Effects of mass transfer on the unsteady free convection flow past an accelerated vertical porous plate with variable suction

An exact analysis of the effects of mass transfer on the flow of a viscous incompressible fluid past an uniformly accelerated vertical porous and non-porous plate has been presented on taking into account the free convection currents. The results are discussed with the effects of the Grashof number Gr, the modified Grashof number Sc, the Schmidt number Sc, and the suction parametera for Pr (the Prandtl number)=0.71 representating air at 20°C.Nomenclature a suction parameter - C species concentration - C species concentration at the free stream - g acceleration due gravity - Gc modified Grashof number (vg^*(C –C )/U ₀ ³ ) - Pr Prandtl number (C _p/K) - T temperature of the fluid near the plate - T dimensionless temperature near the plate ((T-T )/(T -T )) - U(t) dimensionless velocity of the plate (U/U ₀) - v normal velocity component - v ₀ suction/injection velocity - x, y coordinate along and normal to the plate - v kinematic viscosity (/gr) - C _p specific heat at constant pressure - C _w species concentration at the plate - C non-dimensional species concentration ((C-C )/(C _w -C )) - Gr Grashof number (g(T _w -T )/U ₀ ³ ) - D chemical molecular diffusivity - K thermal conductivity - Sc Schmidt number (/D) - T _w temperature of the plate - T free stream temperature - t time variable - t dimensionless time (tU ₀ ² /) - U ₀ reference velocity - u velocity of the fluid near the plate - u non-dimensional velocity (u/U ₀) - v dimensionless velocity (v/U ₀) - v ₀ non-dimensionalv ₀ (v ₀ /U₀)=–at^–1/2 - y dimensionless ordinate (yU ₀/) - density of the fluid - coefficient of viscosity     

On the relative roles of unipolar and mixed-polarity fields

Away from plages, solar magnetic fields may be classified as unipolar or as of mixed polarity, though the distinction is strictly arbitrary. The dividing line used here is 0.4 ¦B _minor/B _major¦ 1, where average fields of major and minor polarities are measured over large areas. Some of their statistical properties and cyclical variations are detailed. In unipolar regions, 3 B _major 50 G, B _minor 0.1 B _major, and ¦B¦ 1.1 B _major. In regions of mixed polarity, 3.5 ¦B¦ 10 G.Below latitudes of ± 60°, mixed polarities predominate for about 5 yr around sunspot minimum. For several years around sunspot maximum, unipolar fields fill the 20°–40° zone completely, and occupy about 75% of the 0°–20° and 40°–60° zones.The polar unipolar fields are weak on the whole (Bmajor 4 G for 6 typical days in 1976–79), with small regions having stronger fields at times, probably not exceeding B _major = 10 G. Again B _minor 0.1 B _major. There is no direct way at present of measuring properties of polar mixed fields, such as may occur around sunspot maximum, but by inference ¦B¦ 2 to 5 G.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

A momentum distribution of high energy particles around Crab pulsar

We determine the momentum distribution of the relativistic particles near the Crab pulsar from the observed X- and -ray spectra (10³10⁹ eV), provided that the curvature radiation is responsible for it. The power law spectrum for the relativistic electrons,f() ^–5, reproduces a close fit to the observed high-energy photon spectrum. The theoretically determined upper limit to the momentum (due to radiation damping), _M 8×10⁶, corresponds to the upper cut-off energy of the -ray spectrum, 10⁹ eV. The lower limit to the momentum, _m 1.8×10⁵, is chosen such that flattening of the X-ray spectrum below 10 keV is simulated. The number density of these electrons is found to be much higher than the Goldreich-Julian density. We also discuss pulse shape and polarization of high-energy photons. The extremely high density of particles and the steep momentum spectrum are difficult to understand. This may imply that another, more efficient, mechanism is in operation.     

Mean free paths and diffusion coefficients for energetic protons at small heliodistances calculated using Helios 1 and 2 data

Pitch angle scattering of energetic particles (100 MeV) in the interplanetary medium are studied using Helios 1 and 2 magnetometer and plasma data during 1976 near the minimum of solar activity. An IMF configuration was used in the computer experiments which allowed the pitch angle diffusion coefficient, D and hence the parallel mean free path, to be determined. The radial mean free path was found to vary as _r r ^-0.9 between 0.4 and 1 AU, but between 0.3 and 0.4 AU it decreases significantly. To reconcile our value of _r at 1 AU, lying between 0.01 and 0.02 AU, with the average prompt solar proton event profile, an increasing value of _r at lower radial distances would be required.     

