Influence of free convection on the heat transfer from hot-wire probes

The present paper is concerned with the experimental determination of the influence of free convection on the heat transfer from horizontal hot-wire probes in cross flow. Free convective flow can be minimized under microgravity conditions and its quantity can be determined by comparisons with terrestrial investigations. Using the developod experimental setups it is possible to investigate all three flow regimes, i.e. pure free, mixed, and pure forced convection (o Re 0.1). It was found that the influence of free convection is limited to Reynolds numberRe 0.0045, independent of the overheat ratio. Based on the findings of Collis and Williams [12], the influence of free convection can be neglected forRe > 0.58 ·Gr ^1/3. The fluid properties are computed at the film temperature. Two correlations are established for the regime of pure forced convection. End losses to the supports were determined in a vacuum experiment, thus allowing comparison with theoretical investigations. The difference between the analytical computations and the measurement data in the range 0 Re 0.02 is caused by the three-dimensional heat transfer occuring in the experimental investigation.Die vorliegende Untersuchung beschäftigt sich mit der experimentellen Bestimmung des Einflusses der freien Konvektion auf den Wärmeübergang bei quergeströmten Hitzdrahtsonden. Die freie Konvektionsströmung kann unter Schwerelosigkeit minimiert und ihre Größe durch den Vergleich mit terrestrischen Untersuchungen ermittelt werden. Mit den erstellten Experimentaufbauten ist es möglich, alle drei Strömungsbereiche von der rein freien über die gemischte bis zur rein erzwungenen Konvektion zu untersuchen (0 <>Re 0,1). Dabei wurde festgestellt, daß der Einfluß der freien Konvektion sich unabhängig von Überhitzungsverhältnis nur bis zu einer ReynoldszahlRe 0,0045 erstreckt. Zur Vernachlässigung der freien Konvektion kann in Anlehnung an Collins und Williams folgende Beziehung angegeben werden:Re > 0,58 ·Gr ^1/3. Die Stoffwerte werden bei der Filmtemperatur berechnet. Für den Bernich der rein erzwungenen Konvektion wurden zwei Korrelationer aufgestellt. Durch einen Vakuumversuch wurden die Endverluste in die Haltespitzen ermittelt und somit der Vergleich mit theoretischen Arbeiten ermöglicht. Der Unterschied zwischen den analytischen Berechnungen und den Meßdaten im Bereich 0 Re 0,02 ergibt sich durch den dreidimensionalen Wärmeübergang bei der experimentellen Untersuchung.The authors acknowledge the support of the Deutsche Agentur für Raumfahrtangelegenheiten (DARA) under grant number 50-QV 8898-1.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Unsteady forced convection heat/mass transfer from a flat plate

Numerical methods are used to investigate the transient, forced convection heat/mass transfer from a finite flat plate to a steady stream of viscous, incompressible fluid. The temperature/concentration inside the plate is considered uniform. The heat/mass balance equations were solved in elliptic cylindrical coordinates by a finite difference implicit ADI method. These solutions span the parameter ranges 10 Re 400 and 0.1 Pr 10. The computations were focused on the influence of the product (aspect ratio) × (volume heat capacity ratio/Henry number) on the heat/mass transfer rate. The occurrence on the plates surface of heat/mass wake phenomena was also studied.     

Flow of rarefied gas past a sweptback cylinder

A numerical investigation has been made of the hypersonic flow of a rarefied monatomic gas past the windward part of the side surface of an infinite circular cylinder. The calculation was made by direct statistical Monte Carlo modeling for freestream Mach number M_t8=20, ratio of the surface temperature of the body to the stagnation temperature equal to t_tw =T _tw/T _t0 = 0.03, sweep angle 75°, and Reynolds number Re_t0 30.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1992.     

Slow flow of gas past a strongly heated sphere in the presence of blowing and evaporation from its surface

A solution is obtained to the problem of the simultaneous influence of blowing (evaporation) and large temperature differences on the flow past a sphere and on the force acting on it with allowance for the Burnett thermal stresses in the momentum equation. It is assumed that the Reynolds numbers calculated using the blowing velocity and the velocity of the oncoming flow, respectively, have the order Re_w 1 and Re 1. The temperature difference is determined by the boundary conditions, namely, a constant temperature T_w T on the surface of the sphere (VT/T 1). The problem is solved by the method of matched asymptotic expansions with respect to the small parameter Re. The equations reduce to a system of ordinary differential equations, which are solved numerically by the orthogonal sweep method [1]. It is found that under certain conditions the drag of the sphere can become negative.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–134, July–August, 1982.I thank O. G. Fridlender for valuable advice and interest in the work.     

