Neutrally buoyant bubbles used as flow tracers in air

Research has been performed to determine the accuracy of neutrally buoyant and near-neutrally-buoyant bubbles used as flow tracers in an incompressible potential flowfield. Experimental and computational results are presented to evaluate the quantitative accuracy of neutrally buoyant bubbles using a commercially available helium bubble generation system. A two-dimensional experiment was conducted to determine actual bubble trajectories in the stagnation region of a NACA 0012 airfoil at 0° angle of attack. A computational scheme evaluating the equation of motion for a single bubble was also used to determine the factors which affect a bubble's trajectory. The theoretical and computational analysis have shown that neutrally buoyant bubbles will trace complex flow patterns faithfully in the flowfield of interest. Experimental analysis revealed that the use of bubbles generated by the commercially available system to trace flow patterns should be limited to qualitative measurements unless care is taken to ensure neutral buoyancy.Nomenclature a _c centripetal acceleration - c model chord - c _D bubble drag coefficient - D bubble diameter - g acceleration due to gravity - g _v acceleration due to gravity vector - h trajectory deviation normalization parameter - K nondimensional inertia parameter, - m _f mass of fluid - m _p mass of bubble - p static pressure - r radial distance, bubble radius - R gas constant - Re free-stream Reynolds number, - Re _p bubble slip Reynolds number, - S cross-sectional area of sphere - T temperature - t time - u streamwise velocity component - U free-stream velocity - v _f fluid velocity vector - V _p bubble velocity vector - x _p bubble position vector - y _b bubble trajectory y/c - y _s streamline y/c - model angle of attack - bubble solution surface tension - potential vortex strength - _bfs bubble solution density - fluid density - bubble density - bubble wall thickness - fluid viscosity     

The motion of gas bubbles rising through stagnant liquid

The scope of the present over-all investigation comprises the determination of rise velocity, path, shape and drag characteristics of gas bubbles rising freely in water in terms of bubble size.The material presented in this first report is limited to the measurement of the rise velocity of air bubbles of various sizes and the determination of their paths.A new experimental technique is introduced which consists essentially of measuring the time interval when the bubble traverses two light beams. An electronic system including two phototubes, a pulse amplifier and shaper, and an electronic timer has been used. Also a photographic technique has been used to determine the bubble's size, shape and path.The results show that the type of motion may be predicted from the value of the Reynolds number at which the motion takes place. Five types of motion were observed in the experiments: 1. Rectilinear motion, 2. motion on a helical path, 3. first plane then helical motion, 4. plane motion, 5. rectilinear motion with rocking.The rise velocity of the bubbles which move on those paths varies with the heightz. All the bubbles studied are accelerated to a maximum velocity in a few cm above the capillary tube. For the bubbles smaller than 4.2 mm, this maximum velocity decreased to an asymptotic value far from the capillary tube.

---

Zusammenfassung Das Ziel dieser gesamten Untersuchungen ist die Bestimmung der Steiggeschwindigkeit sowie charakteristischer Daten der Bahn, der Form und des Widerstandes von frei in Wasser aufsteigenden Gasblasen in Abhängigkeit von der Blasengröße. Dieser erste Bericht handelt nur von der Messung der Aufstiegsgeschwindigkeit von Luftblasen verschiedener Größe und der Bahnbestimmung.Eine neue, hier angewandte Experimentiertechnik besteht im wesentlichen in der Messung des Zeitintervalls zwischen den Durchgängen der Blase durch zwei Lichtstrahlen. Es wurde ein elektronisches System benutzt, bestehend aus zwei Photozellen, Verstärker, Umformer und Zeitnehmer. Größe, Form und Bahn der Blasen wurde photographisch bestimmt.Die Ergebnisse zeigen, daß die Bewegungsform sich im wesentlichen aus dem Wert der Reynolds-Zahl voraussagen läßt. Fünf verschiedene Formen wurden beobachtet: 1. gradlinige Bewegung, 2. spiralige Bewegung, 3. anfangs ebene, später spiralige Bewegung, 4. ebene Bewegung, 5. gradlinige Schaukelbewegung.Die Aufstiegsgeschwindigkeit längs dieser Bahnen ändert sich mit der Höhe. Alle beobachteten Blasen werden innerhalb einiger Zentimeter oberhalb der Kapillare auf eine Maximalgeschwindigkeit beschleunigt. Für Blasen unter 4,2 mm Durchmesser sinkt diese Maximalgeschwindigkeit auf einen asymptotischen Wert ab, der in größerer Entfernung von der Kapillare erreicht wird.

