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.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

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An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

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Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

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     

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     

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.     

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.     

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.     

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.     

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     

Eine kanonische Zustandsgleichung für Kohlendioxid

Zusammenfassung Es wird eine kanonische Zustandsgleichung für Kohlendioxid in der Form des Helmholtz-Potentials mitgeteilt, die mit einem Verfahren aufgestellt wurde, das die gleichzeitige Approximation verschiedenartiger Zustandsgrößen erlaubt. Zur Ermittlung der Vorgabewerte für die Approximation wurden Meßwerte sowie Werte bereits vorliegender Gleichungen verwendet. Außerdem werden einige Temperaturfunktionen für Zustandsgrößen an den Grenzkurven und im idealen Gaszustand angegeben. Der Verlauf einiger Zustandsgrößen von Kohlendioxid wird mit dem entsprechenden Verlauf bei Wasser bzw. Wasserdampf verglichen. Es zeigt sich eine überraschend gute qualitative Übereinstimmung.

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A canonical equation of state for carbon dioxide

A canonical equation of state for carbon dioxide in the form of the Helmholtz function is presented which was established by means of a method allowing the simultaneous fitting of different properties. The data points which have to be fitted are based on experimental values as well as on values of still existing equations. Several temperature functions for properties along the boundary lines and in the ideal gaseous state are given. The behaviour of some properties of state of carbon dioxide is compared with that of water substance. A surprisingly good qualitative agreement is shown.

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Formelzeichen und definierte Werte A Matrix - a_ij Gleichungskoeffizienten - B Vektor - c_p isobare spezifische Wärmekapazität - c_v isochore spezifische Wärmekapazität - f spezifische freie Energie (Helmholtz-Funktion) - h spezifische Enthalpie - i, j Laufvariable - k Isentropenexponent - p Druck - t Celsius-Temperatur (t=T–T₀ mit T₀=273.15 K) - v spezifisches Volumen - z=pv/(RT) Realfaktor - _h isenthalper Drosselkoeffizient - _T isothermer Drosselkoeffizient - Dichte - IMAX obere Grenze der Laufvariablen i - JMAX_i obere Grenze der Laufvariablen j (abhängig von i) - JMIN_i untere Grenze der Laufvariablen j (abhängig von i) - R Gaskonstante - T thermodynamische oder Kelvin-Temperatur - W Bewertungsfaktor - Anstieg der Dampfdruckkurve - =P/Pk reduzierter Druck - =T/T_k reduzierte Temperatur - =–1 transformierte reduzierte Dichte - G=1/–1 transformierte reduzierte Temperatur - =f_k/Pk reduzierte spezifische freie Enthalpie - =/k reduzierte Dichte - I=R T_{k k}/Pk reduzierte Gaskonstante Indizes k kritischer Zustand - tr Zustand am Tripelpunkt - sub Sublimationszustand - s Sättigungszustand - flüssiger Sättigungszustand - gasförmiger Sättigungszustand - * Zustand beim Normdruck p^*=1 atm - o idealer Gaszustand bei p=0 oder p=p^* Herrn Professor Dr. Romano Gregorig gewidmet zum 65. Geburtstag.     

Effect of nose shape on hypersonic flow past slender blunt bodies at an angle of attack

We examine some characteristics of hypersonic flow past slender blunt bodies of revolution at a small angle of attack 1, where is the relative body thickness. It is shown that, within the framework of hypersonic theory, for a correct-consideration of the effect of the conditions in the transitional section between the nose and the lateral surface it is necessary, in the general case, to specify the circumferential distribution of the force effect for the nose and the mass of the gas. For small , the effect of the nose, just as in two-dimensional flows [1–4], shows up only through its drag coefficient c_x, for =0. On this basis, the similarity law [1–4] for flow past such bodies, with arbitrary form of the lateral surface and differing in the shape of the nose blunting, which is valid over the entire disturbed region, with the exception of a small vicinity of the nose, is extended to the case in question.The notation r₀ and L maximum nose radius and characteristic body length - V, M, and density, velocity, Mach number, and adiabatic exponent of the gas in the approaching stream - , V²i, and V²p density, enthalpy, and pressure - x, r, and coordinate system of the cylindrical body with its center at the transitional section between the nose and the side surface - Vu, Vv, and Vw corresponding velocity components     