Слектры релятивистских эиектронов и изиучения, формируемьЫе комптоновским рассеянием лри отличном от нуля потоке

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Effects of Hall current on hydromagnetic free-convective flow through a porous medium

An analysis of the effects of Hall current on hydromagnetic free-convective flow through a porous medium bounded by a vertical plate is theoretically investigated when a strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an angle to the vertical direction. The influence of Hall currents on the flow is studied for various values of .Nomenclature c _p specific heat at constant pressure - e electrical charge - E Eckert number - E electrical field intensity - g acceleration due to gravity - G Grashof number - H ₀ applied magnetic field - H magnetic field intensity - (j _x , j _y , j _z ) components of current densityJ - J current density - K permeability of porous medium - M magnetic parameter - m Hall parameter - n _e electron number density - P Prandtl number - q velocity vector - (T, T _w , T ) temperature - t time - (u, v, w) components of the velocity vectorq - U ₀ uniform velocity - v ₀ suction velocity - (x, y, z) Cartesian coordinates Greek Symbols angle - coefficient of volume expansion - _e cyclotron frequency - frequency - dimensionless temperature - thermal conductivity - coefficient of viscosity - magnetic permeability - kinematic viscosity - mass density of fluid - _e charge density - electrical conductivity - _e electron collision time     

(Super) cosmological solution for the Kasner model

In the theory of supergravity (N=1), the supersymmetric version of general relativity, and for the Kasner cosmological model (Bianchi type I) we find a non-trivial solution (for the metric and spinor-vector) under the most simple assumption =₁₁ + ₂₂; ₁₂+₂₁=0 and for a special choosed gaugeN=1,N ^j=0, ₀=0. This method could be also applied to other cosmological metrics and extended to enlarged Grassmann basis.O. Obregón was partially supported by the Alexander von Humboldt Stiftung.     

Unsteady free-convective flow and mass transfer in a rotating porous medium with Hall current,I

The Hall effect on the unsteady hydromagnetic free-convection resulting from the combined effects of thermal and mass diffusion of an electrical-conducting liquid through a porous medium past an infinite vertical porous plate in a rotating system have been analysed. The expressions for the mean velocity, mean skin friction, and mean rate of heat transfer on the plate are derived. The effects of magnetic parameterM, Hall parameterm, Ekman numberE, and permeability parameterK ^* on the flow field are discussed with the help of graphs and tables.Nomenclature C _p specific heat at constant pressure - C the species concentration inside the boundary layer - C _w the species concentration at porous plate - C the species concentration of the fluid at infinite - C dimensionless species concentration - D chemical molecular diffusivity - E Ekman number - Ec Eckert number - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - H ₀ applied magnetic field - (J _x, J_y, J_z) components of current density - M magnetic parameter - m Hall parameter - P Prandtl number - q _m mean rate of heat transfer - Sc Schmidt number - t time - t dimensionless time - T temperature of fluid - T _w temperature of the plate - T temperature of fluid at infinite - T dimensionless temperature - (u, v, w) components of the velocityq - w ₀ suction velocity - (x, y, z) Cartesian coordinates - z dimensionless coordinate normal to the plate Greek symbols coefficient of volume expansion - * coefficient of thermal expansion with concentration - frequency - dimensionless frequency - k thermal conductivity - K ^* permeability parameter - dinematic viscosity - density of the fluid in the boundary layer - coefficient of viscosity - _e magnetic permeability - angular velocity - electrical conductivity of the fluid - _m mean skin friction - _mn mean skin friction in the direction ofx - _mv mean skin friction in the direction ofy     

Theory of the Trojan asteroids,IV

In a previous publication (1977) the author has constructed a family () of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies ₁ and ₂. In terms of thecritical masses m_j(₂), or the associatedcritical energies _j ² (m), is the intersection of the intervals ofshallow resonance, of the form. Inasmuch as the intervals |₂– _j ² |<_j ofdeep resonance aredisjoint, it follows that (1) the disjointed family () embraces the tadpole branch, 0²1, lying in: and (2) despite the clustering of _j ² (m) atj=, the family () includes, for ²=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL ₃.In a similar manner, the family () can be extended to the horseshoe branch 1<_{2 2} ² .     