---

Nomenclature C _D Drag coefficient - D _h Helix diameter - R _h Helix radius - R Radius of curvature of the spherical cap - R _e Reynolds number determined in terms of the equivalent bubble diameter (=(vd _e)/ ₁) - d _e Equivalent bubble diameter - g Acceleration of gravity - h Helix pitch - m Mass - r Bubble radius - v Bubble velocity - v _z Bubble rise velocity - Acceleration - ₁ Dynamic viscosity of the liquid - _g Dynamic viscosity of the gas - ₁ Kinematic viscosity of the liquid - ₁ Density of the liquid - _g Density of the gas - Helix slope defined by tg =h/D _h     

Bouncing and Coalescence of Bubble Pairs Rising at High Reynolds Number in Pure Water or Aqueous Surfactant Solutions

The encounter of bubble pairs of O(1 mm) in both pure water and aqueous surfactant solutions was studied experimentally. In pure water, two equally sized bubbles were found to coalesce if the Weber number, W = V² R/, based on the velocity of approach, V, was below a critical value, W_cr = 0.18, where and are the density and surface tension of the liquid respectively and R the equivalent radius of the bubbles. After coalescence bubbles perform volume and shape oscillations.When W_cr is exceeded, bubbles bounce. After bouncing, bubbles can either coalesce or separate without coalescing. This was found to depend on the Weber number, based on the rise velocity U, We = U² R/. If this number was below a critical value, bubbles coalesced after bouncing. The relative motion of the bubbles was found to be damped out by acoustic damping due to surface oscillations rather then by viscosity.If We was above a critical value, which was close to that for path instability of a single bubble (We = 3.3), the bubbles separated after bouncing. This is probably caused by shedding of vortices which dominate the relative motion of the bubbles. This mechanism may cause bubbles in bubbly flows not aggregating in horizontal planes, as was found in calculations based on potential flow theory. For modelling bubbly flows it will therefore be essential to incorporate the influence of vorticity.When surfactants are added to the water it was found that bubbles are prevented to coalesce above a critical concentration, which is nearly identical to that of single rising bubbles. Above this critical concentration, bubbles behave as rigid spheres and trajectories cannot be predicted by potential flow theory.     

Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell

An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. In the first part, the bubble velocity U relative to the fluid velocity at infinity is determined for small surface tension T by determining a transcendentally small correction to the asymptotic series solution. It is found that for any relative bubble velocity U in the interval (U _c , 2), solutions exist at a countably infinite set of values of T (which has zero as its limit point) corresponding to the different branches of bubble solutions. The value of U _c decreases monotonically from 2 to 1 as the bubble area increases from 0 to . However, for a bubble of an arbitrarily given size, as T 0, a solution exists on any given branch with the relative bubble velocity U satisfying the relation 2–U=cT ^2/3, where c depends on the branch but is independent of the bubble area. The analytical evidence further suggests that there are no solutions for U>2. These results are in agreement with earlier analytical results for a finger.In Part II an analytic theory is presented for the determination of the linear stability of the bubble in the limit of zero surface tension. Only the solution branch corresponding to the largest possible U for given surface tension is found to be stable, while all the others are unstable, in accordance with earlier numerical results.This research has been supported by National Science Foundation Grant DMS-8713246. Partial support was also provided by the NASA Langley Research Center (NAS1-18605) while the author was in residence at the Institute of Computer Applications in Science and Engineering.     

Modeling of bubble detachment in reduced gravity under the influence of electric fields and experimental verification

