Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

Hypersonic flow over a narrow delta plate at large angles of attack

The problem of hypersonic flow over a flat delta plate with a high sweepback anglex at angles of attack close to /2 is solved using a numerical algorithm based on transition to the conical solution. The existence of conical flow at /2 with the velocity vector directed towards the apex of the plate is established. Values ofC _p/sin² and the thickness of the shock layer in the plane of symmetry of the plate are given as functions of the hypersonic similarity parameterk=tan tanx. A comparison of the calculated and experimental data shows that they are in good agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 183–185, September–October, 1992.     

On the indentation of an inhomogeneous plastic solid

Summary The problem considered here is that of the indentation of a semi infinite, inhomogeneous rigid-plastic solid by a smooth, flat ended punch under conditions of plane strain. It is assumed that the yield stress of the solid k(x, y) has the form k ⁰+k(x, y) where k ⁰ is a constant and is small. A perturbation method of solution developed by Spencer [1] is used, and general results are obtained for arbitrary values of k(x, y). Some particular cases are then considered.     

The Most Unstable Profiles of a Plane-Parallel Flow in a Channel

It is well known (after Rayleigh) that a plan-parallel flow in a channel can be unstable only if the basic velocity profile U(z) possesses inflection points. The profile determines (via the Rayleigh equation) the maximal increment c _i of small perturbations and 'eigenvalues' c (see Equation (1)). The increment and the imaginary parts c _i of the eigenvalues c provide a quantitative characterization of the basic stability properties of the flow. Here we find some best possible bounds for these values. The bounds are determined by the following parameters: wave number ; enstrophy of the basic flow; width of the channel L. A similar approach can be applied to models of atmosphere, ocean, plasma etc.     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

The swirling radial free jet

Summary A general similarity solution suggested by Watson for the problem of the laminar, radial, free-jet with swirl has been previously discussed by Riley who also calculated the order to which the solution was valid. That problem is considered in more detail here and higher order terms are given. It is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ⁿ (ln ) ^m must be adopted in order to satisfy the boundary conditions. It is also shown that the general similarity solution may be obtained from the asymptotic series representing the general case with swirl for certain special values of the free constants and also for the no-swirl or free-jet problem. The asymptotic series is given to order ^–13 for the case of swirl and to order ^–29 when there is no swirl.     

Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

Die Eigenschaften von Wasser und Wasserdampf nach „The 1968 IFC Formulation“

Zusammenfassung Diese Arbeit enthält Druck-Temperatur-Diagramme für 6 spezifische Zustandsgrößen und 16 erste Ableitungen und zusammengesetzte Größen von Wasser und Wasserdampf, die nach einem Gleichungssystem berechnet wurden, das unter dem Namen The 1968 IFC Formulation for Scientific and General Use von der 6. Internationalen Konferenz für die Eigenschaften des Wasserdampfes angenommen wurde. Einige Konsequenzen der thermodynamischen Konsistenz, das Verhalten im kritischen Gebiet und bei sehr kleinen Drücken werden diskutiert. Ferner werden die kinematische Viskosität und die Temperaturleitfähigkeit, sowie eine Beziehung zwischen dynamischer Viskosität und isenthalpem Drosselkoeffizienten angegeben.

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This paper contains pressure-temperature diagrams for 6 properties and 16 first derivatives and combined terms for water and steam. These were calculated from a system of equations accepted by the 6^th International Conference on the Properties of Steam, and called The 1968 IFC Formulation For Scientific and General Use. Some consequences of thermodynamic consistency, and the behaviour in the critical region and at very small pressures are discussed. Further, the kinematic viscosity and the thermal diffusivity and a relation between the dynamic viscosity and the throttling coefficient at constant enthalpy are given.

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Bezeichnungen (s. auch Tabelle 1) k Temperaturleitfähigkeit:k=/c _p - p Druck - r spezifische Verdampfungsenthalpie:r=h –h - T thermodynamische oder Kelvin-Temperatur - t Celsius-Temperatur - dynamische Viskosität - Wärmeleitfähigkeit - v kinematische Viskosität:=/ - Dichte:=1/v Indices und Sättigungswerte des Dampfes und der Flüssigkeit Differenz der Sättigungswerte, z. B. h=h –h     

Die Torsionsspannungen in Wellen mit tiefen Längsnuten

Übersicht MitF(x, y) als Spannungsfunktion einer Welle ohne Nut und(, y) als Potentialfunktion des Quelle-Senke-Systems erhält man Spannungsfunktionen(, y) =F(x, y) –(, y) für Wellen mit tiefen Längsnuten. Es wird gezeigt, daß sich damit die Schubspannungen in den Läufern von Schraubenverdichtern ermitteln lassen.

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Shearing stresses in shafts with deep longitudinal grooves

Summary The stress functions(, y) of shafts with deep longitudinal grooves may be represented by(, y) =F(x, y) –(, y) whereF(x, y) is the stress function of a cylindrical shaft without grooves and(, y) denotes the potential function of the source-sink system. It is shown that the shearing stresses in rotors of screw-compressors may be obtained in this way.

     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - 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 - v traditional superficial volume averaged 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 - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Regularity of invariant manifolds

Under mild conditions it is proved that an invariant submanifold ofX 0<1 for the equationdx/dt+Ax=f(x), A sectorial,fC'(X ,X),0<1, is a submanifold ofX ¹ as well. In addition, conditions are given for the semiflow of the equation to extend fromX toX and a new inertial manifold theorem is proved for the scalar reaction diffusion equation.     

Extended Korn's inequalities and the associated best possible constants

We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H¹ () or its subspaces, a domain in ⁿ representing an isotropic elastic body—certain specific cases are called Korn's inequalities. Sufficient (and necessary) conditions on the Lamé moduli for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Evolution of the Diffusion Mixing Layer of Two Gases upon Interaction with Shock Waves

A mathematical model of mechanics of a twovelocity twotemperature mixture of gases is developed. Based on this model, evolution of the mixing layer of two gases with different densities under the action of shock and compression waves is considered by methods of mathematical simulation in the onedimensional unsteady approximation. In the asymptotic approximation of the full model, a solution of an initialboundary problem is obtained, which describes the formation of a diffusion layer between two gases. Problems of interaction of shock and compression waves with the diffusion layer are solved numerically in the full formulation. It is shown that the layer is compressed as the shock wave traverses it; the magnitude of compression depends on shockwave intensity. As the shock wave passes from the heavy gas to the light gas, the mixing layer becomes overcompressed and expands after shockwave transition. The wave pattern of the flow is described in detail. The calculated evolution of the mixinglayer width is in good agreement with experimental data.     

Injection of ideal fluid from a slot into a free stream

A cold gas is injected from a slot into a free stream of hot gas. In a simple model this leads to a two-fluid free boundary problem with the jump relation |u^-|²–|u⁺|² = ( constant) on the free boundary {u=0}, where u is the stream function. We prove that for any (–1, ) there exists a unique solution (Q, u) where Q is the flux of the injected fluid. Various properties of the solution u and of the free boundary are established.     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.