Approximate relationships for the determination of the pressure on the surface of a sphere or cylinder with an arbitrary mach number in the free stream

Numerical calculations have been made [1–4] of the pressure distribution over the surface of a sphere or cylinder during transverse flow in the range 0 /2, where is the angle reckoned from the stagnation point along the meridional plane, and on the basis of these results simple analytical equations have been proposed in order to determine the pressure for arbitrary Mach numbers M in the free stream. The gas is assumed to be ideal and perfect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 185–188, March–April, 1985.     

Experimental investigation of separation of turbulent boundary layer in vicinity of a step

The results of an experimental investigation of the separation of a turbulent boundary layer in the vicinity of a step on a flat plate at M = 2 and 3, and Re = U/v = (26–66)·10⁶ m^–1 are given. The step height was varied from 3 to 16 mm, which corresponded to the range of relative heights 1.1 h/ 7.6, where is the thickness of the boundary layer at the point at which the pressure starts to increase in front of the step. The obtained data for the pressure distribution in front of the step, and on its face and top surface, and the results of probe measurements in the separation and adjacent regions provide a more accurate scheme of the flow. The obtained data are compared with the results of other investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 73–80, September–October, 1977.We express our thanks to A. M. Kharitonov for valuable comments made during the discussion of this work, and also to M. A. Gol'dfel'd for kindly providing the experimental data for axisytnmetric steps.     

Zum Einfluß der Porengeometrie auf den Druckverlust bei der Durchströmung von Porensystemen

Zusammenfassung Es wird das Strömungsverhalten newtonscher und nicht-newtonscher Fluide in Kanälen mit variierendem Querschnitt untersucht, deren Konturen aus einer analytischen Betrachtung geordneter Kugelpackungen hergeleitet werden. Die untersuchten Kanäle sind den zugeordneten Kugelpackungen insofern ähnlich, als sich deren kreisförmige Strömungsquerschnitte in Fließrichtung in der gleichen Weise ändern wie die Porenquerschnitte entsprechender Packungstypen.Auf der Suche nach geeigneten Kennzahlen zur Beschreibung des Strömungsproblems für newtonsche Fluide werden verschiedene Darstellungsarten der Widerstandskennlinien = f(Re) diskutiert. Hierbei zeigen die Modellkanäle im Bereich der schleichenden und der turbulenten Strömung das gleiche Widerstandsverhalten wie die entsprechenden Packungsarten, wenn die Kennzahlen undRe analog mit dem hydraulischen Durchmesser und der nachDupuit definierten mittleren Kanalgeschwindigkeit gebildet werden. Während sich bei der Durchströmung von Kugelpackungen der Übergang von der schleichenden zur turbulenten Strömung allmählich vollzieht, ist jedoch bei der untersuchten Kanalströmung ein charakteristisches Übergangsverhalten festzustellen.Mit Hilfe einer geeignet definierten Durchmesserkenngröße, die sich theoretisch aus der Kanalgeometrie berechnen läßt und einer von diesem Durchmesserparameter abhängig gewählten Kanalgeschwindigkeit kann das Widerstandsverhalten aller Kanäle für newtonsche und nicht-newtonsche Fluide im Bereich der schleichenden Strömung durch die bekannte Hagen-Poiseuille-Gleichung beschrieben werden. Zur Bestimmung der scherabhängigen Viskosität nicht-newtonscher Fluide wird ein modifiziertes vonChmiel undSchlümmer vorgeschlagenes Berechnungsverfahren herangezogen, wobei die oben angeführten Geschwindigkeits- und Durchmesserparameter zur Definition einer charakteristischen SchergeschwindigkeitD _rep verwendet werden.Darüber hinaus lassen sich die Widerstandskennlinien für newtonsche und wenig elastische nichtnewtonsche Fluide im BereichRe 30 undRe 1200 durch die Gleichung = 64/Re + C_t/Re^0,1 beschreiben; hierbei hängt der FaktorC _t, für den eine halbempirische Berechnungsgleichung angegeben wird, mit 0,39 C _t 1,7 allein von der Kanalgeometrie ab. Demgegenüber zeigen Fluide mit ausgeprägten viskoelastischen Eigenschaften fürRe > 1 ein deutlich überhöhtes Druckverlustverhalten, das von der Lösungskonzentration und von der Kanalgeometrie beeinflußt wird. Gestützt auf die experimentellen Ergebnisse kann an Hand theoretischer Überlegungen gezeigt werden, daß bis zuRe 300 das auf viskoelastischen Phänomenen beruhende Widerstandsverhalten neben den genannten Kennzahlen undRe durch eine ÄhnlichkeitskennzahlDe = _fl D _rep charakterisiert werden kann, wobei _fl eine fluidspezifische Relaxationszeit bedeutet.

