Various reciprocal theorems and variational principles in the theories of nonlocal micropolar linear elastic mediums

In the first part of our paper, we have extended the concepts of the classical convolution and the convolution scalar product given by I. Hlaváck and presented the concepts of the convolution vector and the convolution vector scalar product, which enable us to extend the initial value as well as the initial-boundary value problems for the equation with the operator coefficients to those for the system of equations with the operator coefficients.In the second part of this paper, based on the concepts of the convolution vector and the convolution vector scalar product, two fundamental types of reciprocal theorems of the non-local micropolar linear elastodynamics for inhomogeneous and anisotropic solids are derived.In the third part of this paper, based on the concepts and results in the first and second parts as well as the Lagrange multiplies method which is presented by W. Z. Chien, four main types of variational principles are given for the nonlocal micropolar linear elastodynamics for inhomogeneous and anisotropic solids. These are the counterparts of the variational principles of Hu-Washizu type, Hellinger-Reissner type and Gurtin type in classical elasticity as well as Hlaváck type and Iesan type in local micropolar and nonlocal elasticity. Finally, we have proved the equivalence of the last two main variational principles which are given in this paper.     

Natural convection heat transfer through an enclosed horizontal layer of supercritical carbon dioxide

In natural convection heat transfer through a thin horizontal layer of carbon dioxide, maxima in the equivalent thermal conductivities are obtained in the vicinity of the respective pseudocritical temperatures at pressures of 75.8, 89.6 and 103.4 bar. The maxima are the more pronounced, the closer the critical point is approached.Comparison of experimental results with Nusselt equations shows good agreement except for the immediate vicinity of the pseudocritical temperature.In visual observations a distinct change in flow structure appears in the immediate vicinity of the pseudocritical temperature. A steady state polygon pattern and a boiling-like action could not be observed in this geometry.

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Zusammenfassung Beim Wärmetransport durch freie Konvektion in einer dünnen waagerechten Schicht von Kohlendioxid ergaben sich Maxima der scheinbaren Wärmeleitfähigkeit in der Nähe der pseudokritischen Temperaturen bei Drükken von 75,8, 89,6 und 103,4 bar. Die Maxima sind um so ausgeprägter, je mehr man sich dem kritischen Punkt nähert.Ein Vergleich der Versuchsergebnisse mit Nusseltbeziehungen ergibt gute Übereinstimmung außer in unmittelbarer Umgebung der pseudokritischen Temperatur. Direkte Beobachtungen der Konvektionsmuster zeigen in unmittelbarer Umgebung der pseudokritischen Temperatur eine deutliche Strukturänderung. Ein stationäres Zellmuster und siedeähnliche Vorgänge konnten in dieser Anordnung nicht beobachtet werden.

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Nomenclature A area of the heating or cooling plate - C constant in the correlation - g acceleration of gravity - h heat transfer coefficient - k thermal conductivity of fluid in the gap - k _e equivalent thermal conductivity - m, n exponents of dimensionless numbers - q heat flux - T _C,PC absolute temperature; critical C, pseudocritical PC - Gr Grashof numberg ( _h– _c) ³/ ² - Nu Nusselt numberh/k - Pr Prandtl number/ - thermal diffusivity - coefficient of volume expansion - width of gap - _c,h temperature of cooling (c)-, heating (h)-plate - _m arithmetic mean temperature ( _c+ _h)/2 - kinematic viscosity - _c,h fluid density at the temperature of the cooling (c)- or heating (h)-plate - heat flow rate through the gap     

An experimental study on the two-dimensional opposed wall jet in a turbulent boundary layer: Change in the flow pattern with velocity ratio

