Analysis of a method and a system to determine heat transfer and properties of fluids in the absence of temperature measurements and a modified Fourier Equation

The technique to determine by capacitance measurements heat transfer, thermal transport and dielectric properties of fluids introduced recently is now analyzed for a simple system of spherical geometry. The temperature distribution under programmed heat input to a fluid annulus between solid walls is computed by finite difference method for the determination of the capacitance time function of the arrangement. A system of heavy wall structure and heated long enough will produce a capacitance-time curve which is a function of thermal conductivity only. Thermal diffusivity is of influence in thin wall systems. The capacitance change of a heavy wall arrangement is related to the thermal conductivity of the test fluid by a modified Fourier equation. This equation describes the heat flow through the fluid layer but includes the thermal expansion of the solid walls. The change of geometry with T is therefore accounted for. For other multicomposite structures the Fourier equation must be further modified by including the thermal expansion of all materials of the structure and possibly also their compressibilities.

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Zusammenfassung Die kürzlich eingeführte Methode der Bestimmung von Wärmeübergang, thermischen Transport und dielektrischen Größen mittels Kapazitäts-Zeit-Messung wird analysiert für ein einfaches kugeliges System. Die Temperaturverteilung in der Flüssigkeit im Kugelspalt zwischen zwei festen Körpern wird für konstante Wärmezufuhr von außen mittels der Differenzmethode bestimmt und daraus die Kapazitäts-Zeit-Funktion ermittelt. Es wird gezeigt, daß die Kapazitäts-Zeit-Kurve nur eine Funktion der Wärmeleitzahl ist für den Fall dickwandiger Anordnungen. Für dünnwandige Systeme wird sie auch abhängig von der Temperaturleitzahl. Es wird eine modifizierte Fourier-Gleichung eingeführt, die den Wärmetransport durch die Flüssigkeit beschreibt, dabei aber die Änderung der Geometrie der Schicht berücksichtigt, die sich wegen der thermischen Ausdehnung der festen Wände bei der Einstellung der Temperaturdifferenz ergibt. Für andere mehrschichtige Körper muß die Fourier-Gleichung weiterhin modifiziert werden durch Berücksichtigung der thermischen Ausdehnungskoeffizienten aller beteiligten Materialien und möglicherweise auch ihrer Kompressibilitäten.

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Nomenclature A average cross-sectional area of fluid layer - A coefficient matrix - B matrix defined by Eq. (20) - B₀ geometric constant of fluid layer (A/L) at reference temperature - C capacitance of arrangement - C_i, C_r capacitance of layer of fluid i and reference fluid at temperature T - capacitances at reference temperature - C_H, c_l specific heats of outer and inner wall - FA...FE constants defined in Eqs. (13 ... 17) - L thickness of fluid layer - M_H, M_L mass of outer and inner wall - P power input to the system - R constant defined by Eq. (24) - T temperature - T_ref reference temperature - T (O, t), T (L, t) temperatures of outer and inner wall at time t - T _i ⁿ , T _i+0 ^n+m temperatures at location i and time n (m=number of t's; 0=number of x's) - T temperature difference across fluid layer - T apparent temperature difference - t_h, T_l temperature increases of outer and inner wall - T_max temperature change of system from one to another thermal equilibrium condition a thermal diffusivity - k, k_i, k_r thermal qonductivity of fluids and of fluid i and reference fluid - q heat flow through fluid layer - ^rh,^rl inner radius of outer wall and outer radius of inner wall - ^rOH,^rOL radii at reference temperature - t time - t time interval - x coordinate - ¯x vector of unknown T_i ⁿ⁺¹ - x length interval Greek symbols linear thermal expansion coefficient - _H, _L linear thermal expansion coefficient of materials of outer and inner wall - dielectric constant - _i, _ref dielectric constant of fluid i and reference fluid - ₀ permittivity of free space - multiplyer of conduction Eq. (7) in finite difference form - time needed to establish quasi-steady state conditions in the system heated by a constant power input In honor of Prof. Dr. E. Schmidt to his 80th Birthday.     

A numerical method for the evaluation of Fourier integrals

This paper suggests the use of spline function interpolation in the evaluation of Fourier integrals. At the same time, the numerical results of some common functions by various interpolation methods and a simplified method of construction of spline function for various boundary conditions are also presented.     

The Fourier pseudo-spectral method for the SRLW equation

In this paper we present a Fourier pseudo-spectral method with a restraint operator for the SRLW equation. We prove the stability of the schemes and give optimum error estimates.     

Variational solutions for the two-stream mixing of power-law fluids

The variational principle of Lebon-Lambermont, originally proposed for Newtonian fluids, is seen to be applicable to generalized Newtonian fluids. As an example, it is applied to obtain approximate solutions of the laminar boundary-layer equations for the two-stream mixing of power-law fluids. The flow along a flat plate is obtained as a particular case when the consistency of one of the fluids diverges.     