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Summary In this paper, the flow of Newtonian and non-Newtonian fluids in channels with varying diameter along the longitudinal axis is investigated. The contures of these channels are calculated by means of the analysis of systematical packings of spheres. The investigated channels are similar to the appointed packings as their circular cross-sections vary in the same way as the cross-sections of the pores in the corresponding types of packings.In search of appropriate characteristic numbers for the flow problem of Newtonian fluids different kinds of representation of the friction factor — Reynolds number correlation, = f(N_Re) are discussed. If the characteristic numbers andN _Re are built analogously to the hydraulic diameter and the flow rate in pores, according toDupuit's definition the models show the same pressure-drop characteristics as the corresponding types of packing, in creeping flow as well as in fully developed turbulent flow. In contrast to the flow through packed beds, where the transition from laminar to turbulent flow occurs continuously, the flow through the channels shows a remarkably different transition behaviour forN _Re 600.By means of an appropriately defined characteristic diameter which is to be calculated theoretically from the channel geometry, and by means of a characteristic flow rate, which depends on this diameter, the flow characteristics of all channels can be described by the well known Hagen-Poiseuille equation, for creeping flows of Newtonian as well as non-Newtonian liquids. For the determination of the shear-rate dependent viscosity of non-Newtonian fluids, a modified method, proposed byChmiel andSchümmer, is used, i.e. the shear rateD _rep is defined with the above specified exoressions of the diameter and the characteristic flow rate.Furthermore, the friction factor — Reynolds number correlation for Newtonian and inelastic or slightly elastic non-Newtonian liquids can be desribed in the rangeN _Re 30 andN _Re 1200 by the equation = 64/N_Re + C_t/N _Re ^0.1 where the factorC _t, for which a semi-empiric equation is given, depends only on the geometry of the channels, with values of 0.39 C _t 1.7. In contrast to this liquids with strong viscoelastic properties show atN _Re > 1 a pronounced excess pressure drop, which depends on the concentration of the polymer solution and the channel geometry. Based on experimental results it is demonstrated by theoretical arguments that up toN _Re 600 the flow characteristics depending on viscoelastic phenomena can be described with the characteristic numberN _De = _fl D _rep, in addition to andN _Re. Here _fl signifies the relaxation time of the polymer solution.

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A _ges Gesamtfläche - A _f,A _p Porenquerschnitt - A _i benetzte Oberfläche - A _s Fläche - A ₀ Anströmquerschnitt - c Formfaktor - C ₁,C ₂ Konstanten im Widerstandsgesetz - d Durchmesser - d _k Partikeldurchmesser - d _p Porendurchmesser, Kanaldurchmesser - D Schergeschwindigkeit - D _rep repräsentative Schergeschwindigkeit - h, hydr hydraulisch (als Index) - i Zählindex, Packungsindex - j Zählindex - k Potenzgesetzkonstante - KZ Koordinatenzahl - l Länge - l ₀ Kugellagenabstand, Periodizitätsfaktor - L Höhe eines porösen Körpers - l _e,L _e effektive Länden - MK Modellkanal - n Potenzgesetz-Exponent - p Druck - P Bezeichnung für Packung - r Radius - S spezifische Oberfläche - t Zeit - V Volumen, Gesamtvolumen - V _f Hohlraumvolumen - V _s Feststoffvolumen - Volumenstrom - w Geschwindigkeit - w ₀ Anströmgeschwindigkeit - x,y,z kartesische Koordinaten - z Hauptfließrichtung - reziproke Permeabilität - Trägheitskoeffizient - _f Flächenlückengrad - _v Volumenporosität - dynamische Viskosität - _rep repräsentative Viskosität - ₂₉₃ dynamische Viskosität bei 293 K - Temperatur - Relaxationszeit - µ Umlenkfaktor - Dichte - Schubspannung - Durchmesserverhältnis - D/Dt Substantielle Ableitung - über die Fläche gemittelter oder arithmetisch gemittelter Wert einer Funktionf - f über das Volumen gemittelter Wert einer Funktionf - II _d zweite Invariante des Deformationsgeschwindigkeitstensors Erster Teil einer von der Abteilung Chemietechnik der Universität Dortmund genehmigten Dissertation; auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Berlin vom 8.–10. Mai 1978.Mit 27 Abbildungen und 5 Tabellen     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.