A simple model for predicting bubble volume and shape at detachment in reduced gravity under the influence of electric fields is described in the paper. The model is based on relatively simple thermodynamic arguments and relies on and combines several models described in the literature. It accounts for the level of gravity and the magnitude of the electric field. For certain conditions of bubble development the properties of the bubble source are also considered. Computations were carried out for a uniform unperturbed electric field for a range of model parameters, and the significance of model assumptions and simplifications is discussed for the particular method of bubble formation. Experiments were conducted in terrestrial conditions and reduced gravity (during parabolic flights in NASAs KC-135 aircraft) by injecting air bubbles through an orifice into the electrically insulating working fluid, PF5052. Bubble shapes visualized experimentally were compared with model predictions. Measured data and model predictions show good agreement. The results suggest that the model can provide quick engineering estimates concerning bubble formation for a range of conditions (both for formation at an orifice and boiling) and such a model reduces the need for complex and expensive numerical simulations for certain applications. a Major axis of spheroid (m) - a _m Measured bubble height (m) - b Minor axis of spheroid (m) - b _m Measured bubble width (m) - A, B, C, F Parameters of the Kumar-Kuloor model - a/b Computed aspect ratio - a _m /b _m Measured aspect ratio - D Orifice diameter (m) - E Magnitude of the electric field (V/m) - g Gravitational acceleration (m/s²) - g _t Terrestrial gravity (g _t = 9.81 m/s²) - N _w Electrical Weber number - p Pressure (Pa) - Q Volume flow rate (m³/s) - r Radius of the spherical bubble (m) - R Radius of curvature at the tip of the bubble (m) - t Time (s) - t Time interval (s) - T Temperature (°C) - U Electrical potential (V) - u Velocity (m/s) - V Volume (m³) - x, y Dimensionless coordinates of the Cartesian coordinate system - x, y Scaled coordinates, Cheng-Chaddock model - X, Y Dimensional coordinates of the Cartesian coordinate system - Characteristic wave number (m^–1) - Eötvös number - Absolute dielectric permittivity (F/m) - Contact angle (deg.) - Gibbs free energy (J) - Surface tension (N/m) - Dynamic viscosity (Pa s) - Density (kg/m³) - cr Critical value - d Detachment - eq Equilibrium - g Gas - K Refers to the Kumar-Kuloor model - l Liquid - m Measured value - t Terrestrial     

Studies on the drag and shape of gas bubbles rising through a stagnant liquid

In connection with our investigations on the motion of gas bubbles rising through a stagnant liquid, experiments were conducted to determine the drag and shape of air bubbles rising freely in water. — A new experimental technique which consists of phototubes, light beams and an electronic circuit has been used to measure the velocity of rise, and the drag coefficient has been determined by equating the buoyancy with the drag. — The size and the deformation of the bubble were determined by a photographic technique. — The results show that, for bubbles rising with a rectilinear motion, theoretical solutions for the drag must be compared with the drag coefficient calculated with the maximum rise velocity, and the results are in good agreement with theory. — There is no wall effect on the rise velocity of the bubble if the ratio of the medium diameter to the equivalent bubble diameter is greater than 18. — Weber number determined according to the maximum velocity is a significant parameter involved in the determination of the shape of air bnbbles. The critical Weber number above which bubbles are not spherical is 0.62, and the bubble surface oscillations begin at the Weber number of 3.70.

---

Zusammenfassung In Verbindung mit unseren Untersuchungen über die Bewegung von Gasblasen, die in einer ruhenden Flüssigkeit aufsteigen, wurden auch Versuche zur Bestimmung des Widerstandes und der Form von frei in Wasser aufsteigenden Luftblasen angestellt. — Zur Messung der Aufstiegsgeschwindigkeit wurde eine neuartige Versuchsanordnung benutzt, bestehend aus Photozellen, Lichtstrahlen und einem elektronischen Schaltkreis. Der Widerstandskoeffizient wurde aus dem Vergleich des Auftriebs mit dem Widerstand ermittelt. Größe und Verformung der Blasen wurden photographisch bestimmt. — Die Versuche zeigen, daß für Blasen mit gradliniger Bahn die theoretische Lösung für den Widerstand mit dem Widerstandskoeffizienten verglichen werden muß, der aus der maximalen Aufstiegsgeschwindigkeit berechnet wird. Die Übereinstimmung zwischen Theorie und Versuch ist gut. — Ein Einfluß der Gefäßwand auf die Aufstiegsgeschwindigkeit ist nicht vorhanden, wenn der Gefäßdurchmesser mehr als 18mal so groß ist wie der äquivalente Blasendurchmesser. — Die Weber-Zahl, gebildet mit der größten Aufstiegsgeschwindigkeit, ist maßgebend für die Blasenform. Die kritische Weber-Zahl, oberhalb deren die Blasen nicht mehr kugelig sind, beträgt 0,62, und bei einer Weber-Zahl von 3,70 beginnt die Blasenwand zu schwingen.

---

Nomenclature C _D Drag coefficient - Drag coefficient determined forv _{z max} - Drag coefficient determined forv _{z ave} - D Drag - Re Reynolds number determined in terms of the equivalent bubble diameter (Re=vd _e/ _l) - We Weber number determined in terms of the equivalent bubble diameter (We=d _{e l} v ²/) - a Major axis of the ellipse - b Minor axis of the ellipse - d _e Equivalent bubble diameter - g Acceleration of gravity - h Helix pitch - m Mass - y Acceleration - _l Dynamic viscosity of the liquid - _g Dynamic viscosity of the gas - _l Kinematic viscosity of the liquid - _l Density of the liquid - _g Density of the gas - Helix slope defined by tg=h/(D _h) - Surface tension - x Bubble deformation (x=a/b) This research is sponsored by the Turkish Scientific and Technical Research Council.     

