Explicit representation of the flow parameters of an incompressible fluid on a near-elliptic contour

The theory of conformai mapping of similar domains [4] is used to obtain a simple form of the velocity (pressure) distribution on an arbitrary smooth profile representing a variation of an elliptic contour. The expression for the velocity can be used, in particular, to establish the characteristics of wavy profiles, to determine more accurately the cavity shape at low cavitation numbers, which is known to be almost elliptical [5], in film boiling problems [6], etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 96–102, July–August, 1988.     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

Degenerate Conformally Invariant Fully Nonlinear Elliptic Equations

We establish Liouville theorems for , entire solutions and locally Lipschitz entire weak solutions to general degenerate conformally invariant fully nonlinear elliptic equations of second order. For applications to local gradient estimates of solutions of general conformally invariant fully nonlinear elliptic equations of second order, see [20].     

Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter \({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\) and the eccentricity \({e \in [0, 1)}\) . We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed \({e\in [0, 1)}\) , we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are ?1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail.     

Development characteristics of drag-reducing surfactant solution flow in a duct

Development characteristics of dilute cationic surfactant solution flow have been studied through the measurements of the time characteristics of surfactant solution by birefringence experiments and of the streamwise mean velocity profiles of surfactant solution duct flow by a laser Doppler velocimetry system. For both experiments, the concentration of cationic surfactant (oleylbishydroxymethylethylammonium chloride: Ethoquad O/12) was kept constant at 1000 ppm and the molar ratio of the counter ion of sodium salicylate to the surfactants was at 1.5. From the birefringence experiments, dilute surfactant solution shows very long retardation time corresponding to micellar shear induced structure formation. This causes very slow flow development of surfactant solution in a duct. Even at the end of the test section with the distance of 112 times of hydraulic diameter form the inlet, the flow is not fully developed but still has the developing boundary layer characteristics on the duct wall. From the time characteristics and the boundary layer development, it is concluded that the entry length of 1000 to 2000 times hydraulic diameter is required for fully developed surfactant solution flow.List of abbreviations and symbols A₁, A₂ Coefficients for time constant fitting [-] - B Breadth of the test duct [m] - C₁, C₂ Coefficients for time constant fitting [-] - D Pipe diameter [m] - D_H Hydraulic diameter [m] - g Impulse response function [Pa] - H Width of the test duct [m] - n Index of Bird-Carreau model [-] - Re Reynolds number (=U_mD_H/) - Re_D Pipe Reynolds number (=U_mD/) - Re_x Streamwise distance Reynolds number (=U₀x/) - T Absolute temperature [K] - t Time [s] - t_a Retardation time [s] - t_b Build-up time [s] - t_x Relaxation time [s] - t_x1, t_x2 Relaxation time for double time constant fitting [s] - t Time constant in Bird-Carreau model [s] - U Time mean velocity [m/s] - U_m Bulk mean velocity [m/s] - U_max Maximum velocity in a pipe [m/s] - U₀ Main flow velocity [m/s] - u Friction velocity [m/s] - x, y Coordinates [m] - Shear rate [s^–1] - Mean shear rate [s^–1] - n Birefringence [-] - 99% boundary layer thickness [m] - Solution viscosity [Pa·s] - _P, _S Surfactant and solvent viscosity [Pa·s] - ₀, Zero and infinite viscosity of Bird-Carreau model [Pa·s] - Characteristic time in Maxwell model [s] - Water kinematic viscosity [m²/s] - Density [kg/m³] - Solution shear stress [Pa] - _P, _S Surfactant and solvent shear stress [Pa] - Time in convolution [s]     