Simultaneous measurements of Ca+ K,Ca+ H,Hα, Hβ, and He D3 emission in prominences

Simultaneously measured line intensities of Ca⁺ K and H, H, H and He D₃ are presented and compared with recent model calculations by Heasley and Milkey (1978). The model calculations are compatible with the observations. There is indication that the observed branching of the relation E(Ca⁺ K and H) E(H) depends on the widths of the Ca⁺-lines.     

Apparent radii and other parameters for 416 B5 V-F5 V stars of the catalogue of the geneva observatory

Apparent radius, visual brightness, effective temperature and absolute radius for 416 B5 v-F5 v stars of the catalogue of the Geneva Observatory (Rufener, 1976) have been determined.Twenty-eight stars, anomalous in log versus (m _v)₀ diagrams, have been singled out. A good correlation for seven stars, in common with the list of Hanbury Brownet al. (1974), has been found. Similar parameters determined for 279 B5 v-F5 v stars of two preceding papers (Fracassiniet al., 1973, 1975) have allowed us to determine the averaged diagrams logq _v/q, logR/R and logT _e versus (B-V)₀ for 695 B5 v-F5 v stars.Moreover, in the present paper a good correlation logq _v/q versus logR/R and careful relation M _v=–7.40logR/R +3.31 for B5 v-F5 v stars have been determined. Plain correlations between logR/R and blanketing parameterm ₂ for some spectral types seem to point out that there arereal differences in the absolute radii of stars of thesame spectral type, in agreement with recent researches on the HR diagram (Houck and Fesen, 1978).Systematic differences between double (spectroscopic and visual) and single stars are found. In particular, the averaged relation m ₂ versus logR/R shows that A2 v-F5 v double stars may have a higher metallicity indexm ₂ and smaller absolute radii than single stars. Finally, the diagram logv sini versus logR/R confirms some properties of binary systems found by other researchers (Huang, 1966; Plavec, 1970; Levato, 1974; Kitamura and Kondo, 1978).Thesis for the degree in Applied Physics.     

Multiple expansion of the tidal potential

The Earth tidal deformation causes an additional gravitational potential. Its effect on the Moon orbital motion has been studied by several authors.In this contribution, we develop this additional potential without specifying the inertial frame chosen.For this purpose, we use the properties of the representation of rotation groups in 3 dimensions space. We finally obtain the interaction potential between the distorted Earth and the Moon which is a necessary preliminary to the study of the evolution of the Earth-Moon system.Nomenclature T.R.O Tide raising object - (, , ) Spherical coordinates of the T.R.O. - (J, _E ) Earth spin axis orientation. _E is the longitude of the ascending node of Earth's equator on thexy-plane - (a ,I ,e , , ,M ) Elliptics elements of the T.R.O     

Optically thick accretion onto black holes

Spherically symmetric, steady-state, optically thick accretion onto a nonrotating black hole with the mass of is studied. The gas accreting onto the black hole is assumed to be a fully ionized hydrogen plasma withn ₀=10⁸ cm^–3 andT ₀=10⁴ K far from the black hole, and a new approximate expression for the Eddington factor is introduced. The luminosity is estimated to beL=1.875×10³³ erg s^–1, which primarily arises from the optical surface (1) ofT10⁴ K. The accretion flow is characterized by 1 and (v/c)10. In the optically thin region, the flow remains isothermal, and the increase of temperature occurs at 1. The radiative equilibrium is strictly realized at (v/c)10.     

Statistical survey of planetary nebulae: Distances,masses, and distribution in the galaxy