Simultaneous PIV and pulsed shadow technique in slug flow: a solution for optical problems

A recent technique of simultaneous particle image velocimetry (PIV) and pulsed shadow technique (PST) measurements, using only one black and white CCD camera, is successfully applied to the study of slug flow. The experimental facility and the operating principle are described. The technique is applied to study the liquid flow pattern around individual Taylor bubbles rising in an aqueous solution of glycerol with a dynamic viscosity of 113×10^–3 Pa s. With this technique the optical perturbations found in PIV measurements at the bubble interface are completely solved in the nose and in annular liquid film regions as well as in the rear of the bubble for cases in which the bottom is flat. However, for Taylor bubbles with concave oblate bottoms, some optical distortions appear and are discussed. The measurements achieved a spatial resolution of 0.0022 tube diameters. The results reported show high precision and are in agreement with theoretical and experimental published data.Symbols D internal column diameter (m) - g acceleration due to gravity (m s^–2) - l _w wake length (m) - Q _v liquid volumetric flow rate (m³ s^–1) - r radial position (m) - r * radial position of the wake boundary (m) - R internal column radius (m) - U _s Taylor bubble velocity (m s^–1) - u _z axial component of the velocity (m s^–1) - u _r radial component of the velocity (m s^–1) - z distance from the Taylor bubble nose (m) - Z * distance from the Taylor bubble nose for which the annular liquid film stabilizes (m) Dimensionless groups Re Reynolds number ( ) - N _f inverse viscosity number ( ) Greek letters liquid film thickness (m) - liquid kinematic viscosity (m² s^–1) - liquid dynamic viscosity (Pa s) - liquid density (kg m^–3)     

Flow characteristics of gas-liquid-particle mixing in a gas-stirred ladle system with throughflow

An experimental study is performed on air-liquid-particle mixing, resulting from an air-particle mixture injected into a liquid flowing through a slender ladle. Flow visualization combined with image processing is employed to investigate the bubble and particle behavior at the nozzle outlet. Effort is directed to particle discrimination in both the liquid and the bubbles to determine particle distribution, which affects the mixing performance of gas bubbles, solid particles and liquid. A real-time movement of bubble and particle behavior can be visualized by means of image processing with the use of a slow-motion video recording. It is disclosed that the particles injected through the nozzle may stick on the inner surface of the gas bubble, break through the bubble surface, or mingle with the gas stream to form a two-phase jet, depending on the particle-to-gas mass flow rate ratio. It is observed that when a solid-gas two-phase jet penetrates deeper in the horizontal direction, the particles and bubbles rise along the vertical sidewall and simultaneously spread in the transverse direction, thus promoting a better liquid-particle mixing. The application of the slow-motion video recording results in quantitative evaluations of both the penetration depth of particles or of gas-particles from the injection nozzle and the velocity distribution along the sidewall.List of symbols B Width of water vessel, m - B _n Nozzle location on bottom surface of water vessel, m - d _o Diameter of a gas-particle injection nozzle, m - H Height of water vessel, m - H _n Nozzle location on vertical surface of water vessel, m - L Penetration length of particles or of particles and gas from the nozzle, m - Q _g Volumetric flow rate of gas, m³/s - Q _l Volumetric flow rate of water, m³/s - Q _s Volumetric flow rate of particle, m³/s - Re _g Gas Reynolds number based on inner diameter of the air-particle injection nozzle - t Time, sec. - W Thickness of water vessel, m - x Transverse coordinate, m - y Longitudinal coordinate, m - Mass flow rate ratio of particles to gas Visiting scholar on leave from the Mechanical Engineering Department, Kagoshima University, Kagoshima, JapanThe work reported was supported by the National Science Foundation under the Grant No. CTS-8921584     

The mathematical modelling of thermal radiation in high temperature fluidized bed

The aim of this study is composed of two parts. One of them is to calculate the radiation heat flux and the other is to determine the overall heat transfer coefficient for the gas-fluidized bed. The radiative heat transfer model is developed for predicting the total heat transfer coefficients between submerged surfaces and fluidized beds for several working temperatures. The role of radiation heat transfer in the overall heat transfer process at an immersed surface in a gas-fluidized bed at high temperatures is investigated. Analytical results are compared with the previously done experiments and a good agreement between the two, is obtained.