Results of the measurement of flow properties in a two-dimensional turbulent wall jet which is injected into the turbulent boundary layer in the direction opposite to that of the boundary layer flow are presented by varying the ratio of the jet issuing velocity to the mainstream velocity . This flow includes the flow separation and the recirculating flow, and so it requires the magnitude and direction of instantaneous velocity be measured. In the present work, a tandem hot-wire probe is manufactured and its characteristics are examined experimentally. With the use of this probe the change in the penetration length of the jet with the velocity ratio is investigated. It is clarified that two regimes of flow patterns exist: in the low velocity ratio the penetration length of the jet changes little with , and in the high velocity ratio it goes far from the nozzle with increasing . Streamlines, turbulence intensity contours and static pressure contours are presented in the two typical velocity ratios corresponding to each of two flow patterns, and they are compared.List of symbols b ₀ nozzle width (Fig. 1) - , e mean and fluctuating output voltages of hot-wire anemometer - p, p mean static pressure, p = p – p _o - p ₀ static pressure in undisturbed mainstream - p _w , p _w mean wall pressure, p _w = p _w – p _o - U ₀ mainstream velocity - U _j jet velocity at the nozzle exit - , u mean and fluctuating velocity components in x-direction - u _e effective cooling velocity - x distance along the wall from nozzle exit - x _pmax, x _pmin positions where the wall pressure has maximum and minimum values respectively - x _s penetration length of jet - y distance from the wall - forward flow fraction - ₁, ₂ yaw and pitch angles of flow direction (Fig. 4) - velocity ratio, = U _j /U _o - density of the fluid - nondimensional stream function The authors wish to express their appreciation to Prof. Toshio Tanaka of Gifu University for his advice in the course of the experiment. They also would like to thank the Research Foundation for the Electrotechnology of Chubu which partly supported this work.     

A non-dualistic unified field theory of gravitation,electromagnetism and spin

The wisdom of classicalunified field theories in the conceptual framework of Weyl, Eddington, Einstein and Schrödinger has often been doubted and in particular there does not appear to be any empirical reason why the Einstein-Maxwell (E-M) theory needs to be geometrized. The crux of the matter is, however not whether the E-M theory is aesthetically satisfactory but whether it answers all the modern questions within the classical context. In particular, the E-M theory does not provide a classical platform from which the Dirac equation can be derived in the way Schrödinger's equation is derived from classical mechanics via the energy equation and the Correspondence Principle. The present paper presents a non-dualistic unified field theory (UFT) in the said conceptual framework as propounded by M. A. Tonnelat. By allowing the metric formds ²=g dx ^v x ^v and the non-degenerate two-formF=(1/2> _l) dx ^vdx ^vto enter symmetrically into the theory we obtain a UFT which contains Einstein's General Relativity and the Born-Infeld electrodynamics as special cases. Above all, it is shown that the Dirac equation describing the electron in an external gravito-electromagnetic field can be derived from the non-dualistic Einstein equation by a simple factorization if the Correspondence Principle is assumed.     

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

Summary A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

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Zusammenfassung Eine Zustandsgleichung vom Integraltyp mit einer deformationsabhängigen und faktorisierten Gedächtnisfunktion wird zur Beschreibung der Zeitabhängigkeit der Schubspannung, der ersten Normalspannungsdifferenz und, unter Verwendung des spannungsoptischen Gesetzes, auch des Auslöschungswinkels und der Strömungsdoppelbrechung einer Polystyrol-Schmelze bei Scherströmungen herangezogen. Die theoretischen Voraussagen werden mit Messungen verglichen. Die Nichtlinearität des viskoelastischen Verhaltens, repräsentiert durch die sogenannte Dämpfungsfunktion, wird durch eine einfache Exponentialfunktion mit nur einem Parametern angenähert. Die Dämpfungskonstanten kann, wie auch ein diskretes Relaxationszeitspektrum der Schmelze, aus der Frequenzabhängigkeit der Speicher- und Verlustmoduln bestimmt werden.

