Showing papers on "Spherical shell published in 2011"

Reynolds stress and heat flux in spherical shell convection

Abstract: Context. Turbulent fluxes of angular momentum and enthalpy or heat due to rotationally affected convection play a key role in determining differential rotation of stars. Their dependence on latitude and depth has been determined in the past from convection simulations in Cartesian or spherical simulations. Here we perform a systematic comparison between the two geometries as a function of the rotation rate. Aims. Here we want to extend the earlier studies by using spherical wedges to obtain turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime in order to study whether restricted geometry introduces artefacts into the results. In particular, we want to clarify whether the sharp equatorial profile of the horizontal Reynolds stress found in earlier Cartesian models is also reproduced in spherical geometry. Methods. We employ direct numerical simulations of turbulent convection in spherical and Cartesian geometries. In order to alleviate the computational cost in the spherical runs, and to reach as high spatial resolution as possible, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number or inverse Rossby number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in accordance with earlier numerical results, but in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong “banana cells”. Their effect in the spherical case does not appear to be as large. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. Longitudinal heat flux is always in the retrograde direction. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor-Proudman balance.

Effects of low-viscosity post-perovskite on thermo-chemical mantle convection in a 3-D spherical shell

Abstract: [1] Numerical simulations of thermo-chemical, multi-phase mantle convection in a 3-D spherical shell are performed to determine how a low viscosity of post-perovskite affects dynamics and structures in the deep mantle Low-viscosity post-perovskite weakens the deepest part of slabs, allowing them to more effectively spread over the core-mantle boundary (CMB), and it also results in a greater volume of basalt segregating, both of which increase the size of dense chemical piles, the horizontal lengthscale of regions of pooled slab material, and the steepness of piles' edges (in composition and phase), consistent with the existence of steep, sharp-sided edges found in seismic analyses CMB heat flux is strongly enhanced in regions of low-viscosity post-perovskite (consistent with a theoretical prediction) and both CMB and surface heat flux are increased on average by a low-viscosity of post-perovskite, which could have important implications for the evolution of Earth's core and mantle

Microscale Glass-Blown Three-Dimensional Spherical Shell Resonators

Abstract: This paper introduces a new paradigm for design and batch fabrication of isotropic 3-D spherical shell resonators. The approach uses pressure and surface tension driven plastic deformation (glassblowing) on a wafer scale as a mechanism for creating inherently smooth and symmetric 3-D resonant structures. The feasibility of the new approach was demonstrated by fabrication and characterization of Pyrex glass spherical shell resonators with millimeter-scale diameter and average thickness of 10 μm . Metal electrodes cofabricated along with the shell were used to actuate the two dynamically balanced four- and six-node vibratory modes. For 1-MHz glass-blown resonators, the relative frequency mismatch Δf/f between the two degenerate four-node wineglass modes was measured as 0.63% without any trimming or tuning. For the higher order six-node wineglass modes, the relative frequency mismatch was only 0.2%, demonstrating the potential for precision manufacturing. The intrinsic manufacturing symmetry enabled by the technology may inspire new classes of high-performance 3-D MEMS for communication and inertial navigation.

Instabilities in magnetized spherical Couette flow.

Abstract: We report three-dimensional numerical simulations of the flow of an electrically conducting fluid in a spherical shell when a magnetic field is applied. Different spherical Couette configurations are investigated by varying the rotation ratio between the inner and the outer sphere, the geometry of the imposed field, and the magnetic boundary conditions on the inner sphere. Either a Stewartson layer or a Shercliff layer, accompanied by a radial jet, can be generated depending on the rotation speeds and the magnetic-field strength, and various nonaxisymmetric destabilizations of the flow are observed. We show that instabilities arising from the presence of boundaries present striking similarities with the magnetorotational instability (MRI). To this end, we compare our numerical results to experimental observations of the Maryland experiment [D. R. Sisan et al., Phys. Rev. Lett. 93, 114502 (2004)], which claimed to observe MRI in a similar setup.

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

Abstract: This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load–deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.

Two-dimensional packing of soft particles and the soft generalized Thomson problem

Abstract: We perform numerical simulations of a model of purely repulsive soft colloidal particles interacting via a generalized elastic potential and constrained to a two-dimensional plane and to the surface of a spherical shell. For the planar case, we compute the phase diagram in terms of the system's rescaled density and temperature. We find that a large number of ordered phases becomes accessible at low temperatures as the density of the system increases, and we study systematically how structural variety depends on the functional shape of the pair potential. For the spherical case, we revisit the generalized Thomson problem for small numbers of particles N ≤ 12 and identify, enumerate and compare the minimal energy polyhedra established by the location of the particles to those of the corresponding electrostatic system.

Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties

Abstract: This paper focuses on non-Fourier hyperbolic heat conduction analysis for heterogeneous hollow cylinders and spheres made of functionally graded material (FGM). All the material properties vary exponentially across the thickness, except for the thermal relaxation parameter which is taken to be constant. The cylinder and sphere are considered to be cylindrically and spherically symmetric, respectively, leading to one-dimensional heat conduction problems. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin’s numerical inversion method. The transient responses of temperature and heat flux are investigated for different inhomogeneity parameters and relative temperature change values. The comparisons of temperature distribution and heat flux between various time and material properties are presented in the form of graphs.

The influence of pick-up ions on space plasma distributions

Abstract: The addition and incorporation of highly ordered distributions of pickup ions can increase the ordering of space plasmas, decreasing their entropy, and driving them away from equilibrium. In this study, we present a model for describing the entropy decrease of heliospheric plasmas caused by the pick-up ions and their highly ordered arrangement. We consider a plasma flow consisting of two proton populations: (1) a solar wind distribution that is in quasi-equilibrium and (2) a spherical shell distribution of pick-up protons. Because all protons are indistinguishable, they equally share a hybrid state, where the hybrid probability distribution is given by the normalized sum of the distributions of the solar wind and pick-up protons. We derive an analytical formulation of the thickness of the pick-up proton shell distribution. Then we show that the entropy of the hybrid state depends on a dimensionless fluctuation number, {Theta}, which is a measure of the organizing role of pick-up protons. This number characterizes and compares two types of fluctuations, the thermal motion of solar wind protons and the bounded motion of pick-up protons that forms the spherical shell distribution. We find that the entropy variation between the hybrid and original states is proportional to the negativemore » logarithm of {Theta}. When the two competing fluctuations balance each other, {Theta} {approx} 1, the pick-up protons have no effect on the entropy of solar wind protons. Remarkably, while sunward of the termination shock pick-up ions act to slightly increase entropy, throughout the inner heliosheath their effect is to decrease the entropy dramatically.« less

Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment

Abstract: We present measurements performed in a spherical shell filled with liquid sodium, where a 74-mm-radius inner sphere is rotated while a 210-mm-radius outer sphere is at rest. The inner sphere holds a dipolar magnetic field and acts as a magnetic propeller when rotated. In this experimental setup called ``Derviche Tourneur Sodium'' (DTS), direct measurements of the velocity are performed by ultrasonic Doppler velocimetry. Differences in electric potential and the induced magnetic field are also measured to characterize the magnetohydrodynamic flow. Rotation frequencies of the inner sphere are varied between $\ensuremath{-}$30 Hz and $+$30 Hz, the magnetic Reynolds number based on measured sodium velocities and on the shell radius reaching to about 33. We have investigated the mean axisymmetric part of the flow, which consists of differential rotation. Strong super-rotation of the fluid with respect to the rotating inner sphere is directly measured. It is found that the organization of the mean flow does not change much throughout the entire range of parameters covered by our experiment. The direct measurements of zonal velocity give a nice illustration of Ferraro's law of isorotation in the vicinity of the inner sphere, where magnetic forces dominate inertial ones. The transition from a Ferraro regime in the interior to a geostrophic regime, where inertial forces predominate, in the outer regions has been well documented. It takes place where the local Elsasser number is about 1. A quantitative agreement with nonlinear numerical simulations is obtained when keeping the same Elsasser number. The experiments also reveal a region that violates Ferraro's law just above the inner sphere.

Effects of stratification in spherical shell convection

Abstract: We report on simulations of mildly turbulent convection in spherical wedge geometry with varying density stratification. We vary the density contrast within the convection zone by a factor of 20 and study the influence of rotation on the solutions. We demonstrate that the size of convective cells decreases and the anisotropy of turbulence increases as the stratification is increased. Differential rotation is found to change from anti-solar (slow equator) to solar-like (fast equator) at roughly the same Coriolis number for all stratifications. The largest stratification runs, however, are sensitive to changes of the Reynolds number. Evidence for a near-surface shear layer is found in runs with strong stratification and large Reynolds numbers.

Convection patterns in a spherical fluid shell.

Abstract: Symmetry-breaking bifurcations have been studied for convection in a nonrotating spherical shell whose outer radius is twice the inner radius, under the influence of an externally applied central force field with a radial dependence proportional to 1/r(5). This work is motivated by the GeoFlow experiment, which is performed under microgravity condition at the International Space Station where this particular central force can be generated. In order to predict the observable patterns, simulations together with path-following techniques and stability computations have been applied. Branches of axisymmetric, octahedral, and seven-cell solutions have been traced. The bifurcations producing them have been identified and their stability ranges determined. At higher Rayleigh numbers, time-periodic states with a complex spatiotemporal symmetry are found, which we call breathing patterns.

