Towards realistic flow modelling. Creation and evaluation of two-dimensional simulated porous media: An image analysis approach

This work is part of an attempt to quantify the relationship between the permeability tensor (K) and the micro-structure of natural porous media. A brief account is first provided of popular theories used to relate the micro-structure toK. Reasons for the lack of predictive power and restricted generality of current models are discussed. An alternative is an empirically based implicit model whereinK is expressed as a consequence of a few pore-types arising from the dynamics of depositional processes. The analytical form of that implicit model arises from evidence of universal association between pore-type and throat size in sandstones and carbonates. An explicit model, relying on the local change of scale technique is then addressed. That explicit model allows, from knowledge of the three-dimensional micro-geometry to calculateK explicitly without having recourse to any constitutive assumptions. The predictive and general character of the explicit model is underlined. The relevance of the change of scale technique is recalled to be contingent on the availability of rock-like three-dimensional synthetic media. A random stationary ergodic process is developed, that allows us to generate three-dimensional synthetic media from a two-dimensional autocorrelation functionr( _x , _y ) and associated probability density function measured on a single binary image. The focus of this work is to ensure the rock-like character of those synthetic media. This is done first through a direct approach:n two-dimensional synthetic media, derived from single set ( ,r( _x , _y )) yieldn permeability tensorsK _i-1,n ⁱ (calculated by the local change of scale) of the same order. This is a necessary condition to ensure thatr( _x , _y ) and carry all structural information relevant toK. The limits of this direct approach, in terms of required Central Process Unit time and Memory is underlined, raising the need for an alternative. This is done by comparing the pore-type content of a sandstone sample andn synthetic media derived fromr( _x , _y ) and measured on that sandstone-sample. Achievement of a good match ensures that the synthetic media comprise the fundamental structural level of all natural sandstones, that is a domainal structure of well-packed clusters of grains bounded by loose-packed pores.     

Quasi-static thermal deformations of a sphere by radiation from a point source

Summary In this paper the quasi-static temperature and stress distributions set up in an elastic sphere by radiation from a point source at a finite distance from the centre of the sphere and out-side it, have been discussed. The temperature boundary condition has been taken in the general form involving an arbitrary function of time. The final solutions have been obtained in terms of series involving Legendre polynomials. Numerical calculations have been done on IBM 1620 Computer and a desk calculator. The results have been represented in graphs.Notation the del operator - u the displacement vector - T the excess of temperature over that at state of zero stress and strain - , Lamé's constants - /2(+) Poisson's ratio - coefficient of linear expansion - 2(1+) - a radius of the sphere - d distance of the point source from the centre of the sphere - d _o a/d - K coefficient of thermal conductivity - h heat transfer coefficient of the surface     

Transmission of Rayleigh waves over the surface of a heterogeneous medium

Summary The frequency equation of Rayleigh waves propagating over the free surface of an isotropic, perfectly elastic, heterogeneous semi-infinite medium with material properties varying as = ₀ e ^az , = ₀ e ^az , = ₀ e ^az (a>0) has been obtained. Solution of the frequency equation in closed form is obtained in two cases (i) =0, (ii) =, and the Rayleigh wave dispersion curves for phase and group velocities drawn. In both the cases the medium yields single Rayleigh modes which cannot propagate below certain cut-off frequencies. It is found that in case (i), <c<c ₀ and 0.87500 <c _g <c ₀, and in case (ii), 1.03082 <c<c ₁ and 0.90850 <c _g <c ₁, wherec andc _g denote phase nad group velocities respectively, is the constant shear wave velocity of the mediumc ₀ andc ₁ are the corresponding Rayleigh wave velocities of the homogeneous medium of the same Poisson's ratio. The motion of the surface particles is found to be retrograde elliptical as in the homogeneous case, but the ratic of the major and minor axes now becomes frequency dependent and is plotted against frequency. In both the cases (i) and (ii), the ratio starts at a lower value at the cut-off frequency and approaches the corresponding value of the homogeneous medium at high frequencies.     

