Hydromagnetic free convection and mass transfer flow of non-Newtonian fluid through a porous medium bounded by an infinite vertical limiting surface with constant suction

An analysis of a two-dimensional steady-free convection and mass transfer flow of an incompressible, viscous, and electrically conductive non-Newtonian fluid through a porous medium bounded by a vertical infinite limiting surface (plane wall) has been presented in the presence of a transverse magnetic field. Approximate solutions to the coupled nonlinear equations governing the flow are derived and expression for the velocity, temperature, concentration, the rate of heat transfer, and the skin-friction are derived. Effects of Gr (Grashof number), Gm (modified Grashof number),M ^* (non-Newtonian parameter),N (magnetic parameter), and permeabilityK of the porous medium on the velocity, the skin-friction and the rate of heat transfer are discussed when the surface is subjected to a constant suction velocity.     

Comparative effect of viscous dissipation on heat transfer in mixed convection flow: UWT and UHF cases

The effect of viscous dissipation on mixed convection flow about a rotating sphere with a prescribed uniform surface heat flux (UHF) has been investigated analytically. Merk's type of series expansions is used to obtain the heat transfer rate and the skin-friction coefficients. The results are presented for Pr=1;B=0, 1; ^*0, 0.5, 1 and various values of the dissipation parameter Ec^* at various angular positions. As in the case of uniform wall temperature (UWT), the heat transfer rate decreases with viscous dissipation. It is also observed that for an equivalence viscous dissipation effect, heating by UHF yields larger Nusselt number than heating by UWT.     

Heat and mass transfer of a viscous heat generating fluid with Hall currents

A study of natural convection in hydrodynamic flows of a viscous heat generating fluid in the presence of Hall currents and variable suction has been carried out. The governing equations for the magnetohydrodynamic fluid flow and heat transfer are solved. The effects of Hall currentm and heat source parameter on the velocity and temperature distributions are discussed.     

Effects of the external temperature-dependent heat sources on the unsteady free convective and mass-transfer hydromagnetic flow in the stokes problem

There have been considered the effects of external temperature-dependent heat sources and mass transfer on free convection flow of an electrically conducting viscous fluid past an impulsively starting infinite vertical limited surface in presence of a transverse magnetic field as considered. Solutions for the velocity and skin-friction, in closed form are obtained by using the Laplace transform technique and the results obtained for various values of the parametersS _c (Schmidt number),M (Hartmann number), andS (Strength a Source or Sink) are given in graphical form. The paper is concluded with a discussion on the obtained results.     

Magnetic field transfer by two-dimensional convection and solar ‘semi-dynamo’

A comparison of the equations for the magnetic field transfer and for the heat transfer by two-dimensional turbulent convection of a conducting compressible medium shows the magnetic field to be transported as a scalar admixture provided it is parallel to the convective rolls. At high magnetic Reynolds numbers the field strength in a convective zone varies proportionally to the density of the medium.A study of the distribution and amplification of the poloidal field in the two-dimensional convection zone of the Sun lying under the supergranulation, together with the processes of field pumping and amplification in other zones, reveals the importance of considering generation mechanisms of thesemi-dynamo type where the amplifying field is excited independently by weak e.m.f.'s of non-electric origin with no feedback which would otherwise produce MHD self-excitation of the field.An illustrative calculation of the solar poloidal field maintained by a weak Coriolis e.m.f. acting in a thin external layer of the convective envelope yields for the general near-polar field, if one somehow takes into account (1) field pumping by three-dimensional supergranulation, (2) field transfer and amplification by two-dimensional convection, and (3) ohmic diffusion of the field into a stable core, a value of the order of 10^–1 gauss.     

Onset of Convection,Heat Flow and Thickness of the Europa‘s ice Shell

The observational evidence given by Galileo spacecraft about Europa supports an icy rigid layer of several kilometers over another ductile layer of ice in convection, which floats over an internal ocean of liquid water. Before the onset of convection, heat is transmitted into the crust by conduction. The heat flow analysis in the potentially convective layer gives values higher than those obtained previously by tidal dissipation models, and suggests that the ice may be limited to a thin layer of ∼4 km total thickness. This revised version was published online in July 2006 with corrections to the Cover Date.     

Free convection effects on the oscillatory flow in the Stokes's problem past an infinite porous vertical limiting surface with constant suction,II

In this work we present the two-dimensional free convection flow of an incompressible viscous fluid past an infinite vertical limiting surface (porous wall) for the Stokes's problem when the fluid is subjected to a constant suction velocity. The flow is normal to the porous wall and the free stream oscillates about a mean value. As the mean steady flow has been presented in Part I, only the solutions for the transient velocity profiles, transient temperature profiles, the amplitude and the phase of the skin friction and the rate of heat transfer are presented in this work. As in the case of mean steady flow, the influence of the Grashof numberG and Eckert numberE on the unsteady flow field is discussed for air (P=0.71) and water (P=7) and for the cases of externally heating and cooling the porous limiting surface by free convection currents.     