Two-Dimensional Packing of Soft Particles and the Soft Generalized Thomson Problem

Abstract: We perform numerical simulations of purely repulsive soft colloidal particles interacting via a generalized elastic potential and constrained to a two-dimensional plane and to the surface of a spherical shell. For the planar case, we compute the phase diagram in terms of the system's rescaled density and temperature. We find that a large number of ordered phases becomes accessible at low temperatures as the density of the system increases, and we study systematically how structural variety depends on the functional shape of the pair potential. For the spherical case, we revisit the generalized Thomson problem for small numbers of particles N <= 12 and identify, enumerate and compare the minimal energy polyhedra established by the location of the particles to those of the corresponding electrostatic system.

Gravastars and black holes of anisotropic dark energy

Abstract: Dynamical models of prototype gravastars made of anisotropic dark energy are constructed, in which an infinitely thin spherical shell of a perfect fluid with the equation of state p = (1 − γ)σ divides the whole spacetime into two regions, the internal region filled with a dark energy fluid, and the external Schwarzschild region. The models represent “bounded excursion” stable gravastars, where the thin shell is oscillating between two finite radii, while in other cases they collapse until the formation of black holes. Here we show, for the first time in the literature, a model of gravastar and formation of black hole with both interior and thin shell constituted exclusively of dark energy. Besides, the sign of the parameter of anisotropy (pt − pr) seems to be relevant to the gravastar formation. The formation is favored when the tangential pressure is greater than the radial pressure, at least in the neighborhood of the isotropic case (ω = −1).

Effects of stratification in spherical shell convection

Abstract: We report on simulations of mildly turbulent convection in spherical wedge geometry with varying density stratification. We vary the density contrast within the convection zone by a factor of 20 and study the influence of rotation on the solutions. We demonstrate that the size of convective cells decreases and the anisotropy of turbulence increases as the stratification is increased. Differential rotation is found to change from anti-solar (slow equator) to solar-like (fast equator) at roughly the same Coriolis number for all stratifications. The largest stratification runs, however, are sensitive to changes of the Reynolds number. Evidence for a near-surface shear layer is found in runs with strong stratification and large Reynolds numbers.

Acoustic radiation force on a spherical contrast agent shell near a vessel porous wall--theory.

Abstract: Contrast agent microshells (CAMSs) are under intensive investigation for their wide applications in biomedical imaging and drug delivery. In drug delivery applications, CAMSs are guided to the targeted site before fragmentation by high-intensity ultrasound waves leading to the drug release. Prediction of the acoustic radiation force used to nondestructively guide a CAMS to the suspected site is becoming increasingly important and gaining attention particularly because it increases the system efficiency. The goal of this work is to present a theoretical model for the time-averaged (static) acoustic radiation force experienced by a CAMS near a blood vessel wall. An exact solution for the scattering of normal incident plane acoustic waves on an air-filled elastic spherical shell immersed in a nonviscous fluid near a porous and nonrigid boundary is employed to evaluate the radiation force function (which is the radiation force per unit energy density per unit cross-sectional surface). A particular example is chosen to illustrate the behavior of the time-averaged (static) radiation force on an elastic polyethylene spherical shell near a porous wall, with particular emphasis on the relative thickness of the shell and the distance from its center to the wall. This proposed model allows obtaining a priori information on the static radiation force that may be used to advantage in related as drug delivery and contrast agent imaging. This study should assist in the development of improved models for the evaluation of the time-averaged acoustic radiation force on a cluster of CAMSs in viscous and heat-conducting fluids.

Artificial hip joint consisting of multi-layer shell core composite structural components

Abstract: An artificial hip joint consisting of multi-layer shell core composite structural components includes an artificial acetabular bone (1) and an artificial femoral head (2) which are mutually matched with each other. The artificial acetabular bone (1) has a multi-layer shell core composite structure and is constituted of a ceramic acetabular bone lining (1-1), transitional layers (1-2, 1-3), an acetabular bone shell made of a porous metal or a porous alloy or a porous toughened ceramic (1-4). The artificial femoral head (2) has a multi-layer shell core composite structure and is constituted of a ceramic spherical shell layer (2-1), a transitional layer (2-5) and a toughened ceramic inner core (2-2).The artificial acetabular bone lining and the artificial femoral head spherical shell layer of the hip joint have high rigid, anti-corrosion and anti-wear performance. The artificial acetabular bone shell layer and the femoral head inner core layer have high tough, shockresistant performance. The transitional layer adopts gradient composite material whose composition is between the shell layer and the inner core layer material, which has functions such as increasing bonding strength between the shell layer and the inner core layer and reducing interface stress between the shell layer and the inner core layer.