Plane entry flows and energy estimates for the navier-stokes equations

One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the end effect, and the celebrated Saint-Venant's Principle of the theory of elasticity.In this paper, I consider the two-dimensional analog of the problem treated in [4] with a view to providing a more explicit formulation of the energy approach to entry flow problems. The flow development in a semi-infinite channel with parallel-plates is analyzed within the framework of the stationary Navier-Stokes equations. Introduction of a stream function leads to a formulation in terms of a boundary-value problem for a single fourth order nonlinear elliptic equation. In the case of Stokes flow, this problem is formally equivalent to a boundary-value problem for the biharmonic equation considered by Knowles [5] in the analysis of Saint-Venant's Principle in plane elasticity. The main result is an explicit estimate which establishes the exponential spatial flow development and leads to an upper bound for an appropriately defined entrance length. These results are obtained using differential inequality techniques analogous to those developed in investigation of Saint-Venant's Principle.     

PIV measurements of velocities and concentrations of wood fibres in pneumatic transport

A method for simultaneous measurement of the concentration and velocity of wood fibres suspended in air was developed. The velocity of the wood fibres was measured by the use of particle image velocimetry (PIV). The concentration of wood fibres was measured using the raw images from the PIV equipment as input data. An image processing procedure was used to determine the volume fraction of the fibre particles in the images. The method gave good qualitative and quantitative results for low volume fractions of fibres; for higher volume fractions the quantitative results were unsatisfactory.Latin letters C Concentration of fibres [g/m³] - d Diameter of fibre [m] - M_w Water mass [kg] - M_f Fibre mass [kg] - m Calibration mass flow [kg/s] - m₂₅ Calibration mass flow at C=25 g/m³ [kg/s] - n Fan rpm [-] - t Thickness of light sheet [m] - t Time between laser pulses [s] - U_i Velocity component in i-direction [m/s] - v Streamwise velocity [m/s] - v_average Average streamwise velocity [m/s] - x_i Particle displacement in i-direction [m] Greek letters _f Volume fraction of fibres [-] - _average Average volume fraction of fibres [-] - Area fraction of fibres in image [-] - Density of fibre particle [kg/m³]     

A new method for measuring time constants of a thermocouple wire in varying flow states

A new measuring method is suggested for determining the time constant of a thermocouple wire to be applied for the measurement of the true fluid temperatures in varying flow states. Based on the techniques of internal heating which are commonly used to measure mean time constants, we extend the existing method to measure instantaneous time constants continuously. A method of measurement and analysis is presented and verified experimentally.List of Symbols A _s surface area [m²] - c specific heat [J/kg K] - D diameter [m] - h heat transfer coefficient [W/m² K] - I current [A] - k thermal conductivity [W/m K] - L length [m] - r resistance per unit length [/m] - T temperature [°C] - t time [s] - t _c characteristic time to reach uniform state [s] - u velocity of stream [m/s] - V volume [m³] - x axial coordinate [m] - thermal diffusivity [m²/s] - normalized temperature (TT )/(T _RT )) - density [kg/m³] - time constant [s] - angular velocity [rad/s] - a amplitude - i initial condition - j junction of thermocouple - R reference point - surrounding The work was supported by Turbo and Power Machinery Research Center at Seoul National University and the authors are grateful to Mr. M. H. Yang for his assistance in the experiment.     

Addendum to “Concentration and Lack of Observability of Waves in Highly Heterogeneous Media”

In [1] we introduced a class of 1?d wave equations with rapidly oscillating Hölder continuous coefficients for which the classical boundary observability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low regularity coefficients. The object of this addendum is to further analyze this issue. As we will see, the argument in [1] only provides sharp counter-examples to the Strichartz estimates when the coefficient ρ belongs to L ^∞. We carefully analyze these counter-examples for Hölder continuous coefficients. We also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough.     

Über die Brauchbarkeit des Newtonschen Abkühlungsgesetzes im Lichte der Fourierschen Wärmeleitungstheorie

Zusammenfassung Es wird gezeigt, daß instationäre Wärme und Stoffübertragungsvorgänge mit sehr guter Näherung in einfacher Weise durch Verwendung zeitabhängiger Wärme- bzw. Stoffübergangskoeffizienten berechnet werden können. Am Beispiel eines Ionenaustauschprozesses, dessen strenge mathematische Behandlung nur numerisch möglich wäre, wird die Brauchbarkeit der Näherung nachgewiesen.