Heat flow, lenticulae spacing, and possibility of convection in the ice shell of europa

Two opposing models to explain the geological features observed on Europa’s surface have been proposed. The thin-shell model states that the ice shell is only a few kilometers thick, transfers heat by conduction only, and can become locally thinner until it exposes an underlying ocean on the satellite’s surface. According to the thick-shell model, the ice shell may be several tens of kilometers thick and have a lower convective layer, above which there is a cold stagnant lid that dissipates heat by conduction. Whichever the case, from magnetic data there is strong support for the presence of a layer of salty liquid water under the ice. The present study was performed to examine whether the possibility of convection is theoretically consistent with surface heat flows of ∼100-200 mW m⁻², deduced from a thin brittle lithosphere, and with the typical spacing of 15-23 km proposed for the features usually known as lenticulae. It was obtained that under Europa’s ice shell conditions convection could occur and also account for high heat flows due to tidal heating of the convective (nearly isothermal) interior, but only if the dominant water ice rheology is superplastic flow (with activation energy of 49 kJ mol⁻¹; this is the rheology thought dominant in the warm interior of the ice shell). In this case the ice shell would be ∼15-50 km thick. Furthermore, in this scenario explaining the origin of the lenticulae related to convective processes requires ice grain size close to 1 mm and ice thickness around 15-20 km.     

Solar-wind model including the effects of rotation,magnetic fields,and anisotropic heat conduction

A time-independent solar-wind model is considered in the case of spherical symmetry and of radial magnetic field at the sun's surface. The energy equation includes besides the usual terms also the heat conduction and magnetic-energy convection (Poynting vector) terms. The dependence of the thermal conductivity on the magnetic field is taken into account. Numerical integrations of the basic equations were performed under the following assumptions: (i) close to the sun the magnetic field is the dominant azimuthal term and solid-body rotation is enforced; (ii) beyond the Alfvénic point the terms quadratic inB are neglected. The model leads to azimuthal velocity at earth between 0.6 and 2.7 km/sec, to radial velocity at earth between 350 and 500 km/sec, and to angular momentum loss of 5×10¹⁸ cm²/sec per unit mass of gas leaving the solar equator. The dependence of the solutions on the reduction of the effective thermal conductivity caused by the micro-structures in the solar wind suggests that the conditions at earth may be largely determined by a transition region in the solar wind, in which the conduction régime changes into an almost adiabatic flow.Presented at the Trieste Colloquium on Mass Loss from Stars, September 12–16, 1968.     

Thermal-orbital evolution of Io and Europa

Coupled thermal-orbital evolution models of Europa and Io are presented. It is assumed that Io, Europa, and Ganymede evolve in the Laplace resonance and that tidal dissipation of orbital energy is an internal heat source for both Io and Europa. While dissipation in Io occurs in the mantle as in the mantle dissipation model of Segatz et al. (1988, Icarus 75, 187), two models for Europa are considered. In the first model dissipation occurs in the silicate mantle while in the second model dissipation occurs in the ice shell. In the latter model, ice shell melting and variations of the shell thickness above an ocean are explicitly included. The rheology of both the ice and the rock is cast in terms of a viscoelastic Maxwell rheology with viscosity and shear modulus depending on the average temperature of the dissipating layer. Heat transfer by convection is calculated using a parameterization for strongly temperature-dependent viscosity convection. Both models are consistent with the present orbital elements of Io, Europa, and Ganymede. It is shown that there may be phases of quasi-steady evolution with large or small dissipation rates (in comparison with radiogenic heating), phases with runaway heating or cooling and oscillatory phases during which the eccentricity and the tidal heating rate will oscillate. Europa's ice thickness varies between roughly 3 and 70 km (dissipation in the silicate layer) or 10 and 60 km (dissipation in the ice layer), suggesting that Europa's ocean existed for geological timescales. The variation in ice thickness, including both convective and purely conductive phases, may be reflected in the formation of different geological surface features on Europa. Both models suggest that at present Europa's ice thickness is several tens of km thick and is increasing, while the eccentricity decreases, implying that the satellites evolve out of resonance. Including lithospheric growth in the models makes it impossible to match the high heat flux constraint for Io. Other heat transfer processes than conduction through the lithosphere must be important for the present Io.     

