spherical shell
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Abstract To study properties of magnetohydrodynamic (MHD) convection and resultant dynamo activities in proto-neutron stars (PNSs), we construct a “PNS in a box” simulation model and solve the compressible MHD equation coupled with a nuclear equation of state (EOS) and simplified leptonic transport. As a demonstration, we apply it to two types of PNS model with different internal structures: a fully convective model and a spherical-shell convection model. By varying the spin rate of the models, the rotational dependence of convection and the dynamo that operate inside the PNS is investigated. We find that, as a consequence of turbulent transport by rotating stratified convection, large-scale structures of flow and thermodynamic fields are developed in all models. Depending on the spin rate and the depth of the convection zone, various profiles of the large-scale structures are obtained, which can be physically understood as steady-state solutions to the “mean-field” equation of motion. Additionally to those hydrodynamic structures, a large-scale magnetic component of  ( 10 15 ) G is also spontaneously organized in disordered tangled magnetic fields in all models. The higher the spin rate, the stronger the large-scale magnetic component grows. Intriguingly, as an overall trend, the fully convective models have a stronger large-scale magnetic component than that in the spherical-shell convection models. The deeper the convection zone extends, the larger the size of the convective eddies becomes. As a result, rotationally constrained convection seems to be more easily achieved in the fully convective model, resulting in a higher efficiency of the large-scale dynamo there. To gain a better understanding of the origin of the diversity of a neutron star’s magnetic field, we need to study the PNS dynamo in a wider parameter range.

We find the analytical form of inertial waves in an incompressible, rotating fluid constrained by concentric inner and outer spherical surfaces with homogeneous boundary conditions on the normal components of velocity and vorticity. These fields are represented by Galerkin expansions whose basis consists of toroidal and poloidal vector functions, i.e., products and curls of products of spherical Bessel functions and vector spherical harmonics. These vector basis functions also satisfy the Helmholtz equation and this has the benefit of providing each basis function with a well-defined wavenumber. Eigenmodes and associated eigenfrequencies are determined for both the ideal and dissipative cases. These eigenmodes are formed from linear combinations of the Galerkin expansion basis functions. The system is truncated to numerically study inertial wave structure, varying the number of eigenmodes. The largest system considered in detail is a 25 eigenmode system and a graphical depiction is presented of the five lowest dissipation eigenmodes, all of which are non-oscillatory. These results may be useful in understanding data produced by numerical simulations of fluid and magnetofluid turbulence in a spherical shell that use a Galerkin, toroidal–poloidal basis as well as qualitative features of liquids confined by a spherical shell.

This paper presents the process of developing a new procedure for estimating the buckling capacity of spherical shells. This procedure is based entirely on the assumptions included in the standard mentioned, EN-1993-1-6 and also becomes a complement of EDR5th by unifying provisions included in them. This procedure is characterized by clarity and its algorithm is characterized by a low degree of complexity. While developing the procedure, no attempt was made to change the main postulates accompanying the dimensions of the spherical shells. The result is a simple engineering approach to the difficult problem of determining the buckling capacity of a spherical shell. In spite of the simple calculation algorithm for estimating the buckling capacity of spherical shells, the results obtained reflect extremely accurately the behavior of real spherical shells, regardless of their geometry and the material used to manufacture them.

This study focused on the three-dimensional surface crack growth of a spherical pressure shell. Eight maraging steel 18Ni (250) samples were fabricated and tested, and the fatigue crack growth rate curves were obtained. Considering the influence of plastic closure effect and sample thickness on crack growth, the fitting formula of fatigue crack growth only related to materials was obtained. Based on the three-dimensional crack closure theory and the strip yield model, a three-dimensional surface crack growth model of spherical pressure shell was established. By using a self-written program and FRANC3D, the three-dimensional surface crack growth simulations of the spherical shell were completed. The influence of the initial shape ratio and initial depth of the crack on the crack growth and the fatigue life of the spherical shell was analyzed.