Showing papers on "Spherical shell published in 2013"

Consistent scaling laws in anelastic spherical shell dynamos

Abstract: Numerical dynamo models always employ parameter values that differ by orders of magnitude from the values expected in natural objects. However, such models have been successful in qualitatively reproducing properties of planetary and stellar dynamos. This qualitative agreement fuels the idea that both numerical models and astrophysical objects may operate in the same asymptotic regime of dynamics. This can be tested by exploring the scaling behavior of the models. For convection-driven incompressible spherical shell dynamos with constant material properties, scaling laws had been established previously that relate flow velocity and magnetic field strength to the available power. Here we analyze 273 direct numerical simulations using the anelastic approximation, involving also cases with radius-dependent magnetic, thermal, and viscous diffusivities. These better represent conditions in gas giant planets and low-mass stars compared to Boussinesq models. Our study provides strong support for the hypothesis that both mean velocity and mean magnetic field strength scale as a function of the power generated by buoyancy forces in the same way for a wide range of conditions.

Inertial waves in a differentially rotating spherical shell

Abstract: We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in with the Ekman number , as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

Nonlocal elasticity theory for radial vibration of nanoscale spherical shells

Abstract: This paper presents the radial vibration of nanoscale spherical shells based on the nonlocal elasticity theory. The shell is considered elastic, homogeneous and isotropic. The nonlocal differential equation of radial motion is derived in terms of radial displacement. The relation between the nonlocal and local frequencies is also investigated. Considering the small-scale effect, the general characteristic equation for radial vibration of spherical shell is obtained by applying boundary conditions. Moreover, the characteristic equations for two special cases are presented. To demonstrate the accuracy of the present formulation, theoretical calculations of the fundamental frequency have been compared with those available in the literature and a good agreement is achieved. The variations of the frequencies with the nonlocal parameter, radius ratio and Poisson's ratio are also examined. It is observed that the frequencies are affected when the size effect is taken into consideration.

Microparticles confined to a nematic liquid crystal shell

Abstract: A seminal paper [D. R. Nelson, Nano Lett., 2002, 2, 1125.] has proposed that a nematic coating could be used to create a valency for spherical colloidal particles through the functionalization of nematic topological defects. Experimental realizations however question the complex behaviour of solid particles and defects embedded in such a nematic spherical shell. In order to address the related topological and geometrical issues, we have studied micrometer-sized silica beads trapped in nematic shells. We underline the mechanisms that strongly modify the texture of the simple (particle-free) shells when colloidal particles are embedded. Finally, we show how the coupling between capillarity and nematic elasticity offers new ways to control the valence and directionality of shells.

Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions

Abstract: Based on the Reissner-Naghdi’s thin shell theory, this paper concentrates on the free vibration of a joined cylindrical-spherical shell with elastic-support boundary conditions by a domain decomposition method. The joined shell is first separated from the elastic-support boundary and then subdivided into some cylindrical and spherical shell segments along the axis of revolution. The elastic-support boundary is regarded as a combination of distributed linear springs and can be treated as a special interface as well as the interface between two adjacent shell segments. Through the variational operation of the whole energy functional, the discretized equations of motion are derived by the expansions of the displacement field for each shell segment with Fourier series and Chebyshev orthogonal polynomials in the circumferential and longitudinal direction, respectively. To analyze the numerical convergence and precision of the present method, a number of case studies have been conducted and the solutions are compared with those derived by ANSYS and those presented in the previous literature to confirm the reliability and accuracy.

Libration-induced mean flow in a spherical shell

Abstract: We investigate the flow in a spherical shell subject to a time harmonic oscillation of its rotation rate, also called longitudinal libration, when the oscillation frequency is larger than twice the mean rotation rate. In this frequency regime, no inertial waves are directly excited by harmonic forcing. We show however that it can generate through non-linear interactions in the Ekman layers a strong mean zonal flow in the interior. An analytical theory is developed using a perturbative approach in the limit of small libration amplitude $\epsilon$ and small Ekman number $E$. The mean flow is found to be at leading order an azimuthal flow which scales as the square of the libration amplitude and only depends on the cylindrical-radius coordinate. The mean flow also exhibits a discontinuity across the cylinder tangent to the inner sphere. We show that this discontinuity can be smoothed through multi-scale Stewartson layers. The mean flow is also found to possess a weak axial flow which scales as $O(\epsilon^2 E^{5/42})$ in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations and a good agreement is demonstrated.

