On the numerical solution of a rising sphere in a Newtonian fluid with temperature-dependent viscosity

In this work, we present some numerical results about the problem of a rising hot solid sphere immersed in a Newtonian fluid which viscosity depends on the temperature. The model formulated to solve the problem considers two dimensionless parameters: The Peclet number, Pe and a parameter related with the viscosity, e. Small and large variations on e lead to interesting results segregated into two regimes which exhibit an asymptotic structure.To carry out the computations to solve the proposed model, the element finite method was used along with a non-slip boundary condition for the contact surface between the sphere and the fluid and the results obtained were compared to those shown recently in papers related wherein contact surface has a slip-boundary condition prescribed.En este trabajo, presentamos algunos resultados numéricos sobre el problema de una esfera solida caliente ascendente que se encuentra dentro de un fluído newtoniano cuya viscosidad depende de la temperatura. El modelo formulado para resolver el problema considera dos parámetros adimensionales: El número de Peclet, Pe y un parámetro relacionado con la viscosidad, e. Las pequeñas y grandes variaciones sobre e conducen a interesantesresultados que son divididas en dos regímenes los cuales presentan una estructura asintótica. Para llevar a cabo los cálculos a fin de resolver el modelo propuesto, el método de elemento finito fue usado junto con una condición de frontera no-slip para la superficie de contacto entre la esfera y el fluído y los resultados fueron comparados a aquellos recientemente mostrados en artículos relacionados en donde la superficie de contacto tiene prescrita una condición de frontera slip

Mantle convection and the state of the Earth's interior

During 1983 to 1986 emphasis in the study of mantle convection shifted away from fluid mechanical analysis of simple systems with uniform material properties and simple geometries, toward analysis of the effects of more complicated, presumably more realistic models. The important processes related to mantle convection are considered. The developments in seismology are discussed

Mantle convection and the state of the Earth's interior

During 1983–1986, the four year period covered by this review, emphasis in the study of mantle convection shifted away from fluid mechanical analysis of simple systems with uniform material properties and simple geometries, toward analyzing the effects of more complicated, presumably more realistic models

Non-Newtonian effects in silicate liquids and crystal bearing melts

High-silica volcanic systems are considered to be the most devastating. Their highly viscous properties create a high pressurised non- fluent system which consequently relaxes the stress mostly by exploding through the brittle regime. Even if an explosion is avoided and the magma fl ows, it often generates lava domes at the top of the volcano; which, patiently, accumulate magmas that will rush down the slopes once the yield stress is crossed. Thus, such volcanoes have an explosive nature and often generate catastrophic pyroclastic flows. The modelling of magma ascent inside eruptive conduits is commonly based on fl uid mechanic principles. The difficulty of the approach is however not as much driven by the physical equations of the numerical model as the variability of the parameters of the magma itself. It is well established that the rheology of magma strongly depends on the temperature, the stress, the strain, the chemical composition, the crystal and the bubble contents. In other words, magmatic modelling involves a set of movement equations which call for a comportmental/rheological law. The movement equations give roughly equivalent results through the different models; however magma rheology is poorly controlled. The deformation of highly crystallised dome lavas is key to understanding their rheology and to fixing their failure onset. It is thus essential to adequately understand magma rheology before performing complex numerical models. Here we focus on the well studied Unzen volcano, in Japan, which had a recent period of activity between 1990 and 1995. The dome building eruptions in Unzen generate repeated dome failure and pyroclastic ows. They vary in character and behaviour from effusive domes to brittle pyroclastic events. Since then, the Unzen Scientic Drilling Project, initiated in April 1999, drilled through the volcano and sampled the eruptive conduit. This provides us with rare original samples for study and characterisation. The physico-chemical properties of these valuable samples were determined with an array of devices. Of these a large uniaxial deformation press, which can operate at high load (0 up to 300KN), and temperature (25-1200°C), will be of utmost importance. This press deforms the samples under known parameters and allows us to determine the viscosity of the melt. In this study we investigate the stress and strain-rate dependence on several glasses and Mt Unzen dome lavas. Their rheology has been determined for temperatures from 900 to 1010°C and stresses from 2 to 120 Mpa (60 MPa for crystal bearing melt) in uniaxial compression . This survey aims to distinguish the Non-Newtonian effects perturbing magmatic melts, also known as indicators of the brittle field. Towards our experiments we observed three majors viscosity decrease types: Two were dependant and typical of the solid fraction (Shear thinning & Time weakening effect). The first is instantaneous and on the whole recovered during stress release. The second is time dependent and non-recoverable. The third and last effect observed is attributed to the melt fraction and its self heating under stress (Viscous Heating Effect). We extensively investigated this last eect on pure silicate liquids and crystal bearing melts. Our findings suggest that most of the Non-Newtonians effects observed in silicate melts are linked to a self heating of the sample and can subsequently be corrected with the temperature without involving other laws than a pure viscous material. Moreover we observed that this self heating reorganise the energy distribution within the sample and by localising the strain may favours the formation of shear banding and the apparition of 'hot cracks'. Crystal bearing melts exhibit two more Non-Newtonian eects. The first one, the shear thinning, is typical of that observed in previous experiments on crystal-bearing melts. On crystal free melts, this viscosity decrease is observed at much lower magnitude. We infer that the crystal phase responds elastically to the stress applied and relaxes once the load is withdrawn. The second one, the time weakening effect, appears more complex and this regime depends on the stress (and/or strain-rate) history. We distinguish four different domains: Newtonian, non-Newtonian, crack propagation and failure domains. Each of these domains expresses itself as a dierent regime of viscosity decrease. Due to stress localisation, cracking appears in crystal-bearing melts (intra-phenocryst and/or the in the melt matrix) earlier than in crystal-free melts. For low stresses, the apparent viscosity is higher for crystal-bearing melts (as predicted by Einstein-Roscoe equations). However, while the stress (or strain rate) increases, the apparent viscosity is decreasing to that of the crystal-free melt and could be even lower if viscous heating effects are involved. Consequently, we emphasise that any numerical simulation performed without taking into account the strain-rate dependencies described above would overestimate the apparent viscosity by orders of magnitude. The magma dynamics will appears slower than in reality. Exaggerating the viscosity of a volcanic dynamic system would overestimate the time range available for a potential evacuation of the red zones. Applying a more realistic rheology would improve the early warning tools and improve the safety of the population surrounding volcanic systems

Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface

The dynamics of isolated air bubbles crossing the horizontal interface separating two Newtonian immiscible liquids initially at rest are studied both experimentally and computationally. High-speed video imaging is used to obtain a detailed evolution of the various interfaces involved in the system. The size of the bubbles and the viscosity contrast between the two liquids are varied by more than one and four orders of magnitude,respectively, making it possible to obtain bubble shapes ranging from spherical to toroidal. A variety of flow regimes isobserved,including that of small bubbles remaining trapped at the fluid–fluid interface in a film-drainage configuration.In most cases, the bubble succeeds in crossing the interface without being stopped near its undisturbed position and, during a certain period of time, tows a significant column of lower fluid which sometimes exhibits a complex dynamics as it lengthens in the upper fluid. Direct numerical simulations of several selected experimental situations are performed with a code employing a volume of-fluid type formulation of the incompressible Navier–Stokes equations. Comparisons between experimental and numerical results confirm the reliability of the computational approach in most situations but also points out the need for improvements to capture some subtle but important physical processes, most notably those related to film drainage. Influence of the physical parameters highlighted by experiments and computations, especially that of the density and viscosity contrasts between the two fluids and of the various interfacial tensions, is discussed and analysed in the light of simple models and available theories

Atmospheric Circulation of Exoplanets

We survey the basic principles of atmospheric dynamics relevant to explaining existing and future observations of exoplanets, both gas giant and terrestrial. Given the paucity of data on exoplanet atmospheres, our approach is to emphasize fundamental principles and insights gained from Solar-System studies that are likely to be generalizable to exoplanets. We begin by presenting the hierarchy of basic equations used in atmospheric dynamics, including the Navier-Stokes, primitive, shallow-water, and two-dimensional nondivergent models. We then survey key concepts in atmospheric dynamics, including the importance of planetary rotation, the concept of balance, and scaling arguments to show how turbulent interactions generally produce large-scale east-west banding on rotating planets. We next turn to issues specific to giant planets, including their expected interior and atmospheric thermal structures, the implications for their wind patterns, and mechanisms to pump their east-west jets. Hot Jupiter atmospheric dynamics are given particular attention, as these close-in planets have been the subject of most of the concrete developments in the study of exoplanetary atmospheres. We then turn to the basic elements of circulation on terrestrial planets as inferred from Solar-System studies, including Hadley cells, jet streams, processes that govern the large-scale horizontal temperature contrasts, and climate, and we discuss how these insights may apply to terrestrial exoplanets. Although exoplanets surely possess a greater diversity of circulation regimes than seen on the planets in our Solar System, our guiding philosophy is that the multi-decade study of Solar-System planets reviewed here provides a foundation upon which our understanding of more exotic exoplanetary meteorology must build.Comment: In EXOPLANETS, edited by S. Seager, to be published in the Spring of 2010 in the Space Science Series of the University of Arizona Press (Tucson, AZ) (refereed; accepted for publication

Fundamental studies in geodynamics

Research in fundamental studies in geodynamics continued in a number of fields including seismic observations and analysis, synthesis of geochemical data, theoretical investigation of geoid anomalies, extensive numerical experiments in a number of geodynamical contexts, and a new field seismic volcanology. Summaries of work in progress or completed during this report period are given. Abstracts of publications submitted from work in progress during this report period are attached as an appendix