Hydrodynamic models as applied to the investigation of magnetic plasma stability

In the present paper magnetohydrodynamic models are employed to investigate the stability of an inhomogeneous magnetic plasma with respect to perturbations in which the electric field may be regarded as a potential field (rot E 0). A hydrodynamic model, actually an extension of the well-known Chew-Goldberg er-Low model [1], is used to investigate motions transverse to a strong magnetic field in a collisionless plasma. The total viscous stress tensor is given; this includes, together with magnetic viscosity, the so-called inertial viscosity.Ordinary two-fluid hydrodynamics is used in the case of strong collisions=. It is shown that the collisional viscosity leads to flute-type instability in the case when, collisions being neglected, the flute mode is stabilized by a finite Larmor radius. A treatment is also given of the case when epithermal high-frequency oscillations (not leading immediately to anomalous diffusion) cause instability in the low-frequency (drift) oscillations in a manner similar to the collisional electron viscosity, leading to anomalous diffusion.Notation f particle distribution function - E electric field component - H₀ magnetic field - density - V particle velocity - e charge - m, M electron and ion mass - _i, _e ion and electron cyclotron frequencies - viscous stress tensor - P pressure - r_i Larmor radius - P pressure tensor - t time - frequency - T temperature - collision frequency - collision time - j current density - _i, _e ion and electron drift frequencies - k_x, k_y, k_z wave-vector components - n₀ particle density - g acceleration due to gravity. The authors are grateful to A. A. Galeev for valuable discussion.     

Dissolution of an ellipsoidal bubble in a slightly viscous liquid

The steady-state velocity, the degree of deformation, and the convective-diffusion-limited rate of quasisteady-state growth (or dissolution) are considered for gas bubbles having shapes close to those of spheres or disks. It is assumed that there are no surface-active substances in the liquid. A qualitative agreement is found between the calculated dissolution rate and the experimental data.Notation a radius of the sphere of equivalent volume - u bubble velocity with respect to the still liquid at infinity - kinematic viscosity of the liquid - liquid density - D gas diffusion coefficient in the liquid - surface tension - g gravitational acceleration - d [R=2au/]-Reynolds number - e [P=2au/D]-Peclet number - f [W=2au²/]-Weber number The author thanks V. G. Levich for a discussion of these results.     

Zur einfachen Berechnung von Dephlegmatoren binärer Dampfgemische

Zusammenfassung Die Dephlegmation ist eine nicht-adiabate Rektifikation ohne Rücklauf am Apparatekopf, die durch die Ackermann/Colburn-Drew-Gleichungen beschrieben werden. In diesem Beitrag wird eine vergleichende Analyse von stationären makroskopischen Modellen mit unterschiedlicher Reduktion gegeben.

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On simple calculation procedures of binary mixed vapour dephlegmation

The dephlegmation is a non-adiabatic rectification without reflux at the top of the column, which for calculation can be described by the Ackermann/Colburn-Drew-equations. In this paper a comparing analysis of steady macroscopic models with different degree of model reduction is given.

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Nomenklatur A Austauschfläche pro Apparate- m²/m länge - C Korrekturfunktion - D Diffusionskoeffizient m²/h - Enthalpiestrom J/h - Impulsstrom kmol m/h - N Zahl der theoretischen Trennstufen - N Molstrom kmol/h - T Temperatur °C - Molmasse kg/kmol - L Apparatelänge m - c_p molare Wärmekapazität J/kmol grd - d Durchmesser m - Enthalpiestromdichte J/h m² - g Erdbeschleunigung m/h² - h molare Enthalpie J/kmol - j Impulsstromdichte kmol/h m - n Molstromdichte kmol/h m² - 1 Länge m - u axiale Geschwindigkeit m/h - x Molkonzentration im Fluid kmol/kmol - y Molkonzentration im Dampf kmol/kmol - z Molkonzentration (S. G1.2) kmol/kmol - Differenz - t Kontaktzeit h - Austauschkoeffizient für die J/h m² grd Enthalpie - ß Austauschkoeffizient für die kmol/h m² Komponente - Austauschkoeffizient für den kmol/h m Impuls - Massendichte kg/m² - Zähigkeit kg/m h - _f Rieselfilmdicke m - ^f Wärmedurchgangskoeffizient J/h m² grd Kennzahlen Re u·d·/ - Sc /·D - Sh ··d/·D Indizes a außen - d dampfseitig - f flüssigkeitsseitig - g Phasengrenze - h hydraulisch - i innen - k Kühlmedium - m mittel - o oberes Apparateende - t total - u unteres Apparateende - w Wand - x Komponente an LS im Fluid - y Komponente an LS im Dampf - gültig für große übergehende Molströme     

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.     

The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

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     

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.     

Symmetry-Breaking Bifurcations of Charged Drops

It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient is larger than some critical value (i.e., when <_c). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where ₂=_c. We further prove that the spherical drop is stable for any >₂, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as t provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at =₂ which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.