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Application of Newton's law under consideration of the heat conduction theory by fourier

Unsteady heat- and mass transfer can be calculated approximately in a simple manner by applying time dependent transfer coefficients.-For an ion-exchange process e.g., for which the rigorous mathematical treatment could be done only by the help of numerical methods, the usefulness of the approximation is demonstrated.

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Bezeichnungen A [m²] stoffaustauschende Oberfläche - a [m²/s] Temperaturleitfähigkeit - a [W/m² °K] Wärmeübergangskoeffizient - [m/s] Stoffübergangskoeffizient - C [W/(°K)⁴m²] Strahlungskoeffizient - c [Joule/kg °K] Wärmekapazität - [m²/s] Diffusionskoeffizient - [°K] Temperatur - k [W/m² °K] Wärmedurchgangskoeffizient - [W/m °K] Wärmeleitfähigkeit - n [mmol/m²s] Molenstromdichte der Säureionen - v [m²/s] kinematische Viskosität - P [bar] Druck - [kg/m³] Dichte - [mmol/l] Gesamtionenkonzentration - R [m] Kugelradius - r [m] radiale Koordinate - t [s] Zeit - [s] Relaxationszeit - V [m³] Volumen - w [m/s] Geschwindigkeitsvektor - ¯x [-] Molenbruch der Säureionen in der Wasserphase (auch Konzentration genannt) - ¯y [-] Molenbruch der Säureionen in der Harzphase (auch Konzentration genannt) - [-] Winkelkoordinate - [-] äußere Porosität der Harzkugelschüttung Indizes a äußere Phase (Umgebung) - i innere Phase (Festkörper) - W Wasserphase - H Harzphase - in der äußeren Phase (Umgebung) - 0 zur Zeit null - R an der Kugeloberfläche - - überstrichen=Mittelwert     

Instationäre eindimensionale Wärmeleitung beim Erstarren einer Flüssigkeit mit konstanter Außentemperatur

Zusammenfassung Die Temperaturverteilung in der festen und flüssigen Phase einer erstarrenden Flüssigkeit mit eindimensionaler Wärmeleitung und konstanter Außentemperatur der festen Phase wurde mit Hilfe von Laplace-Transformationen abgeleitet und mit der Neumannschen Losung des gleichen Problems verglichen.Die Übereinstimmung zwischen der Losung im vorliegen Beitrag und der Neumannschen Lösung ist recht gut.

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Transient one dimensional heat transfer in a liquid in solidification and with constant outer surface temperature of the solid phase

The temperature distribution in the solid and liquid phase of a liquid in solidification with one dimensional heat transfer and constant outer surface temperature of the solid phase is laid down by using Laplace transforms.The agreement between the present solution and Neumanns solution of the same problem is very good.

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Formelzeichen h Schmelz- oder Erstarrungswärme [J/kg] - k Erstarrungskoeffizient [m/s^0,5] - k₁ Temperaturleitfähigkeit der festen Phase [m²/s] - k₂ Temperaturleitfähigkeit der fl. Phase [m²/s] - T_o Außentemperatur der festen Phase [K] - T_s Schmelz- Oder Erstarrungstemperatur [K] - T Temperatur der fl. Phase zur Zeit t=0[K] - t Zeit [t] - x Entfernung von der Außenfläche der festen Phase [t] - ₁ Wärmeleitzahl der festen Phase [W/m·k] - ₂ Wärmeleitzahl der fl. Phase [W/m·k] - Dicke der festen Phase, Entfernung der Erstarrungsfront von der Außenfläche der festen Phase [m] - Dichte der fl. Phase [Kg/m³]     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