Comparison of spherical-shell and plane-layer mantle convection thermal structure in viscously stratified models with mixed-mode heating: implications for the incorporation of temperature-dependent parameters

Abstract: SUMMARY Plane-layergeometryconvectionmodelsremainausefultoolforinvestigatingplanetarymantle dynamicsbutyieldsignificantlywarmergeothermsthanspherical-shellsystems.Comparisons of uniform property plane-layer and spherical-shell models have provided insight into the role of geometry on temperature in convecting systems but the inclusion of first-order terrestrial characteristics is needed to quantitatively assess the influence of system geometry on more relevant mantle models. Here, we analyse the mean temperatures of over 160 spherical-shell and plane-layer convection models featuring a uniform upper-mantle viscosity and a lower mantle that increases in viscosity by a factor of 30 or 100. With the imposition of the stratified viscosity, an effective Rayleigh number, Raη, is defined based on the average viscosity of the mantle. We derive equations for the relationship between the mean temperature, θ, Raη and the non-dimensional internal heating rate, H, for both convection in a spherical shell with Earth-like mantle geometry and plane-layer solution domains. These equations predict the mean temperatures in the corresponding systems to an accuracy of a few percent or better. Our equations can be combined to derive the appropriate heating rate for a planelayer convection model to emulate the temperatures in a mixed heating mode spherical-shell convection model with effective Rayleigh number comparable to the Earth’s value, or greater. When comparing cases with the same internal heating rate and effective Rayleigh number, we findthattheincreasedlowermantleviscosityamplifiesthemeantemperatureratiooftheplanelayer and spherical-shell systems relative to isoviscous convection. These findings imply that the disagreement between spherical-shell mantle convection and plane-layer geometry mantle convectionthermalstructuremustbeparticularlyaccountedforinplane-layergeometrymodels featuring variable viscosities.

Meridional trapping and zonal propagation of inertial waves in a rotating fluid shell

Abstract: Inertial waves propagate in homogeneous rotating fluids, and constitute a challenging and simplified case study for the broader class of inertio-gravity waves, present in all geophysical and astrophysical media, and responsible for energetically costly processes such as diapycnal and angular momentum mixing. However, a complete analytical description and understanding of internal waves in arbitrarily shaped enclosed domains, such as the ocean or a planet liquid core, is still missing. In this work, the inviscid, linear inertial wave field is investigated by means of three-dimensional ray tracing in spherical shell domains, having in mind possible oceanographic applications. Rays are here classically interpreted as representative of energy paths, but in contrast to previous studies, they are now launched with a non-zero initial zonal component allowing for a more realistic, localized forcing and the development of azimuthal inhomogeneities. We find that meridional planes generally act in the shell geometry as attractors for ray trajectories. In addition, the existence of trajectories that are not subject to meridional trapping is here observed for the first time. Their dynamics was not captured by the previous purely meridional studies and unveils a new class of possible solutions for inertial motion in the spherical shell. Both observed behaviours shed some new light on possible mechanisms of energy localization, a key process that still deserves further investigation in our ocean, as well as in other stratified, rotating media.

Thermal postbuckling behavior of laminated composite spherical shell panel using NFEM

Abstract: Postbuckling behavior of laminated shell panel in thermal environment is reported in this article. The geometric nonlinearity is introduced in Green–Lagrange sense and the model is developed for the geometrically large translations and/or rotations and the excess thermal deformation of the curved panel based on higher order shear deformation theory (HSDT) using nonlinear finite element. The governing equation of shell panel is derived by minimizing the energy expression. The postbuckling strength in terms of temperature ratio (postbuckling to buckling temperature) of the panel by obtained by a direct iterative method. The results are obtained using the developed model and compared with those of the available published literature. Some of the new results are computed for different parameters such as layup sequences, thickness ratios, amplitude ratios, boundary conditions, aspect ratios, and various curvature ratios and presented.

Amphibious spherical exploration robot

Abstract: An amphibious spherical exploration robot comprises a spherical shell, a paddle, an inner drive mechanism assembly and a connecting piece. The spherical shell is transparent. The inner drive mechanism assembly and the connecting piece are contained in the spherical shell. An exploration camera is arranged in the spherical shell. The paddle is fixed to the outside of the spherical shell, the paddle is immersed in water when the robot moves on the water, and therefore the robot water paddling motion function in water can be achieved. The inner drive mechanism assembly is composed of a straight line motion drive motor, a steering motion drive motor, a clump weight and a supporting part. Under the action of the straight line drive motor, the inner drive mechanism assembly can rotate around a transverse center, so that the inner gravity position of a sphere is changed, and the robot can be in forward or backward straight line motion. Under the action of the steering motion drive motor, the spherical shell rotates in the reverse direction when the clump weight rotates in an accelerating mode around a longitudinal central line, so that the spherical robot can be in pivot steering motion. The amphibious spherical exploration robot can move flexibly on land or on the water surface and execute exploration tasks.

