Influx of two fluids to a completed gallery and well in stratified soils

It often happens that waters from different water-bearing horizons have different mineralization. Salt accumulation is usually found in clayey interlayers. Therefore during pumping from one horizon water may enter that horizon from two neighboring strata or clayey interlayers, and the water pumped out may be a mixture of waters with differing mineralization.The study of Girinskii [1] using the assumption of relative smallness of the thickness of the water-bearing stratum established certain hydraulic criteria for the steady-state two-dimensional unpressurized motion of two streams of different mineralization. However, in considering concrete problems we soon see that for different densities of the liquids (₁₂) in order to determine the four constants of integration it is necessary to establish another condition in addition to the three boundary conditions. Here this condition will be taken to be the possibility of representing the sought quantities in the form of a series in powers of the small parameter = 1 – ₁/₂.The authors wish to thank P. Ya. Polubarinova-Kochina for discussions and advice on this study.     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

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

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

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

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

Eine kanonische Zustandsgleichung für Kohlendioxid

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

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A canonical equation of state for carbon dioxide

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

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

Problem of a jet flowing into the surface of a heavy fluid

This paper is a continuation of [1]. The problem of the nonuniqueness of the angle of incidence of a fine jet into water is considered and the mathematical formulation of the problem is improved. A diagram of the flow is shown in Fig. 1.; the jet is an inviscid, incompressible, weightless fluid of density ₁ flowing from a nozzle onto the surface of a still heavier fluid of density ₂. The problem is two-dimensional.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 82–89, March–April, 1973.     

Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties

The steady laminar boundary layer flow, with an external force, along a vertical isothermal plate is studied in this paper. The external force may be produced either by the motion of the plate or by a free stream. The fluid is water whose density-temperature relationship is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations with , k and variable across the boundary layer. Both upward and downward flow is considered. It was found that the variation of , k and with temperature has a strong influence on mixed convection characteristics.Nomenclature c_p water specific heat - f dimensionless stream function - g gravitational acceleration - Gr_x local Grashof number - k thermal conductivity - Nu_x local Nusselt number - Pr Prandtl number - Pr_a ambient Prandtl number - Re_x local Reynolds number - s salinity - T water temperature - T_a ambient water temperature - T_o plate temperature - u vertical velocity - u_a free stream velocity - u_o plate velocity - v horizontal velocity - x vertical coordinate - y horizontal coordinate - pseudo-similarity variable - nondimensional temperature - dynamic viscosity - _f film dynamic viscosity - _o dynamic viscosity at plate surface - kinematic viscosity - buoyancy parameter - water density - _a ambient water density - _f film water density - _o water density at plate surface - physical stream function     

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

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

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Equilibrium Configurations for a Floating Drop

If a drop of fluid of density ₁ rests on the surface of a fluid of density ₂ below a fluid of density ₀, ₀ < ₁ < ₂, the surface of the drop is made up of a sessile drop and an inverted sessile drop which match an external capillary surface. Solutions of this problem are constructed by matching solutions of the axisymmetric capillary surface equation. For general values of the surface tensions at the common boundaries of the three fluids the surfaces need not be graphs and the profiles of these axisymmetric surfaces are parametrized by their tangent angles. The solutions are obtained by finding the value of the tangent angle for which the three surfaces match. In addition the asymptotic form of the solution is found for small drops.     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Symmetry-Breaking Bifurcations of Charged Drops

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

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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     

Influence of initial conditions on compressible vorticity dynamics

Prompted by the lack of a unique choice of pressure (P) and density () fields for a compressible free vortex and by the observed dependence of turbulence dynamics on initial P and in compressible simulations, we address the effects of initial conditions on the evolution of a single vortex, on the prototypical phenomenon of vortex reconnection, and on two-dimensional turbulence. Two previous choices of initial conditions used for numerical simulations of compressible turbulence have been: (i) both P and uniform (constant initial conditions, CIC), and (ii) uniform with P determined from the Poisson equation (constant density initial conditions, CDIC). We find these initial conditions to be inappropriate for compressible vorticity dynamics studies. Specifically, in compressible reconnection, the effects of baroclinic vorticity generation and shocklet formation cancel each other during early evolution for CDIC, thus leading to almost incompressible behavior. Although CIC captures compressibility effects, it incorrectly changes the initial vorticity distribution by introducing strong acoustic transients, thereby significantly altering the evolving dynamics.Here, a new initial condition, called polytropic initial condition (PIC), is proposed, for which the Poisson equation is solved for initially polytropically related P and fields. PIC provides P and distributions within vortices which are consistent with those observed in shock-wedge interaction experiment and also leads to compressible solutions with no acoustic transients. At low Mach number (M), we show that the effects of all these three initial conditions can be predicted by low-M asymptotic theories of the Navier-Stokes equations. At high M, it is shown here that inappropriate initial conditions may alter the evolutionary dynamics and, hence, lead to wrong conclusions regarding compressibility effects. We argue that PIC is a more appropriate choice.D. Virk acknowledges financial support of the Advanced Study Program at NCAR, Boulder, for 2 years during graduate studies. Part of this work and the writing of the paper have been supported by NSF Grant CTS-9214818.     

Dissolution of an ellipsoidal bubble in a slightly viscous liquid

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

Symmetries and Global Solvability of the Isothermal Gas Dynamics Equations

We study the Cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics, in which the pressure and the density are related by the -law equation p() with =1. Our results complete those obtained earlier for >1. We prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method. The proof relies on compensated compactness arguments and symmetry group analysis. Interestingly, we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density. This property enables us to reduce our problem to that with a small initial density.One symmetry group associated with the linear hyperbolic equations describing all entropies of the Euler equations gives rise to a fundamental solution with initial data imposed on the line =1. This is in contrast to the common approach (when >1) which prescribes initial data on the vacuum line =0. The entropies we construct here are weak entropies, i.e., they vanish when the density vanishes.Another feature of our proof lies in the reduction theorem, which makes use of the family of weak entropies to show that a Young measure must reduce to a Dirac mass. This step is based on new convergence results for regularized products of measures and functions of bounded variation.Acknowledgement P.G.L. and V.S. were supported by a grant from INTAS (01-868). The support and hospitality of the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, where part of this research was performed during the Semester Program Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics (January to July 2003) is also gratefully acknowledged. P.G.L. was also supported by the Centre National de la Recherche Scientifique (CNRS).     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

The Thermodynamic Significance of the Local Volume Averaged Temperature

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

An investigation of laminar radial flow between two parallel discs

Summary Experiments have been carried out to test recent theoretical predictions of the pressure distribution for laminar flow between parallel discs, including inertia effects. The experimental investigation covered the condition where the inertia effects were always completely dominant over the central region of the discs in contrast to other recent experimental work on the problem where the central injection diameter was considerably larger. The present experiments subject the theories to a stringent test, due to the dominance of the inertia effects, and it is found that the inertia effects predicted by the various theoretical analyses are significantly smaller than those shown by the experimental results. It is suggested that the theoretical approach requires further development before it will cover the conditions where the central injection diameter is small.Nomenclature r, y, cylindrical co-ordinates - u velocity in r direction - U _m mean velocity in r direction at radius r - density - coefficient of viscosity - Q volume flow per unit time - 2h gap between parallel discs - p static pressure - R r/h - P h ³ p/Q - R _e Q/h     

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