On the determination of the European gravimetric geoid

Summary The method of the automated computation of the gravimetric deflections of the vertical and of the geoidal heights for the European region is described. The work was carried out during the period 1986–1988 by the Topographic Service of the Czechoslovak Army. The computation applies to 20 sheets of the international map 1:1 000 000 (total area of =16^c, =30^c - see Fig. 1). The mean values of the free-air anomalies for each surface element =5, =7.5, approximately 9 × 9 km, were used with radius of integration of 300 km.     

Relations entre le creusement et le comblement des perturbations et leur déplacement

Résumé On commence par définir le creusement et le comblement d'une fonctionp(, t) du tempst et des points (, ) d'une surface régulière fermée en se donnant, sur cette surface, un vecteur vitesse d'advection ou de transfert tangent à . Le creusement (ou le comblement) est la variation dep sur les particules fictives se déplaçant constamment et partout à la vitesse , A chaque vecteur et pour un mêmep(, ,t) correspond naturellement une fonction creusementC (, ,t) admissible a priori; mais une condition analytique très générale (l'intégrale du creusement sur toute la surface fermée du champ est nulle à chaque instant), à laquelle satisfont les fonctions de perturbation sur les surfaces géopotentielles, permet de restreindre beaucoup la généralité des vecteurs d'advection admissibles a priori et conduit à des vecteurs de la forme: , oùT est un scalaire régulier, () une fonction régulière de la latitude , le vecteur unitaire des verticales ascendantes etR/2 une constante. Ces vecteurs sont donc une généralisation naturelle des vitesses géostrophiques attachées à tout scalaire régulier. Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde, le vecteur d'advection par rapport auquel on doit définir le creusement est précisément une vitesse géostrophique: on a alors ()=sin etT un certain champ bien défini de température moyenne.On déduit ensuite une formule générale de géométrie et de cinématique différentielles reliant la vitesse de déplacement d'un centre ou d'un col d'un champp(, ,t) à son champ de creusementC (, ,t) et au vecteur d'advection correspondant. Cette formule peut être transformée et prend la forme d'une relation générale entre le creusement (ou le comblement) d'un centre ou d'un col et la vitesse de son déplacement, sans que le vecteur d'advection intervienne explicitement. On analyse alors les conséquences de ces formules dans les cas suivants: 1^o) perturbations circulaires dans le voisinage du centre; 2^o) perturbations ayant, dans le voisinage du centre, un axe de symétrie normal ou tangent à la vitesse du centre; 3^o) évolution normale des cyclones tropicaux.Finalement, on examine les relations qui existent entre le creusement ou le comblement d'un champ, le vecteur d'advection et la configuration des iso-lignes du champ dans le voisinage d'un centre.Ces considérations permettent d'expliquer plusieurs propriétés bien connues du comportement des perturbations dans différentes régions.

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Summary The deepening and filling (development) of a functionp(, ,t) of the timet and the points (, ) of a regular closed surface is first of all defined, in respect to a given advection or transfer velocity field tangent to , as the variation ofp on any fictitious particle moving constantly and everywhere with the velocity . For a givenp(, ,t) and to any there corresponds a well defined development fieldC (, ,t). All theseC fields are a priori admissible, but a very general analytical condition of the perturbation fields in synoptic meteorology (the integral of the development fieldC (, ,t) on any geopotential surface vanishes at any moment), leads to an important restriction to advection vectors of the form: , whereT is any regular scalar, () any regular function of latitude, the unit vector of the ascending verticals andR/2 a constant. These vectors are a natural generalisation of the geostrophic velocities attached to any regular scalar. Whenp(, ,t) is the pressure perturbation at sea level, its development must be defined in respect to a geostrophic advection vector belonging to the above defined class of vectors with ()=sin andT a well defined mean temperature field.A general formula of the differential geometry and kinematics ofp(, ,t) is then derived, giving the velocity of any centre and col of ap(, ,t) as a function of the advection vector and the corresponding development fieldC (, ,t). This formula can be transformed and takes the form of a general relation between the deepening (and filling) of a centre (or a col) of ap(, ,t) and its displament velocity, the advection vector appearing no more explicitly. A detailed analysis of the consequences of these formulae is then given for the following cases: 1^o) circular perturbations in the vicinity of a centre; 2^o) perturbations having, in the vicinity of a centre, an axis of symmetry normal or tangent to the velocity of the centre; 3^o) normal evolution of the tropical cyclones.Finally, the relations between the developmentC (, ,t) of a fieldp(, ,t), the advection velocity vector and the configuration of the iso-lines in the vicinity of a centre are analysed.These theoretical results give a rational explanation of several well known properties of the behaviour of the perturbations in different geographical regions.