Modes and instabilities in magnetized spherical Couette flow

Abstract: Several teams have reported peculiar frequency spectra for flows in a spherical shell. To address their origin, we perform numerical simulations of the spherical Couette flow in a dipolar magnetic field, in the configuration of the experiment. The frequency spectra computed from time-series of the induced magnetic field display similar bumpy spectra, where each bump corresponds to a given azimuthal mode number . The bumps appear at moderate Reynolds number ( ) if the time-series are long enough ( rotations of the inner sphere). We present a new method that permits retrieval of the dominant frequencies for individual mode numbers , and extraction of the modal structure of the full nonlinear flow. The maps of the energy of the fluctuations and the spatio-temporal evolution of the velocity field suggest that fluctuations originate in the outer boundary layer. The threshold of instability is found at . The fluctuations result from two coupled instabilities: high-latitude Bodewadt-type boundary layer instability, and secondary non-axisymmetric instability of a centripetal jet forming at the equator of the outer sphere. We explore the variation of the magnetic and kinetic energies with the input parameters, and show that a modified Elsasser number controls their evolution. We can thus compare with experimental determinations of these energies and find a good agreement. Because of the dipolar nature of the imposed magnetic field, the energy of magnetic fluctuations is much larger near the inner sphere, but their origin lies in velocity fluctuations that are initiated in the outer boundary layer.

Folding of an opened spherical shell

Abstract: Thin, doubly curved shells occur commonly in nature and their mechanical properties and modes of deformation are very important for engineering structures of all scales. Although there has been substantial work on the stability and modes of failure of thin shells, relatively little work has been done to understand the conditions that promote continuous large scale deformations. A major impediment to progress in this direction is the inherent difficulty in obtaining analytical expressions for the deformed shapes. In this work we propose a new integrable solution which describes the behavior under load of a thin spherical shell with an opening (aperture) of n-fold axial symmetry. We derive a two-parameter family of approximately isometric, constant positive Gaussian curvature shapes that is in excellent agreement with our experimental results of deformed shells (3D scans of compressed ping-pong balls) and simulations (tethered membrane simulations minimizing the stretching and bending energy). The integrable solutions that describe those shapes have n symmetrically arranged curvature singularities which correspond to cusps of the folded shape. We examine the properties of the folded shells and observe that in the analytic solutions isometric closure is more easily achieved when the singularities lie away from the center of the aperture. We find that when allowed by the geometry of the aperture and the nature of the load, physical shells expel the curvature singularities into the aperture.

An improved formulation for the incompressible Navier-Stokes equations with variable viscosity

Abstract: We present a new formulation of the incompressible Navier-Stokes equations with variable viscosity. By utilizing the incompressibility constraint to remove the trace from the deviatoric stress tensor, we eliminat es econdorder cross-derivatives of the velocity field, simplifying and improvin gt he accuracy of co-located discretization techniques on both structured- and unstructured grids. This formulation improves the performance of SIMPLEtype algorithms that use sequential mass-momentum iterations to enforce incompressibility. A trace-free stress tensor also removes a typical source of net-rotation for simulations employing free-slip boundary conditions in spherical geometry. We implement the new scheme as a modification o fa n existing Boussinesq convection code, which we benchmark against analytical solutions of the Stokes problem in a spherical shell with both constant and radially dependent viscosity, and time-dependent thermal convection at

Multivalent ion effects on electrostatic stability of virus-like nano-shells.

Abstract: Electrostatic properties and stability of charged virus-like nano-shells are examined in ionic solutions with monovalent and multivalent ions. A theoretical model based on a thin charged spherical shell and multivalent ions within the “dressed multivalent ion” approximation, yielding their distribution across the shell and the corresponding electrostatic (osmotic) pressure acting on the shell, is compared with extensive implicit Monte-Carlo simulations. It is found to be accurate for positive or low negative surface charge densities of the shell and for sufficiently high (low) monovalent (multivalent) salt concentrations. Phase diagrams involving electrostatic pressure exhibit positive and negative values, corresponding to an outward and an inward facing force on the shell, respectively. This provides an explanation for the high sensitivity of viral shell stability and self-assembly of viral capsid shells on the ionic environment.