On the basis of empirical (D)-dependency at the frequency of 5 GHz constructed using 15 planetary nebulae with the independently measured distances (10^–171×10^–20 W m^–2 Hz^–1 ster^–1), we evaluated distances of 335 objects. Independent evidence of the correctness of the accepted scale are given. Then(D)-dependency is constructed and it is shown that atD<0.08 pc the mean electron density is higher than the one determined by the Seaton method. We showed that the filling factor diminishes with the increase of the PN diameter (1 atD0.08 pc and 0.2 atD0.4 pc). the ionized mass of 33 PNs is determined. With the diameter increase the ionized mass grows and atD0.4 pc reaches the valueM0.07M . We used the new distance scale when investigating the space distribution of PNs. The mean scale height =130±15 pc and the mean gradient of the change of surface densitym=0.37, which allowed us to estimate the total number of nebulae in the GalaxyN4×10⁴. We divided the PNs according to their velocities (withV _LSR>35 km s^–1 andV _LSR<35 km s^–1) and permitted us to confirm that the PN belong to different sub-systems of the Galaxy. The estimated local formation rate of PNs [=(4.6±2.2)×10^–12 pc^–3 yr^–1] is a little higher than the one of the white dwarfs. That can be explained by a large number of PNs having binary cores, which used in our sample. The statistical estimation of PN expansion velocity showed thatV _ex increases from 5–7 km s^–1 (atD0.03 pc) to 40–50 km s^–1 (atD0.8 pc).     

Isothermal neutron star cores

By considering the relativistic expression for isothermal NS cores,T·e ^/2 = constant, we have shown that some of the standard equations of state, when applied to NS cores, correspond to constancy of some adiabatic exponents. It has been shown that the equation of state,P=KE, corresponds to ₁ = to ₂ = ₃ 1 +K and the equation of state, dP/dE=K, corresponds to ₃ 1 +K. The conditions under which different equations of state represent isothermal cores have been obtained: For isothermal NS, the local temperatureT, can be expressed in terms of pressureP, energy densityE, and rest mass density . For example: (a)P =KE :T = constant × (E/); (b)P=KE :T = constant × (P/); (c) dP/dE =K :T ^K ; (d) = ₂ :T = constant × (P/E); and (e) = ₃ :T = constant × (P/)^1/2. Equation of state corresponding to = ₂ is obtained as:P=E/ln(K/E) and the equation corresponding to = ₃ comes out as:E=P ln(K/P). Core-envelope models can be developed for these two cases. When core equation corresponding to = ₂ or = ₃ is used in the core, we can ensure the continuity of dP/dE at the core-envelope boundary, along with the continuity ofP, E, , and . The parameters of isothermal NS cores corresponding to the cases = ₂ and = ₃, have been obtained. The maximum mass of these NS cores comes out to be 2.7 .     

Central holes in disks of spiral galaxies

On the basis of the solutions obtained in the previous paper, the changes in the scenario of the standard model of the Big Bang are found. The chaos degree (constrainst on fluctuation spectra) is obtained, which could be still preserved by the initially completely chaotic Universe at the time of light elements nucleosynthesist _es. The time boundaries of hadron and lepton eras and the time the electron neutrinos and neutrons become frozen in reactions of weak interaction may be shifted up to 1.4 times. The corresponding temperatures may shift off from the standard ones 0.88 times if the mean-square level of fluctuations is close to unity. If the density of the energy of fluctuations concentrated in the short-wave region of the spectrum is less than 1.5 ^– , the nucleosynthesis leads to a helium abundance coinciding with the observe one. If at the timet _es the maximum of the spectral density of the energy is in the long-wave region, that is _max ct _es the level of the chaos during the period of nucleosynthesis is restricted to 1.76 (where |C _K |² d³ K,C _K is Fourier component of the amplitude of metric fluctuations). In particular, the protogalactic vortical disturbances with a wide spectrum 4 × 10^{3 -1}( = K/K, = /_crit) are compatible with the observed helium abundance.     

The velocity correlation function in cosmic-ray diffusion theory

The concept of velocity correlation functions is introduced and applied to the calculation of cosmic ray spatial diffusion coefficients. It is assumed that the pitch angle scattering coefficient is already known from some other theory, and is reasonably well-behaved. Previous results for the coefficient for diffusion parallel to the mean field are recovered when the velocity-changing mechanism is artificially restricted to pitch angle scattering. The velocity correlation method is then applied to the more general case where there are fluctuations in the local mean field. It is found that the parallel diffusion coefficient is reduced in proportion to the amplitude of the field fluctuations, and that the ratio of the perpendicular to parallel diffusion coefficients cannot be greater than B _x ² /B ₀ ² . It is shown in the appendix that the Liouville form of the scattering equation implies that the Fokker-Planck coefficients ²/t=2D and /t=D /, and that all higher-order coefficients are identically zero.     