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Communication à la 2ème Assemblée de la «Società Italiana di Geofisica e Meteorologia» (Gênes, 23–25 Avril 1954).     

Turbidometric estimations in Mexico city using the Volz sun photometer

Summary Measuring, with the aid of two filters, the instantaneous intensity of the solar radiation in two wave lengths ( _B = 0.44 , _R = 0.64 ) by means of a sun photometer designed byVolz, we carried out determinations of the decadic turbidity coefficientB (=0.5 ) and the wave length exponent of the haze extinction for Mexico City. Observations were made for almost two and a half years (1960 to 1962 period). A seasonal size distribution in both parameters was found. Although the data thus obtained are provenient of a contaminated atmosphere, comparison of our data is made with those found for higher latitudes ofÅngström, Schüepp andVolz. The height of the homogeneous haze layerH _D was calculated showing pronounced variations for a given wind direction. The maximum and minimum values ofB enable us to get, by the first approximation, the aerosol size distribution ofJunge for our latitudes. However, for exceptional very clear days having maximum actinometric intensity of the solar radiation the sensitivity of the microamperimeter in theVolz sun photometer fails.     

Recherches complémentaires sur le bases du calculateur météorologique «Temp»

Résumé La formule de base, traduisant une propriété analytique d'une classe très générale de fonctions, est un corollaire du théorème fondamental démontré dans un mémoire précédent, d'après lequel, étant donnés une fonction continue,p(, ,t) des points (, ) d'une surface régulière fermée et du temps et le champ d'un vecteur vitesse de transfert ou d'advection tangent à et ayant des lignes de flux fermées et régulières, il existe un opérateur spatial, linéaire, non singulierA tel que la fonctionA(p+Const.) soit purement advective par rapport a (sans creusement ni comblement). Ce théorème peut être exprimé par l'équation , où est un opérateur spatial, linéaire et non singulier, fonction deA.La détermination de peut être faite, soit en comparant deux formes différentes de la solution générale de l'équation en , soit en utilisant un raisonnement a priori très simple. On arrive ainsi au résultat pour un certain scalaireu(, ).Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde l'équation résulte aussi, comme nous l'avons montré dans le mémoire précédent, de notre théorie hydrodynamique des perturbations. On montre ici que la même équation peut encore être déduite de l'équation de continuité associée à la condition d'équilibre quasi statique selon la verticale.Comme applications de la formule de base (solution générale de l'équation enM), on étudie les problèmes suivants: 1^o creusement et comblement en général; 2^o creusement et comblement des centres et des cols; 3^o mouvement des centres et des cols; 4^o instabilité d'un champ moyen; 5^o propriétés spatiales des champsp(, ,t) et des vecteurs d'advection analytiques.Après une discussion des erreurs de la prévision d'un champp(, ,t) par la formule de base, du fait des erreurs des observations et du fonctionnement du calculateur, on examine quelques particularités du transfert ou advection d'un champf ₀(, ) par le vecteur . Enfin, le dernier chapitre du mémoire donne des éclaircissements complémentaires sur la structure du calculateur électronique «Temp» (qui effectue automatiquement les opérations mathématiques de la formule de base) et expose l'état actuel de sa construction.