Thermal instability of functionally graded deep spherical shell

Abstract: Thermal instability of deep spherical shells made of functionally graded material (FGM) is studied in this paper. The governing equations are based on the first-order theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the mechanical properties are linear functions of thickness coordinate. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. The analytical solutions are obtained for three types of thermal loadings including the uniform temperature rise (UTR), the linear radial temperature (LRT), and the nonlinear radial temperature (NRT). Results are validated with the known data in literature.

Zonal shear and super-rotation in a magnetized spherical Couette flow experiment

Abstract: We present measurements performed in a spherical shell filled with liquid sodium, where a 74-mm-radius inner sphere is rotated while a 210-mm-radius outer sphere is at rest. The inner sphere holds a dipolar magnetic field and acts as a magnetic propeller when rotated. In this experimental setup called ``Derviche Tourneur Sodium'' (DTS), direct measurements of the velocity are performed by ultrasonic Doppler velocimetry. Differences in electric potential and the induced magnetic field are also measured to characterize the magnetohydrodynamic flow. Rotation frequencies of the inner sphere are varied between $\ensuremath{-}$30 Hz and $+$30 Hz, the magnetic Reynolds number based on measured sodium velocities and on the shell radius reaching to about 33. We have investigated the mean axisymmetric part of the flow, which consists of differential rotation. Strong super-rotation of the fluid with respect to the rotating inner sphere is directly measured. It is found that the organization of the mean flow does not change much throughout the entire range of parameters covered by our experiment. The direct measurements of zonal velocity give a nice illustration of Ferraro's law of isorotation in the vicinity of the inner sphere, where magnetic forces dominate inertial ones. The transition from a Ferraro regime in the interior to a geostrophic regime, where inertial forces predominate, in the outer regions has been well documented. It takes place where the local Elsasser number is about 1. A quantitative agreement with nonlinear numerical simulations is obtained when keeping the same Elsasser number. The experiments also reveal a region that violates Ferraro's law just above the inner sphere.

Small amplitude radial oscillations of an incompressible, isotropic elastic spherical shell

Abstract: The small amplitude radial oscillation and infinitesimal stability about an equilibrium configuration of an arbitrary incompressible, isotropic and homogeneous elastic spherical shell under constant inflation pressure loading is studied for both thick- and thin-walled shells and for a spherical cavity within an unbounded continuum. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and the pressure response. It follows that points at which the pressure is stationary are unstable or neutrally stable. All results are expressed in terms of the shear response function for a general incompressible, isotropic elastic material, and specific results are illustrated for the Mooney—Rivlin and Gent material models, the latter having limited extensibility. The classical neo-Hookean material exhibits results that are lower bounds for both models. A criterion obtained by others to characterize the possible bifurcation from a spherical to an aspherical shape is c...

Adhesion of an Elastic Convex Shell onto a Rigid Plate

Abstract: An elastic spherical shell is compressed between two parallel rigid plates and undergoes large geometrical deformation. The convex surface conforms and adheres to the plates. The shell profile and contact stresses under large deformation are obtained numerically using a finite difference method. A thermodynamic energy balance following the classical Johnson-Kendall-Roberts (JKR) model is established to construct the adhesion mechanics, such that the sum of potential energy of the external load, elastic energy stored in the elastic shell, and surface energy to create new surface is minimized. Interrelationship between applied load, approach distance, contact radius, and deformed profile, as well as the “pull-off” phenomenon, are derived. A comparison is made between this model and existing models for a solid sphere in the literature.

Nonstationary axisymmetric problem of the impact of a spherical shell on an elastic half-space (initial stage of interaction)

Abstract: The supersonic stage of interaction (where the rate of expansion of the contact region is no less than the speed of compression waves) between a Timoshenko-type spherical shell (indenter) and an elastic half-space (foundation) is studied. The expansion of the desired functions in series in Legendre polynomials and their derivatives are used to construct a system of resolving equations. An analytical-numerical algorithm for solving this system is developed. A similar problem was considered in [1], where the original problem was replaced by a problem with a periodic system of indenters.

Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications

Abstract: We study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model uses the two-dimensional Navier-Stokes equations on a rotating three-dimensional spherical surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows rotating around the poles. This work addresses exact stationary and non-stationary solutions associated with the nonlinear Navier-Stokes. The exact solutions in terms of elementary functions for the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact solutions of the Navier-Stokes equations (non-zero viscosity). The obtained solutions are expressed in terms of elementary functions, analyzed, and visualized.