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a _i weight factor of thei-th relaxation time - a _T shift factor - C stress-optical coefficient - n flow birefringence in the shear flow plane - shear relaxation modulus - G() shear storage modulus - () shear loss modulus - H() relaxation time spectrum - h( _t,t ² ) damping function - M _w weight-average molecular weight - M _n number-average molecular weight - n damping constant - p ₁₂ shear stress - p ₁₁ –p ₂₂ primary normal stress difference - t current time - t past time - extinction angle - ( — _i) delta function - time and shear rate dependent viscosity - | ^*| absolute value of the complex viscosity - shear rate - _t,t relative shear strain between the statest andt - memory function - angular frequency - relaxation time - _i i-th relaxation time of the line spectrum - time and shear rate dependent primary normal stress coefficient - s steady-state value - t time dependence - ° linear viscoelastic behaviour With 6 figures and 1 table     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Effect of the Nature of the Interaction between a Gas and a Surface on the Relations for the Knudsen Layer in the Case of Strong Evaporation

The effect of the temperature accommodation coefficient _T on the relations at the Knudsen layer edge is investigated for strong evaporation using the moment method. An explicit expression for the dimensionless density as a function of the temperature and the Mach number M is obtained for 0 < _T < 1. For _T = 0 the entire solution is obtained in explicit form. It is shown that for = 0 and a condensation coefficient << 1 the temperature outside the Knudsen layer changes sharply as M varies from 0 to a certain value much less than unity after which the temperature ceases to depend on . For the model of specular reflection of the molecules from the surface the density and the temperature outside the Knudsen layer are found in explicit form as functions of the Mach number.     

Nonlinear shear-layer instability waves

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are discussed. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium (unsteady) nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise aging of the critical layer into a quasi-equilibrium state. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is shown to be in good agreement with available experimental results. Nonlinear effects occur at smaller amplitudes for the three-dimensional and supersonic cases than in the two-dimensional incompressible case. The instability-wave amplitude evolution is then described by one integro-differential equation with a cubic-type nonlinearity, whose inviscid solution always end in a singularity at finite downstream distance.The US Government has the right to retain a nonexclusive royalty-free license in and to any copyright covering this paper.     

Interaction of an oscillating vortex with a turbulent boundary layer

An oscillating vortex embedded within a turbulent boundary layer was generated experimentally by forcing a periodic lateral translation of a half-delta wing vortex generator. The objective of the experiment was to investigate the possibility that a natural oscillation, or meander, might be responsible for flattened vortex cores observed in previous work, which could also have contaminated previous turbulence measurements. The effect of this forced oscillation was characterized by comparison of measurements of the mean velocities and Reynolds stresses at two streamwise stations, for cases with and without forcing. The Reynolds stresses, especially w, were affected significantly by the forced oscillation, mainly through contributions from the individual production terms, provided the vortex was not too diffuse.List of Symbols a amplitude of forced vortex motion - f frequency of forced vortex generator motion - l vortex generator root chord - L flow length scale - R _Y , R _Z vortex core radial dimensions in vertical and spanwise directions, respectively - R_r vortex circulation Reynolds number R = / - u, v, w instantaneous velocity components in X, Y, Z directions - U, V, W mean velocities; shorthand notation for u, , w - X, Y, Z right-hand Cartesian streamwise, vertical, and spanwise coordinate directions - boundary-layer thickness - overall circulation - air kinematic viscosity - _x streamwise vorticity, _X = W/Y–V/d+t6Z - ( )₀ reference value (measured at X = 10 cm) - ( )c refers to vortex center - ( ) _max maximum value for a particular crossflow plane - ( ) (overbar) time average - ( ) (prime) fluctuating component, e.g., u=U+u     

Modellvorstellung zur turbulenten Filmkondensation strömenden Dampfes

Zusammenfassung Der Wärmeübergang bei turbulenter Film kondensation strömenden Dampfes an einer waagerechten ebenen Platte wurde mit Hilfe der Analogie zwischen Impuls-und Wärmeaustausch untersucht. Zur Beschreibung des Impulsaustausches im Film wurde ein Vierbereichmodell vorgestellt. Nach diesem Modell wird die wellige Phasengrenze als starre rauhe Wand angesehen. Die Abhängigkeit einer Schubspannungs-Nusseltzahl von der Film-Reynoldszahl und Prandtlzahl wurde berechnet und dargestellt.