An integrated multiphase flow sensor for microchannels

The flow regimes of microscale multiphase flows affect the yield and selectivity of microchemical systems, and the heat transfer properties of micro heat exchangers. We describe an integrated optical sensor that uses total internal reflection to detect the structure of multiphase flows in microchannels. The non-intrusive sensor enables detection of individual slugs, bubbles, or drops, and can be used to continuously determine their number and velocity. The sensor performance is modeled using ray-tracing techniques, and tested for several channel geometries. Both gas-liquid and liquid-liquid flows are investigated in microchannels with rectangular and triangular cross-sections. Statistical properties of the flow, derived from the sensor signal, compare favorably to commonly-used dynamic pressure measurements. We demonstrate the integration of the sensor into a planar multichannel microreactor. An existing glass layer used as a waveguide allows us to monitor flows in optically inaccessible channels. This sensor configuration can be integrated into layers of vertically-stacked multichannel microreactors.

List of symbols

Roman symbols a Radius of largest sphere inscribed in channel [m] - A_ch Channel cross-sectional area [m²] - Ca Capillary number [-] - Critical capillary number [-] - d_h Hydraulic diameter [m] - d_sensor Distance prism surface-sensor origin [m] - E₀ Incident light energy [J] - E_r Emerging light energy [J] - f(t_pass) Probability density function (PDF) of slug dwell times [1/s] - f Focal length [m] - f_slug Slug frequency [Hz] - F(t_pass) Probability distribution of slug dwell times [-] - g(t) Arbitrary function of time [-] - h Liquid film thickness [m] - j_G Superficial gas velocity [m/s] - j_L Superficial liquid velocity [m/s] - l Slug length [m] - N Number of samples [-] - n Refractive index [-] - N_c Number of channel corners [-] - n_i Refractive index of incident medium [-] - n_r Number of reflections [-] - n_t Refractive index of transmitting medium [-] - n_slug Number of slugs [-] - p Gas inlet pressure [Pa] - r Reflectance [-] - R_XX(x,) Autocorrelation function [-] - R_Xp(x,) Cross correlation function [-] - r Slug radius at infinite distance from leading slug tip [m] - s Standard deviation of measured slug dwell times [s] - t Time [s] - t Measurement time interval [s] - t_pass Slug dwell time [s] - U_b Slug (bubble) velocity [m/s] - W Bin size of slug dwell time histogram [-] - x Streamwise coordinate [m] - X(x,t) Phase density function [-] - Y Surface tension of the gas-liquid interface [N/m] - Volumetric gas flow rate [m³/s] - Volumetric liquid flow rate [m³/s] - Volumetric oil flow rate [m³/s] - Volumetric water flow rate [m³/s] - z Normal coordinate [m]Greek symbols Void fraction [-] - _c Critical angle for total internal reflection [^°] - _i Incident angle [^°] - Laser wavelength [m] - µ Liquid viscosity [Pa s] - Normalization factor [-] - _h Dimensionless liquid film thickness [-] - _r Dimensionless radius [-] - _x Dimensionless streamwise position [-] - _r Dimensionless slug radius at infinite distance from leading slug end [-] - Standard deviation of the slug dwell time distribution [s] - Time shift [s] - Contact angle [^°]     

Onset of viscous fingering for miscible liquid-liquid displacements in porous media