Tidal Heating and Convection in the Medium Sized Icy Satellites

Surface features of some icy satellites indicate that the satellites are modified due to the internally driven tectonic activity. Convection could be one of the processes responsible for the formation of the surface features. The potential sources of energy inside the satellites are discussed. For the medium sized icy satellites the radiogenic and tidal heat sources seem to be of primary importance. To investigate the problem, a 3D model of convection is developed based on the Navier–Stokes equation, the equation of thermal conductivity, the equation of continuity, and the equation of state. The model includes both the tidal and the radiogenic heating. It can be applied to the homogeneous, non-differentiated medium sized satellites. The 3D formulae for tidal heat generation and stress tensor based on the results of Peale and Cassen (1978) and others are applied. A new dimensionless number C _t is introduced. It measures the relative importance of tidal and radiogenic heat sources. The systematic investigation of the steady-state convection is performed for different values of the Rayleigh number and for 0C _t 1. The results indicate that the convection pattern for low Rayleigh number driven by tidal heating in the medium sized icy satellites consists of two cells. The pattern of tidally driven convection is oriented, that is, the regions of downward motion are situated at the center of the near- and of the far-side of the satellite.     

Hall effects on heat and mass transfer flow through porous medium

Effects of Hall current on free convection and mass transfer flow through a porous medium bounded by a vertical surface has been analysed. The problem is solved analytically. The velocity profiles are shown on graphs. Effects ofm (Hall parameter).K ^* (permeability parameter), and Sc (Schmidt number) on velocity are discussed.     

Free convection effects on the oscillatory flow in the Stokes's problem past an infinite porous vertical limiting surface with constant suction — I

An analysis of Rayleigh's problem (also Stokes's problem) for the flow of a viscous fluid (e.g. of a stellar atmosphere) past an impulsively started infinite, vertical porous limiting surface (e.g. of a star) with constant suction, when the free stream velocity oscillates in time about a constant mean, has been carried out. On solving the coupled non-linear equations in approximate way, expressions for the mean velocity, the mean temperature, the mean skin-friction and the mean rate of heat transfer, expressed in terms of Nusselt number, are obtained. The effects of Grashof numberG, Eckert numberE and Prandtl numberP, on these quantities, is discussed for the cases of an externally heating and cooling of the limiting surface, by the free convection currents, and the variations of them are shown graphically.     

Radiative and Free Convective Effects of a MHD Flow Through a Porous Medium Between Infinite Parallel Plates with Time-dependent Suction

The problem of magnetohydrodynamic free-convection flow, with radiative heat transfer in porour media subject to time-dependent suction of an incompressible and optically transparent medium has been solved making fairly realistic assumption. For a small-time-dependent perturbation of the fluid velocity and temperatures, the nonlinear problem is tackled by asymptotic approximation, giving solutions for steady-flow on which a first-order transient component is superimposed. The effect of heat radiation and free convection on the flow of the fluid is demonstrated analytically and quantitatively. The flow field is seen to be affected mainly by radiation and convection parameters, in addition to the porosity and magnetic factors.     

Thermal Equilibrium States of Europa's Ice Shell: Implications for Internal Ocean Thickness and Surface Heat Flow

Ice-shell thickness and ocean depth are calculated for steady state models of tidal dissipation in Europa's ice shell using the present-day values of the orbital elements. The tidal dissipation rate is obtained using a viscoelastic Maxwell rheology for the ice, the viscosity of which has been varied over a wide range, and is found to strongly increase if an (inviscid) internal ocean is present. To determine steady state values, the tidal dissipation rate is equated to the heat-transfer rate through the ice shell calculated from a parameterized model of convective heat transfer or from a thermal conduction model, if the ice layer is found to be stable against convection. Although high dissipation rates and heat fluxes of up to 300 mWm⁻² are, in principle, possible for Europa, these values are unrealistic because the states for which they are obtained are thermodynamically unstable. Equilibrium models have surface heat flows around 20 mWm⁻² and ice-layer thicknesses around 30 km, which is significantly less than the total thickness of the H₂O-layer. These results support models of Europa with ice shells a few tens of kilometers thick and around 100-km-thick subsurface oceans.     