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A model for turbulent film condensation of flowing vapour

The heat transfer in turbulent film condensation of flowing vapour on a horizontal flat plate was investigated by means of the analogy between momentum and heat transfer. To describe the momentum transfer in the film a four-region model was presented. With this model the wavy interfacial surface is treated as a stiff rough wall. A shear Nusselt number has been calculated and represented as a function of film Reynolds number and Prandtl number.

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Formelzeichen a Temperaturleitkoeffizient - k Mischungswegkonstante - k _s äquivalente Sandkornrauhigkeit - Nu _x lokale Schubspannungs-Nusseltzahl,Nu _x=x_xv/uw - Pr Prandtlzahl,Pr=v/a - Pr _t turbulente Prandtlzahl,Pr _t =_m/_q - q Wärmestromdichte _q - R Wärmeübergangswiderstand - R_f Wärmeübergangswiderstand des Films - Re _F Reynoldszahl der Filmströmung - T Temperatur - U, V Geschwindigkeitskomponenten des Dampfes in waagerechter und senkrechter Richtung - u, Geschwindigkeitskomponenten des Kondensats in waagerechter und senkrechter Richtung - V Querschwankungsgeschwindigkeit des Kondensats und des Dampfes - u _/gtD Schubspannungsgeschwindigkeit an der Phasengrenze für die Dampfgrenzschicht, uD =(/)^1/2 - u _F Schubspannungsgeschwindigkeit an der Phasengrenze für den Kondensatfilm,u _F =(/)^1/2 - u _w Schubspannungsgeschwindigkeit an der Wand der Kühlplatte,u _w =(w/)^1/2 - y Wandabstand - x Wärmeübergangskoeffizient - gemittelte Kondensatfilmdicke - _s Dicke der zähen Schicht der Filmströmung an der welligen Phasengrenze - ₄ Dicke der zähen Schicht der Filmströmung an der gemittelten glatten Phasengrenze - Wärmeleitzahl - dynamische Viskosität - v kinematische Viskosität - Dichte - Oberflächenspannung - _w Wandschubspannung - Schubspannung an der Phasengrenzfläche - _m turbulente Impulsaustauschgröße - _q turbulente Wärmeaustauschgröße Indizes d Wert des Dampfes - w Wert an der Wand - x lokaler Wert inx - Wert an der Phasengrenze Stoffgrößen ohne Index gelten für das Kondensat     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Turbulent boundary layer over a smooth wall with widely separated transverse square cavities

Laser-Doppler velocimetry (LDV) measurements and flow visualizations are used to study a turbulent boundary layer over a smooth wall with transverse square cavities at two values of the momentum thickness Reynolds number (R =400 and 1300). The cavities are spaced 20 element widths apart in the streamwise direction. Flow visualizations reveal a significant communication between the cavities and the overlying shear layer, with frequent inflows and ejections of fluid to and from cavities. There is evidence to suggest that quasi-streamwise near-wall vortices are responsible for the ejections of fluid out of the cavities. The wall shear stress, which is measured accurately, increases sharply immediately downstream of the cavity. This increase is followed by a sudden decrease and a slower return to the smooth wall value. Integration of the wall shear stress in the streamwise direction indicates that there is an increase in drag of 3.4% at bothR .Nomenclature C _f skin friction coefficient - C _fsw friction coefficient for a continuous smooth wall - k height of the cavity - k ⁺ ku / - R Reynolds number based on momentum thickness (U ₁ /v) - Rx Reynolds number based on streamwise distance (U ₁ x/) - s streamwise distance between two cavities - t time - t ⁺ tu ₂ / - U ₁ freestream velocity - mean velocity inx direction - u,v,w rms turbulent intensities inx,y andz directions - u local friction velocity - u _sw friction velocity for a continuous smooth wall - w width of the cavity - x streamwise co-ordinate measured from the downstream edge of the cavity - y co-ordinate normal to the wall - z spanwise co-ordinate - y ⁺ yu / - boundary layer thickness - ₀ boundary layer thickness near the upstream edge of the cavity - _i thickness of internal layer - kinematic viscosity of water - ⁺ zu / - momentum thickness     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

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.     