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Summary The basic formula, expressing an analytical property of a very general class of functions, is a corollary of the fundamental theorem, proved in a previous paper, according to which, given a functionp(, ,t) of the points (, ) of a closed regular surface and of the time, and a transfer or advection velocity vector tangent to and having regular closed streamlines, there is a spatial, linear, non singular operatorA such thatA(p+const.) is a purely advective function in respect to (no deepening). This theorem can be expressed by the equation where is a spatial, linear, non singular operator depending onA.The determination of can be attained, either by the comparison of two different forms of the general solution of the -equation, or by a simple a priori reasonning. The conclusion is thus reached that for a certain scalaru(, ).Whenp(, ,t) is the pressure perturbation at sea level, it was shown, in the preceding paper, that the equation can also be derived from our hydrodynamical perturbation theory. We now show that for this particular case, the same equation is also a consequence of the equation of continuity together with the condition of quasi statical vertical equilibrium.The following problems are then analysed by means of the basic formula: 1^o deepening and filling in general; 2^o deepening and filling of the centres and cols; 3^o motion of the centres and cols; 4^o instability of a mean field; 5^o spatial properties of the analytical fields and advection vectors .The errors in the forecast of a field,p(, ,t) by means of the basic formula, due to the observational and computational errors, are discussed, and some peculiarities of the transfer or advection of a fieldf ₀(, ) by are examined. Finally, complementary points are disclosed on the structure of the electronic computer «Temp» which performs automatically the mathematical operations of the basic formula, and a brief report is given of the present state of its construction.

     

Theory of the piezomagnetic effect in titanomagnetite-bearing rocks

Summary The stress sensitivitiesS _x andS _R of susceptibility and remanence for titanomagnetite-bearing rocks are calculated in terms of magnetostriction constants ₁₀₀ and ₁₁₁ and anisotropy constantsK ₁,K ₂ of the magnetic minerals.S _x andS _R are represented by quite different algebraic expressions but happen to have comparable numerical values over the whole range of titanomagnetite compositions. Both increase strongly with titanium content. This leads to more optimistic calculations of tectonomagnetic effects than with the previously assumed stress sensitivity for pure magnetite.     

UV spectrometry for measuring particulate organic matter in natural water

Ultraviolet (UV) spectra were measured to characterize the dissolved and particulate organic matter in a bog and to investigate the seasonal fluctuation of this organic matter. The optical density spectra in the UV region of dissolved organic matter (DOM) were proportional to ^–4, where is the wavelength of incident light. There was also small absorption on the ^–4 spectra. The optical density at a wavelength of 280 nm was proportional to the concentration of dissolved organic carbon (DOC). The spectra and the dependency of optical density on DOC did not vary throughout the investigation period. The UV spectra of acid extractions of particulate organic matter (POM) showed two absorption peaks at the wavelengths of 220 nm and 335 nm. These peaks are well represented by the sum of two Lorentz functions. The ratio of the absorbances,A ₂₈₀/A ₃₃₅, was dependent on the nitrogen/carbon ratio of POM and the chlorophyll fraction in particulate organic carbon (POC), and not dependent on POC itself. The dependency of POC absorption at 280 nm varied with this ratio which fluctuated seasonally.     

Friction constitutive law with rate and state dependences

A general formula for the Dieterich-Ruina friction constitutive law with rate and state (n-state variables,n=1, 2,...) dependences has been obtained and discussed under the assumption that the slip acceleration a varies ion a linearly with the slip displacement , namelya = a ₀ + (-₀). Wherea ₀, ₀ are initial constants, is the acceleration rate and constant.a ₀ and may be arbitrary constants (positive, negative or zero).The extreme value of frictional resistance and the existence condition of the extreme value, which are very important and govern to some degree the motion process of a frictionally slipping mechanical system, have been analyzed. A critical value _c which is the measure of the velocity weakening and velocity strengthening of the mechanical system, and its properties and the relationship to the extreme problem have been studied. Again, according to the critical value _c, the concepts of light or strong velocity weakening (or strengthening) are introduced.A possibly new phenomenon that frictional resistance may vary in some kind of decayed oscillation is found. Finally, the condition for the smallest frictional resistance for a slipping mechanical system with nonuniform acceleration has been obtained.     