---

Bestimmung der Wärmeübertragungs-Koeffizienten in Gas-Wirbelschichten

Zusammenfassung Diese Untersuchung besteht aus folgenden zwei Teilen: 1. Kalkulation des Radiationswärmeübergangs in Gas-Wirbelschichten. 2. Bestimmung des Wärmeübergangs-Koeffizienten in Gas-Wirbelschichten. Dieses Radiationswärmeübergangsmodell wurde entwickelt, um die Wärmeübertragungs-Koeffizienten zwischen der eingetauchten Oberfläche und der Wirbelschicht bei verschiedener Wärme schätzungsweise zu bestimmen. Es wurde das Verhältnis der Radiationswärmeübertragung in Gas-Wirbelschichten zum totalen Wärmeübergang untersucht. Die Meßwerte wurden mit theoretischen Resultaten verglichen.

---

Nomenclature c (x) specific heat capacity of packet [J/kg K] - c _p specific heat capacity of particle [J/kg K] - c _pg specific heat capacity of gas [J/kg K] - d _p average diameter of the bed particles [m] - f ₀ the fraction of time that a unit surface exposed to the bubble phase - 1–f ₀ the fraction of time that a unit surface exposed to the packet phase - g acceleration due to gravity [m/s²] - h _b heat transfer coefficient for the surface in contact with bubble [W/m² K] - h _bc conduction heat transfer coefficient for the surface/bubble [W/m²K] - h _br radiation heat transfer coefficient for the surface/bubble [W/m²K] - h _p heat transfer coefficient for the surface in contact with packet [W/m²K] - h _pc conduction heat transfer coefficient for the surface/packet [W/m² K] - h _pr radiation heat transfer coefficient for the surface/packet [W/m² K] - h _T total heat transfer coefficient between bed and surface [W/m² K] - k ⁰ thermal conductivity of the emulsion phase for fixed bed [W/m K] - k(x) thermal conductivity of packet [W/m K] - k _e the logarithmic mean of conductivity for first layer in packet [W/m K] - k _g the logarithmic mean of conductivity for the first layer in packet [W/m K] - K extinction coefficient [1/m] - m mass [kg] - n number of layers - p air pressure [pa] - q _pc mean local conduction heat transfer for packet [kW/m²] - q _pr mean local radiation heat transfer for packet [kW/m²] - Q _p average heat flux during packet contact with surface [kW/m²] - Q _b average heat flux during bubble contact with surface [kW/m²] - R gas constant [287.04 J/kg K] - t time [s] - t _g residence time for gas bubble [s] - t _k residence time for packet [s] - T temperature [K] - T _b bed temperature [K] - T _W surface temperature [K] - V _mf minimum fluidization velocity [m/s] - v _t terminal velocity [m/s] - x distance [m] Greek symbols t time increment - x thickness of the layer - emissivity - thermal diffusivity [m²/s] - (x) voidage of fluidized bed - _mf void ratio of the bed at minimum fluidization - ₀ voidage of fixed bed - _g dynamic viscosity of gas [kg/m s] - _g kinematic viscosity of gas [m²/s] - (x) density of packet [kg/m³] - _p density of particles [kg/m³] - _g density of gas [kg/m³] - Stefan-Boltzmann constant [5.66·10^–8 W/m²K⁴] - geometric shape factor for particles Dimensionless numbers Ar Archimedes numberAr=g d _p ³ ( _p – _g ) _g / _g ² - Nu Nusselt numberNu=h·d/k - Re Reynolds numberRe=d _p ·V _mf / _g - Pr Prandtl numberPr=C _{pg g} /k _g      

Analytical approximation for nonlinear adsorbing solute transport and first-order degradation

Under some constraints, solutes undergoing nonlinear adsorption migrate according to a traveling wave. Analytical traveling wave solutions were used to obtain an approximation for the solute front shape,c(z, t), for the situation of equilibrium nonlinear adsorption and first-order degradation. This approximation describes numerically obtained fronts and breakthrough curves well. It is shown to describe fronts more accurately than a solution based on linearized adsorption. The latter solution accounts neither for the relatively steep downstream solute front nor for the deceleration in time of the nonlinear front.Notation A parameter - c concentration [mol/m³] - c ₀ ^* depth-dependent local maximum concentration [mol/m³] - c; c ₀;c _i concentration difference, feed, and initial resident concentrations, respectively [mol/m³] - D pore scale diffusion/dispersion coefficient [m²/yr] - f adsorption isotherm - f derivative off toc - f second derivative off toc - G ^* parameter - K nonlinear adsorption coefficient [mol/m³)^1–n ] - l column length [m] - L _d dispersivity [m] - m parameter - n Freundlich sorption parameter - P function ofc ₀ ^* - _q change inq [mol/m³] - q adsorbed amount (volumetric basis) [mol/m³] - q derivative ofq toc - R nonlinear retardation factor - retardation factor for concentrationc - R _l linear retardation factor - R(z ^*) depth-dependent average retardation factor, for front at depthz ^* - s adsorbed amount (mass basis) [mol/kg] - t time [years] - u parameter - v flow velocity [m] - z ^* downstream front depth [m] - z depth [m] - transformed coordinate [m] - ^* reference point value of [m] - first-order decay parameter [y^–1] - dry bulk density [kg/m³] - volumetric water fraction - parameter     

