualitative form of the neutral curves obtained is determined by the value of the unique parameter of the problem ^r, which is equal to the ratio of the density of the inner cylinder to the density of the fluid. It is noted that if ^r 1, then however small the values of the angular velocities of the cylinders, generally speaking, they cannot be neglected in calculating the natural vibrations of the system. It is shown that if ^r > 1, then in the absence of gravitation the radial play between the cylinder axes leads to the growth of disturbances in the region stable with respect to the Rayleigh criterion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 110–119, September–October, 1991.In conclusion the authors express their gratitude to A. L. Krylov and Yu. N. Avsyuk for drawing their attention to the problem and to Yu. S. Kopysov for discussing the results.     

Messung der Viskosität von Kohlendioxid und Propylen

Zusammenfassung Die Viskosität von Kohlendioxid und Propylen wurde bei Temperaturen zwischen 298 K und 473 K und Drücken zwischen 1 bar und dem zweifachen kritischen Druck mit einem Schwingscheibenviskosimeter gemessen. Dieses unterscheidet sich von den bisher bekannten Schwingscheibenviskosimetern insbesondere durch die optische Einrichtung zur Bestimmung der Winkelamplituden. Die mittleren relativen Fehler der Meßwerte sind vom thermodynamischen Zustandsbereich abhängig und liegen zwischen 0,9 % und 1,6%. Für Interpolationsrechnungen werden einfache Gleichungen angegeben.

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Measurements on the viscosity of carbon dioxide and propylene

Measurements on the viscosity of carbon dioxide and propylene are reported. The experimental investigations have been performed with an oscillating disk viscosimeter at temperatures between 298 K and 473K and pressures from 1 bar up to the twofold critical pressure of each gas. The optical system for reproducing the oscillations of the disk on a scale is modified to the yet known oscillating disk viscosimeters. With respect to the thermodynamic state of the gas an accuracy between 98,4% and 99,1% could be reached. For correlating the measured values two polynomial approximations are reported.

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Formelzeichen b mittlere Spaltbreite; - C_N,C Gerätekonstante - D Scheibenabstand - d Scheibendicke - J Trägheitsmoment - p Druck - R Scheibenradius - T Temperatur - T_kr kritische Temperatur - T_N normierte Temperatur, T_N=T/T_kr - t Zeit - Zeitverhältnis; =t/t_O - logarithmisches Dekrement - dynamische Viskosität - kinematische Viskosität - Kreisfrequenz - Dichte - _kr kritische Dichte - _N normierte Dichte; _N= /_kr Index 0 Vakuum Der Verfasser dankt Herrn Prof. Dr.-Ing. K. Stephan für die grozügige Förderung dieser Arbeit.     

Method of calculating supersonic three-dimensional flow past smooth bodies

A method is suggested in [1] for calculating supersonic flow past smooth bodies that uses an analytic approximation of the gasdynamic functions on layers and the method of characteristics for calculating the flow parameters at the nodes of a fixed grid. In the present paper this method is discussed for three-dimensional flows of a perfect gas in general form for cylindrical and spherical coordinate systems; relations are presented for calculating the flow parameters at the layer nodes, results are given for the calculation of the flow for specific bodies, and results are shown for a numerical analysis of the suggested method. Three-dimensional steady flows with plane symmetry are considered. In the relations presented in the article all geometric quantities are referred to the characteristic dimension L, the velocity components u, v, w and the sonic velocitya are referred to the characteristic velocity W, the density is referred to the density of the free stream, and the pressure p is referred to w².     

Symmetries and Global Solvability of the Isothermal Gas Dynamics Equations

We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the -law equation p() with =1. Our results complete those obtained earlier for >1. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density.One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed on the line =1. This is in contrast to the common approach (when >1) which prescribes initial data on the vacuum line =0. The entropies we construct here are weak entropies, i.e., they vanish when the density vanishes.Another feature of our proof lies in the reduction theorem, which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.Acknowledgement P.G.L. and V.S. were supported by a grant from INTAS (01-868). The support and hospitality of the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, where part of this research was performed during the Semester Program Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics (January to July 2003) is also gratefully acknowledged. P.G.L. was also supported by the Centre National de la Recherche Scientifique (CNRS).