Determination of characteristics of steam reservoirs by Radon-222 measurements in geothermal fluids

The authors conducted a Rn²²² survey in wells of the Larderello geothermal field (Italy) and observed considerable variations in concentrations. Simple models show that flow-rate plays an important part in the Rn²²² content of each well, as it directly affects the fluid transit time in the reservoirs. Rn²²² has been sampled from two wells of the Serrazzano area during flow-rate drawdown tests. The apparent volume of the steam reservoir of each of these two wells has been estimated from the Rn²²² concentration versus flow-rate curves.List of symbols Q Flow-rate (kg h^–1) - Decay constant of Rn²²² (=7.553×10^–3 h^–1) - Porosity of the reservoir (volume of fluid/volume of rock) - ₁ Density of the fluid in the reservoir (kg m^–3) - ₂ Density of the rock in the reservoir (kg m^–3) - M Stationary mass of fluid filling the reservoir (kg). - E Emanating power of the rock in the reservoir (nCi kg _rock ^–1 h^–1). - P Production rate of Rn²²² in the reservoir: number of atoms of Rn²²² (divided by 1.764×10⁷) transferred by the rock to the mass unit of fluid per unit time (nCi kg _fluid ^–1 h^–1). - N Specific concentration of Rn²²² in the fluid (nCi kg^–1) - Characteristic time of the steam reservoir at maximum flow-rate (=M/Q)     

Sur l'étude de l'intensité solaire en Égypte et en particulier de la région spectrale de courtes longueurs d'onde

Résumé Des mesures actinométriques de l'intensité solaire dans les domaines spectraux: <425, <525, >630 m ainsi que celle dans l'ensemble du spectre solaire (totale) ont été effectuées à Giza (lat. 30°N) à l'aide d'un actinomètre Linke-Feussner, pendant une année complète pour tous les jours le ciel étant parfaitement clair.D'après ces mesures on peut conclure que: l'intensité solaire totale en été est inférieure à sa valeur en hiver en raison de l'accroissement de la teneur en vapeur d'eau atmosphérique en été; la distribution spectrale de l'énergie solaire, au cours de l'année peut être considérée comme constante, pour une masse d'air déterminée; l'intensité solaire, arrivant au sol, dans la région spectrale en-dessous de =425 m peut donner une indication des fluctuations de l'épaisseur réduite de l'ozone.

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Summary Actinometric measurements of the solar intensity in the spectral regions: <425, <525, >630 m as well as the total solar intensity for the whole spectrum of the sun have been undertaken at Giza (lat. 30°N) with a Linke-Feussner actinometer, for a whole year and for every day the sky was perfectly clear.From these measurements, one can conclude that: the total solar intensity in summer is less than that in winter because of the increase of atmospheric water vapor content; the spectral distribution of the solar energy, during the year could be considered constant for a certain air mass; the solar intensity, reaching the earth's surface, in the spectral region below =425 m can give an indication to the fluctuations in the reduced thickness of the ozone layer.

     

Propagation of surface waves on a nonhomogenous spherically aeolotropic shell

Summary This paper studies the propagation of Surface Waves on a spherically aeolotropic shell surrounded by vacuum. The elastic constantsc _ij and density of the material of the shell are assumed to be of the form _ij r ^l and _o r ^m respectively, where _{ij o} are constants andl, m are any integers.     

Rayleigh wave dispersion equation for a layered spherical earth with exponential function solutions in each shell

We consider the second-order differential equations ofP-SV motion in an isotropic elastic medium with spherical coordinates. We assume that in the medium Lamé's parameters , r ^p and compressional and shear-wave velocities , r, wherer is radial distance. With this regular heterogeneity both the radial functions appearing in displacement components satisfy a fourth-order differential equation which provides solutions in terms of exponential functions. We then consider a layered spherical earth in which each layer has heterogeneity as specified above. The dispersion equation of the Rayleigh wave is obtained using the Thomson-Haskel method. Due to exponential function solutions in each layer, the dispersion equation has similar simplicity, as in a flat-layered earth. The dispersion equation is further simplified, whenp=–2. We obtain numerical results which agree with results obtained by other methods.     