Time-dependent nucleation theory and the formation of interstellar grains

The theory of homogeneous nucleation is discussed and applied to the formation of interstellar grains. Special attention is given to the thermodynamics of very small clusters (N10); it is found that, at least for carbon and metals, the free energies ofN-mers may be conveniently expressed in terms of a parameter _N which is nearly independent ofN. Estimates of _N from the Lothe-Pound theory disagree with the experimental observations forN<10; it is preferable to simply approximate _N 1/2 ( is known from bulk properties). The kinetic process of nucleation and cluster growth is essentially just a function of two dimensionless parameters; /T and . The domain in which nucleation theory should be valid is delineated. Approximate formulae are given for the critical supercooling, the critical cluster size, and the mean final cluster size; more accurate numerical results are presented graphically. Nucleation of grains in planetary nebulae, novae, and red giant outflows is discussed. The origin of the Pt-rich nuggets found in the Allende meteorite is considered; it is shown that the nuggets could have been formed by homogeneous nucleation in the primordial solar nebula, whereas they couldnot have formed in a supernova explosion.Invited contribution to the Proceedings of a Workshop onThermodynamics and Kinetics of Dust Formation in the Space Medium, held at the Lunar and Planetary Institute, Houston, 6–8 September, 1978.     

Free convection and mass transfer effects on the hydro-magnetic oscillatory flow past an infinite vertical porous plate

Two-dimensional unsteady free convection and mass transfer, flow of an incompressible viscous dissipative and electrically conducting fluid, past an infinite, vertical porous plate, is considered, when the flow, is subjected in the action of uniform transverse magnetic field. The magnetic Reynolds number is taken to be small enough so that the induced magnetic field is negligible. The solution of the problem is obtained in the form of power series of Eckert numberE, which is very small for incompressible fluids. Analytical expressions for the velocity field and temperature field are given, as well as for the skin friction and the rate of heat transfer for the case of the mean steady flow and for the unsteady one. The influence of the magnetic parameter,M, modified Grashof numberG _c , Schmidt numberS _c and frequency , on the flow field, is discussed with the help of graphs, when the plate is being cooled, by the free convection currents (G _r ,E>0), or heated (G _r ,E<0). A comparative study with hydrodynamic case (M=0) and the hydromagnetic one (M0) is also made whenever necessary.List of symbols B₀ applied magnetic field - |B| amplitude of the skin friction - C concentration inside the boundary layer - C concentration in the free stream - C _w concentration at the porous plate - C _p specific heat at constant pressure - D diffusion coefficient - E Eckert number - g _x acceleration due to gravity - G _c modified Grashof number - G _r Grashof number - M magnetic parameter - N _u Nusselt number - P Prandtl number - |Q| amplitude of the rate of heat transfer - S _c Schmidt number - T temperature of the fluid - T _w temperature of the plate - T temperature of the fluid in the free stream - T _r ,T _i fluctuating parts of the temperature profile - u, v velocity components in thex, y directions - u dimensionless velocity in thex direction - u ₀ mean steady velocity - u ₁ unsteady part of the velocity - u _r ,u _i fluctuating parts of the velocity profile - U dimensionless free stream volocity - U ₀ mean free stream velocity - v ₀ suction velocity - x, y co-rodinate system Greek Symbols phase angle of the skin-friction - coefficient of volume expansion - _* coefficient of expansion with concentration - phase angle of the rate of heat transfer - dimensionless co-ordinate normal to the plate - dimensionless temperature - ₀ mean steady temperature - ₁ unsteady part of temperature - k thermal conductivity - v kinematic viscocity - density of fluid in the boundary layer - density of fluid in the free stream - electrical conductivity of the fluid - skin friction - ₀ mean skin friction - frequency - dimensionless frequency     

Noise-Storm Continua: Power Estimates for Electron Acceleration

We use a generic stochastic acceleration formalism to examine the power L_in (erg s^-1) input to non-thermal electrons that cause noise-storm continuum emission. The analytical approach includes the derivation of the Greens function for a general second-order Fermi process, and its application to obtain the particular solution for the non-thermal electron distribution resulting from the acceleration of a Maxwellian source in the corona. We compare L_in with the power L_out observed in noise-storm radiation. Using typical values for the various parameters, we find that L_in 10²³–10²⁶ erg s^-1, yielding an efficiency estimate L_out/L_in in the range 10^-10 10^-6 for this non-thermal acceleration/radiation process. These results reflect the efficiency of the overall process, starting from electron acceleration and culminating in the observed noise-storm emission.