Influence of elastic support on the bending of parallelogram plates

Summary The influence of elastic support on the centre deflections and maximum centre and edge moments in clamped parallelogram shaped plates is examined. A polynomial series is assumed for the deflection function, and by applying Galerkin's process, an approximate solution to the governing differential equation is obtained. Convergence of the results were verified. Results for various skew angles and aspect ratios are presented graphically.Nomenclature 2a, 2b lengths of the sides of the parallelogram - a _mn undetermined parameter in series representing the deflection W - c = cosine - D flexural rigidity of plate =Eh ³/[12(1– ²)] - E Young's modulus - h thickness of plate - k, l, m, n, M, N positive integers - p aspect ratio = b/a - q intensity of uniform load - s = sine - u, v oblique co-ordinates - W lateral deflection - x, y Cartesian co-ordinates - reaction of foundation per unit area for unit deflection - adjustable constant, zero or one - outwardly drawn normal - , dimensionless oblique co-ordinates equal to v/b and u/a respectively - skew angle - Poisson's ratio     

The relaxation of concentrated polymer solutions

The focus of this paper is on the viscoelastic properties of concentrated polymer solutions and polymer melts. Dynamic mechanical measurements were performed on various polystyrene/ethylbenzene solutions with polymer concentrations ranging from 40% up to 100% and temperatures from Tg+30°C up to 70°C (230°C for polymer melts). The basis polymers are two commerical grade polystyrenes (BASF) with M _W = 247 kg/mol and 374 kg/mol, respectively. To avoid solvent loss due to evaporating during the measurements, a special sealing technique was used.A phenomenological model which describes quantitatively the relaxation spectrum of concentrated polymer solutions from the flow regime up to the glass transition regime is developed. The relaxation data of the respective polymer melt and the glass transition temperature of the solution are the only input parameters needed. The temperature dependence is described by a universal, concentration invariant WLF-equation. The relaxation spectra are divided into two parts accounting for the entanglement and the segmental relaxation modes, respectively. The relaxation strength related to the flow and entanglement regime scale with c ^2.3, whereas the segmental relaxation strength does not alter with concentration. All relaxation times change with concentration proportional to c ^3.5. Flow curves can be calculated from these relaxation spectra and thus, our results are useful for engineering applications.Roman Symbols a _T Time temperature superposition shift - factor - a _c Time concentration superposition - shift factor in the flow regime - a _c Time concentration superposition - shift factor in the glassy regime - b _T Modulus temperature superposition - shift factor - b _c Modulus concentration shift factor - in the flow regime - b _c Modulus concentration shift factor - in the glassy regime - B Virial coefficients - c Polymer mass fraction kg/kg - c ₁ WLF-parameter - c2 WLF-parameter K - g Relaxation strength of a relaxation Pa mode - G(t) Relaxation modulus Pa - G Storage modulus Pa - G Loss modulus Pa - GN Plateau modulus of linear flexible Pa polymers - (x) Delta function: (0) = 1, - (x<>0)=0 - h() Damping function - H() Relaxation spectrum Pa - J ₀ ^N Recoverable compliance Pa^–1 - m Mass kg - M _c Critical molecular weight kg/mol - M _e Entanglement molecular weight kg/mol - M _w Weight average molecular weight kg/mol - M Number of datapoints - n Scaling exponent - N Number of discrete relaxation modes - T Temperature °C - T _g Glass transition temperature °C - V Volume 1 Greek Symbols Scaling exponent - _f Thermal expansion coefficient K^–1 - Scaling exponent - Shear deformation - Shear rate s^t–1 - Relaxation time s - _c Characteristic relaxation time of thes Cross model - _e Entanglement relaxation time s - Viscosity Pa s - ₀ Zero shear viscosity Pa s - ₀ First normal stress coefficientPa s2 - Segmental friction coefficient - Frequency rad/s Indices f Flow and entanglement regime - g Glass transition regime - i Count parameter - p Polymer - ref Reference state - s Solvent Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Oscillating convection modes in the surroundings of an air bubble under a horizontal heated wall