The displacement of one fluid by another miscible fluid in porous media is an important phenomenon that occurs in petroleum engineering, in groundwater movement, and in the chemical industry. This paper presents a recently developed stability criterion which applies to the most general miscible displacement. Under special conditions, different expressions for the onset of fingering given in the literature can be obtained from the universally applicable criterion. In particular, it is shown that the commonly used equation to predict the stable velocity ignores the effects of dispersion on viscous fingering.Nomenclature C Solvent concentration - Unperturbed solvent concentration - D _L Longitudinal dispersion coefficient [m²/s] - D _T Transverse dispersion coefficient [m²/s] - g Gravitational acceleration [m/s²] - I _sr Instability number - k Permeability [m²] - K Ratio of transverse to longitudinal dispersion coefficient - L Length of the porous medium [m] - L _x Width of the porous medium [m] - L _y Height of the porous medium [m] - M Mobility ratio - V Superficial velocity [m/s] - V _c Critical velocity [m/s] - V _s Velocity at the onset of instability [m/s] - µ Viscosity [Pa/s] - Unperturbed viscosity [Pa/s] - µ ₀,µ _s Viscosities of oil and solvent, respectively [Pa/s] - Density [kg/m³] - ₀, _s Densities of oil and solvent, respectively [kg/m³] - Porosity - Dimensionless length     

The effect of a nonlinear temperature dependent Prandtl-number on heat transfer of fully developed flow of liquids in a straight tube

If Nu_o is the Nusselt Number for a temperature-independent Prandtl number Pr, and Nu the Nusselt number for a temperature dependent Prandtl number, it is usual to define the correction factor Nu/Nu_o=C. A correction factor which has been defined in this form has, up to now, only been expressed as a function of two characteristic Pr numbers. Thus it was simply assumed that the Pr number was a linear function of the temperature. Fluids with very large Pr numbers show a (T;Pr) relationship which deviates considerably from a linear one. This leads to a very large difference (up to 70%) between the calculated and the measured values of the Nusselt number. In the following study we shall determine a so-called curvature parameter of the (T;Pr) curve and obtain a semi-empirical formula for C. This formula has a deviation from the actual relationship many times smaller than that of the formulae for a linear (T;Pr) relationship.

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Zusammenfassung Ist Nu_o die Nusseltzahl bei temperaturunabhangiger und Nu die Nusseltzahl bei temperaturabhangiger Prandtlzahl Pr, so ist es üblich, mit Nu/Nu_o=C den Korrekturfaktor zu bezeichnen. Ein in dieser Form definierter Faktor C ist bisher als Funktion nur zweier charakteristischer Pr-Zahlen ausgedrückt worden. Es wurde somit nur eine lineare Abhängigkeit von der Pr-Zahl von der Temperatur T vorausgesetzt. Flüssigkeiten mit großen Pr-Zahlen weisen eine (T;Pr)-Charakteristik auf, die sehr stark von der linearen abweicht. Sehr große Abweichungen (bis — 70%) der gerechneten von den gemessenen Nu-Zahlen sind eine Folge davon. In vorliegender Arbeit bilden wir mit einer dritten charakteristischen Pr-Zahl einen sogenannten Krümmungsparameter der Kurve (T;Pr) und leiten eine semiempirische Formel für C ab, die um ein großes Vielfaches kleinere Fehler aufweist, als die Formeln für den linearen (T;Pr)-Verlauf.

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Nomenclature

Material constants c_p specific heat at constant pressure [J/kgK] - k heat conductivity [W/mK] - density [kg/m³] - a temperature diffusivity, a=k/c_p [m²/s] - dynamic viscosity [Ns/m²] - kinematic viscosity [m²/s] Fluid dynamics D inner diameter of the tube [m] - L length of tube [m] - w mean speed of fluid [m/s] Heat transfer h coefficient of heat transfer [W/m²K] - T absolute temperature [K] - T_b bulk temperature (corresponding to the adiabatic mixing temperature) [K] - T_w tube wall temperature [K] - T_f=(T_b+T_w)/2 film temperature [K] - T=T_b-T_w temperature forcing difference of heat transfer [K] Characteristic quantities without dimensions Re=wD/ Reynolds number - Pr=/a Prandtl number - Nu=hD/k Nusselt number - related temperature - related Prandtl number - curvature parameter of the Prandtl number Various - C=Nu_b/Nu_o correction factor according to Eq.(5) - p exponent in Eq.(6), (a), (8) and (16) Indices o corresponding to the quasi-isothermal heat transfer - b,w,f with reference to quantities, including physical properties which correspond to the temperatures T_b, T_wor T_f - Pr,k,, for quantities calculated corresponding to the Prandtl number, the thermal conductivity coefficient, the density or the dynamic viscosity - H,C for heating or cooling exp for quantities calculated from experimental data     

Displacement of water-oil contact in nonuniform stratum

Charnyi [1] has suggested the use of the scheme of limiting anisotropic strata for finding the water-oil contact surface (WOC) moving in a uniform stratum. The self-similar problems of WOC movement were studied in [2], and the numerical solutions are presented in [3]. The numerical integration of the nonlinear equations of the parabolic type obtained in [1] were presented in [4, 5]. A comparison was made in [4] of the results of the numerical solution of the problem of WOC movement for two cases of limiting anisotropy (k_z== and k_z=0) with the experimental data for an Isotropie stratum, which showed that the case k_z= yields results which are very close to the experimental data for the isotropic stratum.In the present investigation the technique suggested in [1] is extended to the nonuniform stratum whose permeability is a function of the distance from the base of the stratum. Only the k_z= case is considered. Nonlinear equations of the parabolic type are derived which define the plane and axisymmetric movement of the WOC in the nonuniform stratum. A possible technique for the numerical integration of the resulting nonlinear parabolic equations is suggested.     

Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity [m²/s] - b temperature coefficient [K^–1], see eq. [4] - c heat capacity [J/kg K] - h slot width [m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. [5] - k thermal conductivity [J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. [3] - M torque [m N] - p pressure [N/m²] - P dimensionless pressure gradient, see eq. [24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. [25] and eq. [26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus [m] - t time [s]; dwell time in the annulus - T temperature [K] - v , v_r, V_z velocity components [m/s] - v ₀ angular velocity at inner wall [m/s] - average velocity inz-direction [m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. [20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor [s^–1] - apparent viscosity [N s/m²], see eq. [3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density [kg/m³] - , kl shear stress tensor [N/m²] - fluidity [m²w/Nw s], see eq. [4] - Gf Griffith number, see eq. [12] - Pe Péclet number, see eq. [13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday     

Flow stability of viscous liquid layer with moment stresses on an inclined plane

The flow stability of a liquid layer on an inclined plane with account for molecular spin [1, 2] has been considered in [3] in the absence of moment stresses within the liquid. It was shown in [3] that molecular spin has a destabilizing effect on the flow. In the following we study the combined effect of molecular spin and internal moment stresses on the behavior of three-dimensional disturbances. The validity of Squire's theorem is established. The flow stability of a layer of relatively long-wave disturbances is studied by the method of sequential approximations [4, 5] under the assumption that the rotational viscosity coefficient _r is significantly smaller than the Newtonian viscosity coefficient .     

Nicht-isotherme Rieselfilmströmung einer hochviskosen Flüssigkeit mit stark temperaturabhÄngiger ViskositÄt

Zusammenfassung Es werden Geschwindigkeitsverteilungen und Filmdickenabnahmen von nichtisothermen NEWTONschen und nicht-NEWTONschen (Potenzansatz) Rieselfilmen mit temperaturanhÄngiger ViskositÄt berechnet, wobei die Temperaturverteilung im Film als linear vorausgesetzt wird. An dicken Rieselfilmen mit Re=10^–4... 10^–2 sind Geschwindigkeitsprofile, Filmdicken und OberflÄchentemperaturen gemessen und daraus die thermische EinlauflÄnge bestimmt worden. Ausgehend von der Penetrationstheorie für eine endlich dicke Platte kann man für diese EinlauflÄnge eine Approximationsformel erhalten, die für Strömungen mit Re < 1000 verwendet werden kann.

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Non-isothermal filmflow of a highly viscous liquid, the viscosity strongly depending on temperature

Velocity distributions and film thicknesses of nonisothermal NEWTONIAN and non-NEWTONIAN (power-law) falling films are computed assuming that the temperature across the film varies linearly. Experimental studies on thick falling films of Re=10^–4...10^–2 had been carried out to measure velocities, film thickness and surface temperature and to calculate the thermal entrance length. One can get for this entrance length a approximation formula which is valid for flows with RePr <1000 by applying the results for the thermal penetration into a material finite plate.

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Bezeichnungen B dimensionsloser Temperaturkoeffizient - ¯B [K] Temperaturkoeffizient (ln)/(1/T) - c_p [J/kgK] spezif. WÄrme bei konst. Druck - Fo FOURIER-Zahl - g [m/s²] Erdbeschleunigung - H dimensionslose Filmdicke - h [m] Filmdicke - m [Pas^2–n] ViskositÄtskoeffizient im Potenzansatz von OSTWALD-DE WAELE - Nu NUSSELT-Zahl - n Flüssigkeitsexponent im Potenzansatz von OSTWALD-DE WAELE - Pr PRANDTL-Zahl (Gl.3.5) - q [W/m²] WÄrmestromdichte - Re REYNOLDS-Zahl (Gl.3.4) - T [K] Temperatur - t [s] Zeit - U dimensionslose Geschwindigkeit (X-Komponente) - u [m/s] Geschwindigkeitskomponente in x-Richtung - X dimensionslose Koordinate (X=x/h₀) - x [m] LÄnge, Koordinate - Y dimensionslose Koordinate (Y=y/h₀) - y [m] Höhe, Koordinate - [W/m²K] WÄrmeübergangskoeffizient - Plattenneigungswinkel gegen Horizontale - [s^–1] Schergeschwindigkeit - dimensionslose Temperatur (Gl.3.3) - [m²/s] TemperaturleitfÄhigkeit (Gl.3.3) - [W/mK] WÄrmeleitfÄhigkeit - [Pas] ViskositÄt - [kg/m³] spezif. Dichte - [Pa] Schubspannung Indizes a scheinbar (apparent) - 0 bei x=0, auch: isotherm - P auf die Penetrationszeit bezogen - s an der OberflÄche - T bei linearer Temperaturdifferenz T - w an der Wand - 99 auf =0,99 bezogen - gemittelt, Mittelwert - thermisch ausgebildet, bei x - proportional - ¯t ungefÄhr - kleiner oder gleich ungefÄhr     

Development of perturbations in plane shock waves

Investigation of the stability of plane shock waves as regards nonuniform perturbations was first performed by D'yakov [1]. He obtained criteria for stability, and showed that perturbations grow exponentially with time in the case of instability. Iordanskii [2] has shown that in the case of stability, the perturbations are attenuated according to a power law. However, the stability criteria of [2] do not agree with the results of [1], Kontorovich [3] has explained the cause of the apparent discrepancies, and asserts the correctness of the criteria of [2]. A power law for the attenuation of perturbations has also been obtained in [4,5] under a somewhat different formulation of the boundary conditions.The Cauchy problem with perturbations is examined in §1 of this paper, results are obtained for cases of practical interest, and the asymptotic behavior is investigated.In §2 the effect of a low viscosity on the development of perturbations is examined. It is shown that when t the amplitude of perturbations is attenuated mainly as exp(-t), where >0 does not depend on the form of the boundary conditions at the shock wave front. The results of §2 were used in processing the experimental data of [6], which made it possible to determine the viscosity of a number of substances at high pressure.In conclusion, the author expresses his gratitude to A. D. Sakharov for valuable advice, and to A. G. Oleinik and V. N. Mincer for useful discussions. The author also thanks G. I. Barenblatt, L. A. Galin, and others who took part in a seminar at the Institute for Problems in Mechanics, for their interesting discussion and valuable comments.