     

On the approximate solution of elastic contact problems for a circular annulus

Paper concerns the Boussinesq and Reissner-Sagoci contact problems for a rigid punch in contact with the plane face of an isotropic elastic half-space over a circular annulusbra. It is shown that a simple combination of the interior and exterior Dirichlet and Neumann solutions for a circle, when expressed in terms of oblate spheriodal coordinates, yields extremely accurate approximate solutions provided that 0b/a0.8.     

A MEMS skin-friction sensor for time resolved measurements in separated flows

A MEMS surface fence sensor for skin-friction measurements in separated flows has been developed and tested successfully in low-speed flows. The new sensor is able to distinguish between forward and reverse flow and features a temporal resolution of up to 1 kHz. Calibrations of the sensor have been obtained in the range of –0.7 N m^-2_w0.7 N m^-2 with a resolution of 0.02 N m^-2. Comparative measurements with the wall-pulsed wire technique in a reverse-flow region show excellent agreement with respect to the mean skin-friction coefficient c_f but also reveal some discrepancies for the fluctuating part c_f.     

Torsion of two bonded layers by a rigid disk

The problem considered in this paper describes the torsion of a homogeneous isotropic elastic layer (0zd ₁) of finite thickness d ₁, perfectly bonded to another elastic layer (-d ₂z0) of finite thickness d ₂. The problem is reduced to the solution of a Fredholm integral equation of the second kind. The solutions are given for some particular cases.

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Sommario In questo lavoro si considera il problema della torsione di uno strato elastico omogeneo ed isotropo (0zd ₁) di spessore finito d ₁, perfettamente incollato ad un altro strato elastico (-d ₂z0) di spessore finito d ₂. II problema é ricondotto alla soluzione di una equazione integrale di Freedholm del secondo ordine. Le soluzioni sono ottenute per alcuni casi particolari.

     

Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature

The purpose of this study is the construction of interpolation formulas for the dependence of Maxwell viscosity, a quantity which is the reciprocal of shear-strain relaxation time , on shear-strain intensity and temperature for several metals: iron, aluminum, copper, and lead. This function was interpolated in various temperature and deformation velocity ranges in accordance with available experimental data for iron (0 10⁷ sec^–1, 200 ° T 1500 °); aluminum (0 10⁷ sec^–1, 300 ° T 900 °); copper (0 10⁵ sec^–1, 300 ° T 1300 °); lead (0 10⁶ sec^–1, 90 ° T 400 °); temperatures in °K.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 114–118, July–August, 1974.     

Spherical couette flow: Transitions and onset of chaos

The results are given of an experimental investigation of the transition to chaos and of the properties of the chaotic regimes in a wide range of Reynolds numbers: 460 Re 3200 7Re₀. Estimates of the probability dimension of the attractors and Lyapunov exponents and the exponential damping of the highest-frequency part of the spectrum indicate a deterministic nature of the chaos in the considered range of Re. It is noted that in one and the same range of Re values the route to chaos is not unique and can depend on the prehistory of the flow development; the simultaneous existence of chaotic and regular regimes for fixed values of the parameters is also noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–18, January–February, 1991.     

Investigation of supersonic viscous flow past a sphere in the presence of subsonic and sonic injection

Supersonic flow past a sphere with a given rate of gas injection along the generator is investigated numerically on the range Re=10²–10⁴. Calculations have been made on the interval 0 90°, where is the angle between the axis of symmetry and the normal to the surface. It is shown that for high subsonic and sonic injection rates it is possible to observe qualitatively new features in the flow structure and in the distribution of the local supersonic flow characteristics around the perimeter of the sphere not previously noted in [9]. In the case of sonic injection the changes in flow structure occur only in the supersonic zone. In the neighborhood of the transition from a subsonic to sonic injection velocity the heat flux has a local maximum, which in absolute value does not exceed the heat flux in the absence of injection. It is shown that there may be qualitative differences in the pressure distribution over the surface of the body with increase in the injection parameter depending on the distribution and value of the injected gas flow rate and, moreover, the number Re.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–89, January–February, 1988.     