The development of different oscillatory modes and their transition into a non-periodic state of convection, initiated by the thermal Marangoni-effect in the vicinity of an air bubble under a horizontal, heated wall, was investigated. In the further surroundings of the air bubble a stably stratified thermal field was maintained. The flow phenomena in the vicinity of the bubble were studied using light sheet and shearing interferometer flow visualization techniques. The observed modes are described with regard to their kinematics. The influence of the Marangoni number and of the bubble geometry on the mode selection is discussed. The boundaries of the different modes and of the non-periodic state are indicated.List of symbols a thermal diffusivity - Bo Bond number, Eq. (4) - c phase velocity, Eq. (6) - g acceleration due to gravity - l characteristic length - Mg Marangoni number, Eq. (1) - n wavenumber - Pr Prandtl number ( = v/a) - r radial coordinate - r _B bubble radius - Ra Rayleigh number ( = ga¦T/r¦l ⁴/va) - Re Reynolds number ( = u _mg l/) - t _p oscillation period - T temperature - T _w wall temperature - u _mg characteristical Marangoni velocity, Eq. (2) - z axial coordinate normal to the heated wall - z _B bubble height Greek letters surface tension - kinematic viscosity - dynamic viscosity Dedicated to Professor Dr.-Ing. Julius Siekmann on the occasion of his 65th birthday     

Viscosity of concentrated protein solutions

The viscosity of solutions of four proteins (Bovine Serum Albumin, Ovalbumin, _s-1 Casein, Lysozyme), brought to the random coil conformation, has been measured over a large concentration range extending into the entanglement region. A master curve is obtained in the dilute and semi-dilute regions with the reduced variables and of Simha and Utracki.By using Graessley's expression for the polymer coil expansion at a given concentration in the semi-dilute region (c ^* c c ^**), a simple equation is established giving the relative viscosity _r as a function of concentrationc: forc ^* c c ^**, ln _r = 2a[]c ^*(c/c ^*)^1/2a – (2a - 1)[]c ^*; wherec ^* is the incipient overlap concentration, [] the intrinsic viscosity, anda the Mark-Houwink exponent for the polymer-solvent considered.This equation fits well the experimental results. The adjustment yields for the parametera values which are comprised between 0.6 and 0.7, as expected, for Bovine Serum Albumin and Ovalbumin, but very close to 0.5 for _s-1 Casein and Lysozyme. This can be explained by the fact that the molecular weights of the two latter proteins are lower than, or very close, the critical molecular weight; the critical molecular weight is estimated to be about 20000.     

An experimental study on vortex breakdown in a differentially-rotating cylindrical container

The vortex breakdown phenomenon in a closed cylindrical container with a rotating endwall disk was reproduced. Visualizations were performed to capture the prominent flow characteristics. The locations of the stagnation points of breakdown bubbles and the attendant global flow features were in excellent agreement with the preceding observations. Experiments were also carried out in a differentially-rotating cylindrical container in which the top endwall rotates at a relatively high angular velocity _t, and the bottom endwall and the sidewall rotate at a low angular velocity _sb. For a fixed cylinder aspect ratio, and for a given relative rotational Reynolds number based on the angular velocity difference _t–_sb, the flow behavior is examined as |_sb/_t| increases. For a co-rotation (_sb/_t>0), the breakdown bubble is located closer to the bottom endwall disk. However, for a counter-rotation (_sb/_t<0), the bubble is seen closer to the top endwall disk. For sufficiently large values of _sb, the bubble ceases to exist for both cases.     

Dependence of the stability criterion of nucleate boiling on the compressibility of the vapor

The article gives the results of a study of the motion of bubbles and their deformation near the heating surface at different pressures. It was observed that, during the time of their growth, the gaseous medium in the bubbles is in a compressed state.Nomenclature R) radius of bubble - R_h) maximul radius of a deformed bubble in the horizontal plane - R_v) maximal radius of a deformed bubble in the vertical plane - ) specific weight - B) universal gas constant - ) surface-tension coefficient - p) pressure - ) edge wetting angle - g) acceleration due to gravity - V) volume - ) molecular weight - C_T) isothermal velocity of sound Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 77–81, July–August, 1971.     