Loss of stability and bifurcation of auto-oscillatory regimes close to Poiseuille flow

Bifurcation of Poiseuille flow in a flat channel is used as an example to analyze the problem of determining variables that permit study of bifurcation of a main steady flow of a viscous incompressible liquid for parameters close to the values of the coordinates of a point on the curve of neutral stability at which the first Lyapunov exponent d₀ vanishes and there is a changeover from subcritical to supercritical bifurcation. For Poiseuille flow, such a point (R₂,₂, where R₂ is the Reynolds number, and ₂ is the wave number, occurs on the lower branch of the neutral curve. In this paper, it is shown by the Lyapunov-Schmidt method that for < ₂ the stable time-periodic solution that bifurcates into the subcritical region loses stability in the case of slight supercriticality, and a fold singularity is formed in the amplitude surface. The nature of this additional bifurcation is determined by the sign of the second Lyapunov exponent d₁. For its calculation, the value of ₂ is fixed, and the bifurcation that occurs when the Reynolds number is changed is considered. A solution is sought in the form of a convergent series in powers of = ((R – R₀)^1/4, = ±1. The condition of solvability, which serves to determine the coefficient of ⁴, makes it possible to determine the value of d₂. This procedure is entirely general and makes it possible to study bifurcation in the neighborhood of a point of degeneracy on the neutral curve in other hydrodynamic problems too.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 41–48, March–April, 1991.     

A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity

The aim of this article is to study the quasistatic evolution of a three-dimensional elastic-perfectly plastic solid which satisfies the Prandtl-Reuss law. The evolution of the field of stresses -which solves a time dependent variational inequality — and that of the field of displacements u, have been described in previous works [15], [26], [35], [36], [37] but it was not shown there that and u satisfy indeed the Prandtl-Reuss constitutive law. In this article we find and u in a class of functions which are sufficiently regular for the Prandtl-Reuss law to make sense and we prove that and u satisfy the constitutive law. This result is attained by considering the elastic-perfectly plastic model as the limit of a family of elastic-visco-plastic models like those of Norton and Hoff. The Norton-Hoff type models which we introduce depend on a viscosity parameter > 0; we study the perturbed models (i.e. > 0 fixed) and then we pass to the limit 0.Dedicated to James Serrin on the occasion of his 60^th Birthday     

Integral technique solutions to a class of simple ablation problems

Summary The integral technique is applied to a class of ablation problems of a semi-infinite solid subjected to a heat flux of the form q = At ^m . The governing equations are highly simplified by normalizing the variables with respect to the values obtained at the onset of ablation and by introducing the asymptotic values obtained for large times. The results are discussed in terms of a parameter expressing the ratio of heat capacities between the heat storable in the solid and the latent heat of ablation.

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Sommario La tecinca integrale viene applicata nella ricerca di soluzioni al problema dell'ablazione di un solido semiinfinito soggetto ad un flusso di calore del tipo q = At ^m . Le equazioni risolventi sono notevolmente semplificate introducendo un a normalizzanzione rispetto ai valori relativi all'inizio del processo ed alla condizione asintotica. I risultati sono espressi in termini di un parametro rapporto delle capacità termiche.

     

Instability of stationary solutions for equations of curvature-driven motion of curves

A study is made for equations of evolving curves on a two-dimensional square domain. It is assumed that a curve moves depending on its curvature, normal vector, and position and is orthogonal to at its end points. Under some conditions, instability of stationary solutions is proved through an eigenvalue analysis.     

Face-Centered and Volume-Centered Discrete Analogs of the Exterior Differential Equations Governing Porous Medium Flow I: Theory

This paper presents a physics-oriented approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcys law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant translation rules. In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the block-centered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.