Generalized thermoelastic longitudinal waves in unbounded medium

Summary In this paper the generalized thermoelastic longitudinal waves and the temperature field set up due to coupling of the displacement and the temperature fields, with heat wave travelling with certain finite velocity, in an unbounded medium are studied. The thermoelastic displacement potential and the temperature field at any point are obtained in terms of the surface integrals involving the potential, the temperature and their normal derivatives.Notation x _i the cartesian coordinate system,i=1,2,3 - n (u _i) the displacement vector - (/x _i) the del operator - _t /t the derivative with respect to time - T _o the temperature corresponding to the natural stat of zero stress and strain - T Absolute temperature - c _e the specific heat - , Lamé's constants - ₀ the density - coefficient of linear thermal expansion - K thermal conductivity coefficient - _kk u - ₀ the relaxation time     

Meridional Turner angles and density compensation in the upper ocean

The World Ocean Atlas 1998 is used to determine the global field of the meridional density ratio R ^hy =T/S, where temperature and salinity changes T and S are evaluated along meridians, in and below the mixed layer. The focus of the analysis is the identification of regions where the R ^hy field matches the values R =2 sometimes suggested as the commonly perceived state of the ocean and R =1, the condition of density compensation. Results are presented through fields of the meridional Turner angle Tu ^hy =arctan(R ^hy ) and through histograms of Tu ^hy for the Pacific, Atlantic and Indian Oceans at the ocean surface and at 300 m depth. At the 300-m depth level, which in the subtropics is representative of conditions in the permanent thermocline, the most frequently encountered values of the meridional density ratio are R ^hy =3.2 in the North and South Pacific, R ^hy =2.0 in the South Atlantic and Indian and R ^hy =1.6 in the North Atlantic Ocean. Conditions in the mixed layer are more variable and show seasonal differences, but R ^hy =2.0 occurs prominently in all ocean regions during winter and in all regions but the Atlantic during summer. Summer values for the Atlantic Ocean are R ^hy =3.2 in the Northern Hemisphere and R ^hy =2.4 in the Southern Hemisphere. Detailed analysis of R ^hy across the Subtropical Front (STF) confirms the most frequently observed values but shows zonal variation along the front in some oceans. Nearly complete density compensation (R ^hy =1) in the mixed layer is encountered in the STF of the eastern North Pacific, the eastern South Pacific and the eastern Indian Ocean. The eastern Indian Ocean south of Australia is also the only region where complete density compensation in the STF occurs below the mixed layer.Responsible Editor: Neville Smith AcknowledgementWe thank Dan Rudnick for helpful comments and discussion during the preparation of this paper.     

Cyclic variations of the initial height of the sporadicE-layer at middle geographical latitudes

Summary The variations of the initialh E _s height are investigated in the solar cycle 1957–1968, deriving the regressive dependency:h E _s =121.4–6·10^–2 R referring to the median monthly values at a solar zenith angle =75°. The similar variations ofh E _s (R) during the day and night are interpreted as a domination of the sporadic layer formation from a redistribution of the day-time ionization and secondary participation of nightly ionizing sources. The analogous cyclich E _s andh E variations confirm this conclusion while the seasonal variations in the state of the sporadic layer show outlined dynamical effects. The comparatively not big cyclic variation in the spatial state of theE-region are considered to confirm the predominating ionizing action of the ultraviolet range (933–1038 Å) in the lower part of theE-region, while the soft X-radiation influences mainly the near maximum part of this region.     

La risoluzione dell'equazione di Laplace nel campo esterno a uno sferoide

Riassunto L'Autore dimostra che, nel sistema di coordinate polari , , , si possono determinare un numeros di funzioni della sola variabile :Q ₁,Q ₃, ....Q _2s–1 tali che la sommatoria delleQ _2i–1/^2i–1 rappresenti il potenzialeV di un geoide di rotazione. La condizione di armonicità determina ciascunaQ (che si riduce a un polinomio nelle potenze di sen ) a meno di una costante arbitraria; si dispone pertanto dis costanti che servono per soddisfare la natura dellaV sulla superficie del geoide. Come esempio l'Autore ha determinato la gravità sul geoide sferico, confermando i risultati delSomigliana, e su uno sferoide generico dove ha ritrovato la relazione diClairaut.