A model for the thermorheological behavior of viscoelastic fluids

Using a power-law ansatz for the temperature dependence of the shear modulus on the level of internal variables, the thermorheological behavior is modeled for viscoelastic fluids of a special group of rheological constitutive equations (rate-type models). The model parameter introduced characterizes thermoelastic contributions. The relation between the model parameter and the physical quantities appearing in deformation processes is discussed. Based on the chosen temperature dependence of the shear modulus, thermodynamically consistent equations like the nonlinear rheological constitutive equation and the temperature equation are derived. The special cases of entirely entropy and energy elastic fluids are also considered. The thermorheological behavior (exo-, - or endothermal processes) of a viscoelastic fluid in a stress-growth experiment followed by relaxation is analyzed with respect to the model parameter.     

Asymptotic profiles for the third grade fluids equations

We study the long time behaviour of the solutions of the third grade fluids equations in dimension 2. Introducing scaled variables and performing several energy estimates in weighted Sobolev spaces, we describe the first order of an asymptotic expansion of these solutions. It shows in particular that, under smallness assumptions on the data, the solutions of the third grade fluids equations converge to self-similar solutions of the heat equations, which can be computed explicitly from the data.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

Smallest scale estimates for the Navier-Stokes equations for incompressible fluids

We consider solutions of the Navier-Stokes equations for incompressible fluids in two and three space dimensions. We obtain improved estimates, in the limit of vanishing viscosity, for the Fourier coefficients. The coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity. This defines the minimum scale, the size of the smallest feature in the flow.The work of Kreiss was supported in part by National Science Foundation under Grant DMS-8312264 and Office of Naval Research under Contract N-00014-83-K-0422.     

求解对称矩阵特征值及特征向量的初等变换法

在力学中有一类量的求解可归结为矩阵特征值和特征向量的求解,而求解矩阵的特征值将要求解高次方程的根,这在数学上将遇到难以克服的困难. 应用初等变换方法,将对称矩阵的特征矩阵对角化. 利用这种方法,同时可求出该矩阵所有的特征值和正交的特征向量,避免了求解高次方程根的困难与把各特征向量正交化的麻烦.     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

An apparent paradox in the mechanics of strings

We show with a simple example that cases of incompatibility (first displayed by Signorini) between the nonlinear and the linearized version of the same problem occur also in contact problems.

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Sommario Si mostra con un esempio elementare che casi di incompatibilità alla Signorini tra i risultati dello studio non-lineare e di quello linearizzato di uno stesso problema si presentano anche in problemi di contatto.

     

Shock structure for heat conducting and viscid fluids

Summary The structure of plane steady shock waves is investigated for a class of theories of heat-conducting and viscid fluids which avoid the well-known difficulties of the Navier-Stokes-Fourier equations for high-frequency sound waves and acceleration waves.It is demonstrated that a unique continuous and stable shock structure exists only for sufficiently low Mach numbers.

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Sommario Si studia la struttura delle onde d'urto piane in una classe di teorie dei fluidi dissipativi che risolvono le note difficoltà delle equazioni di Navier-Stokes-Fourier per le onde sonore di alta frequenza e per le onde d'accelerazione.Si dimostra che una soluzione unica, continua e stabile esiste solo per numeri di Mach sufficientemente piccoli.

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Research supported by C.N.R., G.N.F.M.     

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.     

A representation of the solution of the Gromeko problem for viscous and elastico-viscous fluids

The solution of the Gromeko problem [1] on unsteady flow of a viscous fluid in a long circular pipe is among the few exact solutions of the Navier-Stokes equations. Its effective solution is obtained only when the longitudinal pressure gradient is given as an arbitrary time function. However, in practice we encounter cases when the flow rate is a known time function. This sort of problem arises, in particular, in rheological experiments using viscometers with a given flow rate. In this case the determination of the pressure gradient from the given flow rate leads in the general case to a very unwieldy expression. Below we present an effective solution of this problem for viscous and elasticoviscous media using the method of solving the inlet flow problem for a steady flow of a viscous fluid in a semi-infinite pipe. It is shown that for the case of a viscous fluid these two problems are actually equivalent.     

Exact solutions of the boundary layer equations for power-law fluids

Exact analytic solutions of the equations of the hydrodynamic boundary layer are obtained for pseudoplastic fluids with exponents n=1/5, 1/4, 1/2, 3/5, 5/7 flowing longitudinally over a flat plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 39–42, September–October, 1989.     

On the lubrication approximation for a class of viscoelastic fluids

We present a rigorous derivation of a dimensionally reduced Reynolds type equation for thin film flow lubrication of a class of viscoelastic fluids by employing a perturbation analysis on the upper-convected Maxwell model in natural orthogonal coordinates. This approximation accounts for the viscoelastic and curvature corrections to the classical Reynolds lubrication approximation. Comparison of our approximation with the classical Reynolds approximation suggests that viscoelasticity can have a significant influence on the lubrication characteristics, at least for certain values of the film thickness and of the eccentricity ratios of the journal bearing.