Pulsational instability of yellow hypergiants

Instability of population I (X = 0.7, Z = 0.02) massive stars against radial oscillations during the post-main-sequence gravitational contraction of the helium core is investigated. Initial stellar masses are in the range 65M _⊙ ≤ M _ZAMS ≤ 90M _⊙. In hydrodynamic computations of self-exciting stellar oscillations we assumed that energy transfer in the envelope of the pulsating star is due to radiative heat conduction and convection. The convective heat transfer was treated in the framework of the theory of time-dependent turbulent convection. During evolutionary expansion of outer layers after hydrogen exhaustion in the stellar core the star is shown to be unstable against radial oscillations while its effective temperature is T _eff > 6700 K for M _ZAMS = 65M _⊙ and T _eff > 7200 K for M _ZAMS = 90M _⊙. Pulsational instability is due to the κ-mechanism in helium ionization zones and at lower effective temperature oscillations decay because of significantly increasing convection. The upper limit of the period of radial pulsations on this stage of evolution does not exceed ≈200 day. Radial oscillations of the hypergiant resume during evolutionary contraction of outer layers when the effective temperature is T _eff > 7300 K for M _ZAMS = 65M _⊙ and T _eff > 7600 K for M _ZAMS = 90M _⊙. Initially radial oscillations are due to instability of the first overtone and transition to fundamental mode pulsations takes place at higher effective temperatures (T _eff > 7700 K for M _ZAMS = 65M _⊙ and T _eff > 8200 K for M _ZAMS = 90M _⊙). The upper limit of the period of radial oscillations of evolving blueward yellow hypergiants does not exceed ≈130 day. Thus, yellow hypergiants are stable against radial stellar pulsations during the major part of their evolutionary stage.     

Evaporation and Condensation Processes of Giant Molecular Clouds in a Hot Plasma

2D hydrodynamical simulations are performed to examine the evaporation and condensation processes of giant molecular clouds in the hot phase of the interstellar medium. The evolution of cold and dense clouds (T = 1000 K, n _H = 3 cm^-3,M = 6·10⁴ M_⊙) is calculated in the subsonic stream of a hot tenuous plasma (T = 5 ·10⁶ K, n _H = 6·10^-4cm^-3). Our code includes self-gravity, heating and cooling processes and heat conduction by electrons. The thermal conductivity of a fully ionized hydrogen plasma (κ ∝ T^5/2) is applied as well as a saturated heat flux in regions where the mean free path of the electrons is large compared to the temperature scaleheight. Significant differences occur between simulations with and without heat conduction. In the simulations without heat conduction, the clouds outermost regions is stired up by Kelvin-Helmholtz (KH) instability after only a few dynamical times. This prevents an infiltration of a significant amount of hot gas into the cloud before its destruction. In contrast, models including heat conduction evolve less violently. The boundary of the cloud remains nearly unsusceptible to KH instabilities. In this scenario it is possible to mix the formerly hot streaming gas very effectively with the cloud material. This revised version was published online in July 2006 with corrections to the Cover Date.     

Magnetic convection

In this paper the process of magnetic convection is studied. It is shown that outside of a radius of about 2 × 10⁵ km, magnetic fields in the Sun may be buoyant. Outside this limit strong field regions tend to rise at the expense of weak field regions which tend to sink. Magnetic convection may be important in magnetic stars and even in the solar interior. A recent calculation of the angular velocity of the Sun provides a period of rotation for the solar core of from 0.5 to 5 days. This calculation requires that the magnetic field extract angular momentum from the solar interior. Magnetic convection thus seems to be required, if this calculation is correct. Furthermore, magnetic convection may transfer heat and thereby possibly change the internal temperature structure of the Sun from what would be expected solely by radiation transfer.     

Free convection in hydromagnetic flows of a viscous heat-generating fluid with wall temperature oscillation and Hall currents

A study is made of the free convection in hydromagnetic flows through a porous medium of a heat generating fluid past an infinite vertical porous plate. A strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an alge to the vertical direction. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved analytically. The influence of Hall currents, the permeabilityK and the inclination upon the velocity field are shown in figures.     

Supersonic convection in stellar interiors

According to the result of the numerical simulation on stellar convective envelopes by the traditional mixing length theory, supersonic convection would occur on the top of the convective regions of the yellow-and-red giants and supergiants. This, however, is not self-consistent with the local convection theory itself. This paper is devoted to analyze in details the origin of supersonic convection, and at the same time to present a plot of the supersonic convection region on the H-R diagram based on the evolutionary track of population I stars. The main results of this paper are as follows: in the process of evolution, (1) no supersonic convection occurs in low-mass stars; (2) for intermediate-mass stars, because of the ionization of hydrogen, the supersonic convection occurs chiefly in the region of 3.6< lgT_e <4.0, tilting from the lower left to the upper right on the H-R diagram and extending to the region of red giants for relatively massive stars; (3) for massive stars, the supersonic convection occurs after they deviate from ZAMS, the greater the stellar mass is, the earlier the supersonic convection emerges. For blue supergiants, the supersonic convection occurs at the absorbtion peak (lg T ∼5.3) of Fe and in the secondary ionization region of helium, but for the red- and yellow-giants and supergiants, it occurs in the ionization region of hydrogen.