Concentration-dependent behaviour of the shear viscosity of coal-fuel oil suspensions

A function correlating the relative viscosity of a suspension of solid particles in liquids to their concentration is derived here theoretically using only general thermodynamic ideas, with out any consideration of microscopic hydrodynamic models. This function ( _r = exp (1/2B ^* C ²)) has a great advantage over the many different functions proposed in literature, for it depends on a single parameter,B ^*, and is therefore concise. To test the validity of this function, a least-squares regression analysis was undertaken of available data on the viscosity and concentration of suspensions of coal particles in fuel oil, which promise to be a useful alternative to fuel oil in the near future. The proposed function was found to accurately describe the concentration-dependent behaviour of the relative viscosity of these suspensions. Furthermore, an attempt was made to obtain information about the factors affecting the value ofB ^*, however the results were only qualitative because of, among other things, the inaccuracy of the viscosity measurements in such highly viscous fluids. shear viscosity of the suspension - ₀ shear viscosity of the Newtonian suspending medium - _r = /₀ relative viscosity - solid volume concentration - c solid weight concentration - _m maximum attainable volume concentration of solids - solid volume concentration at which the relative viscosity of the suspension becomes infinite - c _m maximum attainable solid weight concentration - _s density of the solid phase - _l density of the liquid phase - _m density of the suspension - k _n coefficients of theø-power series expansion of _r - { _j } sets of parameters specifying the thermodynamic state of the solid phase of a suspension - T absolute temperature (K) - f (c, T, _j) formal expression for the relative variation of the viscosity with concentration = [1 / (/c)] _T,j - d median size of the granulometric distribution - _B plastic or Bingham viscosity - K consistency factor - n flow index - g ([c _m –c],T, _j ) function including an asymptotic divergence asc tends toc _m , formally describing the concentration dependent behaviour of the shear viscosity of a suspension - A (T, _j) regression analysis parameters - B (T, _j) regression analysis parameters - B ^* (T, _j ) regression analysis parameters     

Transient temperature response of a porous matrix entering a planetary atmosphere

Summary An analysis is made of the transient temperature behavior of a transpiration-cooled porous matrix entering a planetary atmosphere with constant velocity and negative entry angle. The analysis is based on one dimensional heat conduction in a porous plate subjected to a time dependent heat flux at one side and cooled internally by mass injection from a constant temperature reservoir at the opposite side. An exact closed-form solution is obtained and temperature charts are presented for a wide range of Fourier number and coolant flow parameters.Nomenclature A surface area, ft² - C constant, 17,600 Btu/ft^3/2-sec - C _c constant pressure specific heat of coolant, Btu/lb_m-°F - g local gravitational acceleration, ft/sec² - g _c coolant flow parameter, defined by equation (15) - h height of entry above planet surface, ft - K ₁ ratio of local heat flux to stagnation point heat flux - K thermal conductivity of plate material, Btu/sec-ft-°F - L plate thickness, ft - m constant, 3.15 - m _c coolant mass flow rate, lb_m/sec - M _n roots of equation (33) - n constant, 0.50 - N defined by equation (30) - P porosity - q surface heat flux, Btu/ft²-sec - q ₀ surface heat flux at t=0, Btu/ft²-sec, defined by (6) - r distance from planet center, ft - R radius of curvature at stagnation point, ft - t time, sec - T temperature, °F - T _c coolant supply temperature, °F - V velocity, ft/sec - x normal coordinate through plate, ft - y altitude, ft - thermal diffusivity of plate, ft²/sec - atmospheric density decay parameter, 1/23500 ft^–1 - flight path angle relative to local horizontal direction, positive for climbing and negative for descent, deg - dimensionless temperature parameter, defined by (12) - dimensionless distance, defined by (13) - free stream atmospheric density, slug/ft³ - ₀ atmospheric density at reference state, slug/ft³ - Fourier number, defined by (14) - 1/sec, defined by (7) - flight entry parameter, defined by (16)     

Pressure wave breakdown of a gas film in a liquid-filled slit

In [1–4] the results of investigating the breakdown of gas bubbles by medium-intensity pressure waves are presented and various bubble breakdown mechanisms are proposed. It is shown that breakdown may occur as a result of the formation of a cumulative jet on the boundary of the bubble or as a result of instability due to the relative motion of the bubble in the wave. In [5] experimental data on the pressure wave breakdown of a gas film in a liquid on a solid wall are reported. It is shown that at wave amplitudes p/p₀1 a liquid jet is formed at the edge of the gas film. The jet, traveling along the wall, strips off the film and carries it into the surrounding liquid. Below we investigate the pressure wave behavior of a gas film in a liquid-filled slit.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 175–178, September–October, 1992.