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Summary The Author proofs that, in the system of polar coordinates , , , it is possible to determine a numbers of functions only of the variable :Q ₁,Q ₃ ....Q _2s–1 in such a way as to make the summatory of theQ _2i–1/^2i–1 represent the potential function of a rotational geoid. The condition of harmonicity determines, saving an arbitrary constant, each of theQ which is reduced to a polynom developed by the sin powers; therefore one disposes of a number of constants to make use for satisfing theV on the geoid. To illustrate his theory the Author determines the gravity on the spherical geoid, thus confirmingSomigliana's formulas and on a spheroidal on which he pointed outClairaut's relations.

     

Planar charged-particle trajectories in multipole magnetic fields

This paper provides a complete generalization of the classic result that the radius of curvature () of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particles equatorial distance () from the dipole (i.e. ³). Comparable results are derived for the radii of curvature of all possible planar chargedparticle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ⁿ⁼². This result reduces to the classic one in the case of a zonal magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m()–_n^m)] = 0, where _n^m = (1/m) arctan (h_n^m/g_n^m); g_n^m and h_n^m denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis ( = O,) of a tesseral magnetic multipole is a magnetic neutral line if m > I. A further 2_m(n – m) neutral lines exist at the intersections of the 2_m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation P_n^m(cos ) = 0 in the range 0 < 9 < , where P_n^m(cos ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ⁿ⁼² and inversely proportional to cos[m(–)]. Since this last expression vanishes at the 2m equatorial neutral ines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rⁿ⁺², where r denotes radial distance from the multiple, and inversely proportional to P_n^m(cos )/sin . Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral ine (m > 1) or any one of the 2m(n – m) neutral ines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are pressented for low-degree (n 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the non-adiabatic scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.Nomenclature a _i inner sphere radius - a _d outer sphere radius - A ₁ defined by equation (5) - A ₂ defined by equation (6) - B ₁ defined by equation (7) - B ₂ defined by equation (8) - c dimensional factor for bispherical coordinates - C constant in equation (4) - d narrowest distance between the two eccentric spheres - E ² operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates - G modified vorticity, defined in equation (22) - G ^* non-dimensional modified vorticity, defined in equation (28) - h metric coefficient of bispherical coordinate system, defined in equation (18) - k _w thermal conductivity of water - K ₁ defined by equation (9) - K ₂ defined by equation (10) - N _Re Reynolds number=2a _dU/gn - N _Pe,h Peclet number=2a _dU/ - n integer counter - q heat flux - r radius - r ^* non-dimensional radius=r/a _d - S surface area - t time - t ^* non-dimensional time=t/a _d ² - T temperature - T _o temperature at inner sphere surface - T _a temperature at outer sphere surface - T ^* non-dimensional temperature;=(T–T _o)/(T_a–T_o) - u velocity - u _r radial velocity in spherical coordinates - u angular velocity in spherical coordinates - u radial velocity in bispherical coordinates - u angular velocity in bispherical coordinates - U free stream velocity - u _r ^* =u _r/U - u ^* =u /U - u ^* =u /U - u ^* =u /U Greek symbols a ₁ small displacement - vorticity, defined in equation (17) - ^* non-dimensional vorticity, defined in equation (27) - radial bispherical coordinates - _o bispherical coordinate of inner sphere - _a bispherical coordinate of outer sphere - angular coordinate in spherical coordinates - thermal diffusivity - _w thermal diffusivity of water - kinematic viscosity - angular bispherical coordinate - spherical coordinate - streamfunction - non-dimensional streamfunction for spherical coordinates, = /(U a _d ² ) - ^* non-dimensional streamfunction for bispherical coordinates, defined in equation (26)