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.Nomenclature a radius of sphere - d the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - E energy dissipation rate in one-half the liquid volume separating the spheres - E _cell total energy dissipation rate in the cell - E _homogeneous energy dissipation rate in the cell of a hypothetical one phase fluid - E _interaction energy dissipation rate in the cell due to sphere interactions - E _sphere energy dissipation rate in the cell due to the sphere at the cell center - F force on one sphere - h minimum separation distance between two spheres - J (1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - p pressure - W velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers - _i unit vectors in thei-th direction - elongation rate - viscosity of the suspending fluid - _r */ = relative viscosity - * viscosity of the suspension - the total stress tensor - the part of the total stress tensor that vanishes at equilibrium - volume fraction of spheres     

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameter=(Ra) ^1/4/(Re)^1/2, with=Pr/(1+Pr) and=Pr/(1 +Pr)^1/2, is proposed to replace the conventional Richardson number, Gr/Re², for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra ^1/4/(Re ^1/2 Pr ^1/3) forPr 1, and (RaPr)^1/2/(RePr ^1/2 forPr 1. Furthermore, new coordinates and dependent variables are properly defined in terms of, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

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Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameter=( Ra) ^1/4 (Re)^1/2, mit=Pr/(1 +Pr) und=Pr/(1 +Pr)^1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re², ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra ^1/4/(Re ^1/2 Pr ^1/3) fürPr 1 und (RaPr)^1/4/(RePr)^1/2 fürPr 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion von definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Nomenclature C _f local friction coefficient - C _p specific heat capacity - f reduced stream function - g gravitational acceleration - Gr local Grashoff number,g T _w –T )x³/v² - Nu local Nusselt number - Pr Prandtl number,v/ - Ra local Rayleigh number,g T _w –T x ³/( v) - Re local Reynolds number,u x/v - Ri Richardson number,Gr/Re ² - T fluid temperature - T _w wall temperature - T free stream temperature - u velocity component in thex direction - u free stream velocity - v velocity component in they direction - x vertical coordinate measuring from the leading edge - y horizontal coordinate Greek symbols thermal diffusivity - thermal expansion coefficient - mixed convection parameter (Ra)1/4/Re)^1/2 - pseudo-similarity variable,(y/x) - ₀ conventional similarity variable,(y/x)Re ^1/2 - dimensionless temperature, (T–T T _W –T - unified mixed-flow parameter, [(Re) ^1/2 + (Ra)^1/4] - dynamic viscosity - kinematic viscosity - stretched streamwise coordinate or mixed convection parameter, [1 + (Re)^1/2/(Ra) ^1/4]^–1=/(1 +) - density - Pr/(1 + Pr) _w wall shear stress - stream function - Pr/(l+Pr)^1/3 This research was supported by a grand from the National Science Council of ROC     

The unsteady motion of a sphere in an elastico-viscous liquid

Summary Consideration is given to the unsteady motion of a sphere moving under a constant force in elastico-viscous liquid, the sphere being initially at rest. The problem is solved by the familiarLaplace-transform method. It is shown that the presence of elasticity in the liquid increases the velocity of the sphere. This increase is very pronounced for small values of the time, but is not very significant for large values of the time. It is concluded that the time taken to reach the terminal velocity is unaffected by the presence of elasticity in the liquid.Formulae are given which may be of use to experimentalists who use the falling-ball method to determine the zero shear-rate viscosity.     

Flow Above a Heated Plate in an Ascending Current

An analysis of the results of numerical experiments in which the two-dimensional flow near a plate placed across an ascending fluid current was simulated is presented. The plate temperature was higher than that of the fluid. Fluid flows with a Prandtl number 0.25 Pr 7 were considered on the moderate Reynolds and Richardson number ranges 25 < Re 100 and 0 Ri < 20. Under these conditions, two flow patterns were observable, which differed from each other by the intensity of the transverse oscillations of a system consisting of attached twin vortices and the near wake. For different Prandtl numbers, in the (Re, Ri^1/2) plane the pattern stability boundaries were established, together with the distinctive features of pattern-to-pattern transition. It was found that the vortex arrangement in the wake above the heated plate can differ from that in the von Kàrmàn street in the absence of buoyancy     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress