Testing viscometric predictions of the internal viscosity model,using dilute viscous theta solutions

A stress-symmetrized internal viscosity (I.V.) model for flexible polymer chains, proposed by Bazua and Williams, is scrutinized for its theoretical predictions of complex viscosity ^* () = – i and non-Newtonian viscosity (), where is frequency and is shear stress. Parameters varied are the number of submolecules,N (i.e., molecular weightM = NM _s ); the hydrodynamic interaction,h ^*; and/f, where andf are the I.V. and friction coefficients of the submolecule. Detailed examination is made of the eigenvalues _p (N, h ^*) and how they can be estimated by various approximations, and property predictions are made for these approximations.Comparisons are made with data from our preceding companion paper, representing intrinsic properties [], [], [] in very viscous theta solutions, so that theoretical foundations of the model are fulfilled. It is found that [ ()] data can be predicted well, but that [ ()] data cannot be matched at high. The latter deficiency is attributed in part to unrealistic predictions of coil deformation in shear.     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Mass transport and reaction in catalyst pellets

When heterogeneous chemical reactions take place in porous catalysts, mass transport can occur by bulk diffusion, Knudsen diffusion, and convective transport. Previous studies of these phenomena have been largely based on Maxwell's dusty gas model with the convective transport or Darcy flow added to the diffusive transport. This is done in order to satisfy one of the limiting conditions encountered in the study of flow in porous media. A more fundamental approach consists of the use of the method of volume averaging and the general form of the species momentum equation. For an N-component system, this leads to N independent flux relations to be used in conjunction with the volume-averaged species continuity equations.Roman Letters _A (t) surface area of a species body, m² - a _v interfacial area per unit volume, m^-1 - A _e area of entrances and exits for the -phase contained within the averaging volume, m² - A _K area of the - surface contained within the averaging volume, m² - b _A species A body force, N/kg - b mass average body force, N/kg - B inverse tortuosity tensor for bulk diffusion - c total molar concentration, moles/m³ - c _A species A molar concentration, moles/m³ - _A surface concentration of species A, moles/m² - C_A² intrinsic phase average molar concentration, moles/m³ - c _A – C_A², spatial deviation concentration, moles/m³ - c _A mean molecular speed for species A, m/s - binary diffusion coefficient, m²/s - D _A ^{K, eff} Knudsen diffusion coefficient for species A, m²/s - f vector that maps P _A into P _A , m - g gravitational vector, m/s² - G second order tensor that maps N _A into N _A for free molecule flow conditions - H inverse tortuosity tensor for Knudsen diffusion - I unit tensor - j _A c _A u _A ^* , molar diffusive flux, moles/m²s - K Darcy's Law permeability tensor, m² - L macroscopic length scale, m - L _D diffusive length, m - l characteristic length for the -phase, m - l _A mean free path for species A, m - M _A molecular weight of species A, kg/mole - n outwardly directed unit normal vector - n _K unit normal vector directed from the -phase toward the -phase - n outwardly directed unit normal vector at the entrances and exits of the -phase contained within the averaging volume - N _A c _A v _A molar flux of species A, moles/m²s - N _A intrinsic phase average of the species A molar flux, moles/m²s - \~N _A spatial deviation of the molar flux of species A, moles/m²s - p total pressure, N/m² - P p + , total pressure over and above the hydrostatic pressure, N/m² - P _A partial pressure of species A, N/m² - p _A intrinsic phase average partial pressure, N/m² - P_A – p _A, spatial deviation partial pressure, N/m² - P _A p_A + _AA partial pressure of species A over and above the hydrostatic pressure of species A, N/m² - p _ab diffusive force exerted by species B on species A, N/m³ - universal gas constant, N m/moles K - R _A molar rate of production of species A owing to homogeneous chemical reaction, moles/m³s - molar rate of production of species A owing to heterogeneous chemical reaction, moles/m²s - r _A mass rate of production of species A owing to homogeneous chemical reaction, kg/m³s - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - t _A species stress vector, N/m² - T _A species stress tensor, N/m² - T total stress tensor, N/m² - T temperature, K - T spatial average temperature, K - u _A v _A – v, mass diffusion velocity, m/su _A ^* v_A – v^*, molar diffusion velocity, m/s - u _o velocity of the rigid, solid phase relative to some inertial frame, m/s - v _A species velocity, m/s - v mass average velocity, m/s - v ^* molar average velocity, m/s - v _A ^* species velocity of those molecules of species A generated by chemical reaction, m/s - _A (t) volume of a species A body, m³ - averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ - v phase average, mass average velocity, m/s - w arbitrary velocity vector, m/s - x _A c _A /c mole fraction of species A - X _A intrinsic phase average mole fraction - X _A – X _A , spatial deviation mole fraction Greek Letters V/V volume fraction of the -phase - _A sum of all terms in the species A momentum equation that are small compared to the diffusive force, N/m³ - viscosity of the -phase, Ns/m² - _A mass density of species A, kg/m³ - total mass density, kg/m³ - _a species viscous stress tensor, N/m² - total viscous stress tensor, N/m² - tortuosity factor - total body force potential function, Nm/kg - _a species body force potential function, Nm/kg - 3.1416 - _{a a} / mass fraction of species A     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Zur einfachen Berechnung von Dephlegmatoren binärer Dampfgemische

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

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On simple calculation procedures of binary mixed vapour dephlegmation

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

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

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Untersuchung des lokalen Stoffüberganges am laminaren und turbulenten Rieselfilm

Zusammenfassung Der lokale Stoffübergang wurde in Abhängigkeit von der Meßlänge, dem Startort und der Zulaufhöhe gemessen. Der Gültigkeitsbereich der Theorie von Nusselt wird ermittelt. Die Reynolds-Zahl nahm Werte zwischen 3,86 und 2496 an. Die örtlich wirkende Hydrodynamik ist entscheidend für das Anwachsen der örtlichen Sherwood-Zahl. Die Genauigkeit aller Versuchsergebnisse kann auf ± 5% abgeschätzt werden.

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Investigation of the local mass transfer of a laminar and turbulent falling liquid film

The local mass transfer was measured as a function of the measuring length, the starting point and the liquid height above the ring-slot. The range of the Reynolds number was 3,86 Re 2496. The validity of the Nusselt theory and the range of it is shown. The local hydrodynamic is the most important factor of the increase of the local Sherwood number. The accuracy of the measurements is ± 5%.

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Bezeichnungen a Temperaturleitfähigkeit m²/s=/(c_p) - c Konzentration, c=¯c + c kmol/m³ - c_i0 Konzentration im Flüssigkeitskern kmol/m³ - D Diffusionskoeffizient m²/s - EL-NR Elektrodennummer - Fa Faraday-Konstante A s/kgäq=96,5·10⁶ - g Erdbeschleunigung m/s² - i_G Grenzstromdichte A/m² - u Geschwindigkeit in x-Richtung, u= + u - U Umfang des Rohres m - v Geschwindigkeit in y-Rich- m/stung, v=¯v + v - V^* Volumenstrom m³/s - x Lauflänge, Koordinate in m Strömungsrichtung - x_M Meßlänge für den Stoff-Übergang m - x_ST Startort für den Stoff-Übergang m - y Wegkoordinate senkrecht zur Rohroberfläche m - z Wertigkeit der Elektro-denreaktion kgäq/kmol - ZH Zulaufhöhe m - Wärmeübergangskoeffizient W/m²C - Stoffübergangskoeffizient m/s - Filmdicke m - Wärmeleitfähigkeit W/(mC) - kinematische Viskosität m²/s - Re=u/=V^*/U Reynolds-Zahl - Pr=/a=c_p/ Prandtl-Zahl - Sc=/D Schmidt-Zahl - Nu= / Nusselt-Zahl - Sh= /D Sherwood-Zahl - SHL lokale Sherwood-Zahl - SHM mittlere Sherwood-Zahl - - zeitlich gemittelt - örtlich gemittelt Die Durchführung der Arbeit am Institut für Verfahrens — und Kältetechnik der ETH Zürich bei Prof. Dr. P. Grassmann wurde ermöglicht durch Zuschüsse der Kommission zur Förderung der wissenschaftlichen Forschung und meiner Eltern.     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Injectivity and self-contact in nonlinear elasticity

Let be a bounded open connected subset of ³ with a sufficiently smooth boundary. The additional condition det dx vol () is imposed on the admissible deformations : ¯ of a hyperelastic body whose reference configuration is ¯. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W ^1,p (;³), p > 3, that is injective almost everywhere.     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Viscosity of multicomponent gas mixtures of polar gases

Summary The rigorous binary viscosity expression _mix as transformed to the form originally suggested by Sutherland is studied for mixtures involving polar gases. Any attempt to simplify the _ij of the Sutherland viscosity expression turns out to be only approximately successful. A relation for _ij / _ji is however derived, and the procedure suggested for computing _mix on this basis appears to be very successful. The _ij to a large extent are temperature and composition independent and it has been shown that this fact can be utilised with success for predicting _mix values at high temperatures.     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

A new interpretation of viscosity and yield stress in dense slurries: Coal and other irregular particles

Illinois coal was ground and wet-sieved to prepare three powder stocks whose particle-size distributions were characterized. Three suspending fluids were used (glycerin, bromonaphthalene, Aroclor), with viscosities _s that differed by a factor of 100 and with very different chemistries, but whose densities matched that of the coal. Suspensions were prepared under vacuum, with coal volume fractions that ranged up to 0.46. Viscosities were measured in a cone-and-plate over a shear rate range 10^–3–10² s^–1. Reduced viscosity _r = /_s is correlated in the high-shear limit ( ) with/ _M, where _M is the maximum packing fraction for the high-shear microstructure, to reveal the roles of size distribution and suspending fluid character. A new model that invokes the stress-dependence of _M is found to correlate _r well under non-Newtonian conditions with simultaneous prediction of yield stress at sufficiently high; a critical result is that stress and not governs the microstructure and rheology. Numerous experimental anomalies provide insight into suspension behavior.     

Non-Markovian diffusion process in polymers and stretched exponential relaxation

In this contribution, we model the long-time behaviour of the desorption from an LDPE sheet, using non-Markovian random walks. It is shown that the mass of penetrant in the final stage of desorption decays as t ^–m , where m is proportional to the exponent of the probability distribution (t) t ^–(1+u), 0 < v < 1. Furthermore, it is shown that this model may lead to the so-called mechanical stretched exponential relaxation, and that Wagner's memory function can be obtained as a special case.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Boundary conditions for contacts lines in coextrusion flows

A bicomponent coextrusion process is modelled using a 3-D finite element formulation. The layer uniformity problem in coextrusion is addressed by examining the effects of the polymer melt/polymer melt/die wall contact line boundary condition. It has been observed that the less viscous polymer layer will tend to displace the more viscous polymer layer near the die wall. The behaviour of the contact lisle is considered to be either a stick or slip boundary condition. In the stick boundary condition, the contact line does not move from its original position after the two polymer layers meet, A slip boundary condition allows the contact line to move along the die wall. The calculated interfaces which result from different contact line assumptions are determined. Results show that if a stick boundary condition is appropriate for a given fluid/fluid/solid contact line, then a very thin entrained layer of the more viscous polymer melt will be trapped between the less viscous polymer melt and the die wall. Slip boundary conditions would allow complete displacement of the contact line along the die wall. Both slip and stick boundary conditions produce similar interface profiles far away from the die wall for small viscosity ratios. In certain eases, the displacement of the more viscous material by the less viscous material will cease and a static interface structure is produced regardless of die length. Experimental work with polycarbonate melts is compared with the numerical simulations.A. Torres on leave from Investigación y Desarrollo,, C.A. (INDESCA), P.O. Box 10319, Complejo Petroquímico El Tablazo, Maracaibo, 4001, Venezuela.     

Spin-up in a saturated porous medium

It is shown that, on the Brinkman model, spin-up is confined to boundary layers whose thickness is of order k ^1/2, and the spin-up is established in a time of order k/, where k, , and denote permeability, density, porosity and dynamic viscosity, respectively.     

Classical limits for the coefficient of variation for the normal distribution

The exact classical limits for the coefficient of variation c for the normal distribution are derived. The hand-calculating approximated classical limits for c having high accuracy are given to meet practical engineering needs. Using Odeh and Owen's computational method and Brent's algorithm, the tables for the r-upper exact classical limits of coefficient of variation for normal distribution are calculated for the different confidence coefficient , the sample size n=1 (1)30,40,60,120, the sample coefficient of variation =0.01(0.01)0.20. It is shown that if n8, 0.20, then the -upper exact classical limits c_u for c are slightly higher than the exact fiducial limits c_u,F for c. if n>8, 0.20, then c_u–c_u,F<5×10^–6.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Free convection on an arbitrarily inclined plate with uniform surface heat flux

This paper proposed a proper inclination parameter and transformation variables for the analysis of free convection from an inclined plate with uniform surface heat flux to fluids of any Prandtl number. Very accurate numerical results and a simple correlation equation are obtained for arbitrary inclination from the horizontal to the vertical and for 0.001 Pr. Maximum deviation between the correlated and calculated data is less than 1.2%.

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Freie Konvektion an einer beliebig geneigten Platte mit erheblicher Wärmestromdichte an der Oberfläche

Zusammenfassung Für die Berechnung von freier Konvektion von Fluiden mit beliebiger Prandtl-Zahl an einer geneigten Platte mit einheitlicher Wärmestromdichte an der Oberfläche werden ein zweckmäßiger Neigungsparameter und Transformationsvariablen eingeführt. Sehr genaue numerische Ergebnisse und eine einfache Korrelationsgleichung wurden für beliebige Neigungen zwischen der Horizontalen und der Vertikalen und für 0.001Pr erhalten. Die größte Abweichung zwischen Korrelations- und berechneten Daten liegt bei weniger als 1.2%.

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Nomenclature f reduced stream function - g gravitational acceleration - h local heat transfer coefficient - k thermal conductivity - Nu local Nusselt number - p static pressure difference - Pr Prandtl number - q _w wall heat flux - Ra* modified local Rayleigh number,g(q _w x/k)x ³/ - T fluid temperature - T temperature of ambient fluid - u velocity component inx-direction - v velocity component iny-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra* |sin|)^1/5/( Ra* cos)^1/6 - ( Ra* cos)^1/6+( Ra*|sin|)^1/5 - (y/x) - dimensionless temperature, (T–T )/(q _w x/k) - kinematic viscosity - [1+( Ra* cos)^1/6/( Ra*|sin|)^1/5]^–1 - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of inclination measured from the horizontal - stream function - dimensionless static pressure difference, p x ²/ ⁴