Inertial Waves in a Rotating Spherical Shell with Homogeneous Boundary Conditions

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.

Reduction of the Self-dual Yang-Mills Equations to Sinh-Poisson Equation and Exact Solutions

The geometric properties of differential systems are used to demonstrate how the sinh-poisson equation describes a surface with a constant negative curvature in this paper. The canonical reduction of 4-dimensional self dual Yang Mills theorem is the sinh-poisson equation, which explains pseudo spherical surfaces. We derive the B¨acklund transformations and the travelling wave solution for the sinh-poisson equation in specific. As a result, we discover exact solutions to the self-dual Yang-Mills equations.

A Note by K. L. Johnson on the History of the JKR Theory

The history of the following note is as follows. In 2003, I invited Kenneth Johnson to Berlin to give a talk on adhesion in a seminar at the Institute of Mechanics. His lecture on the topic "Mechanics of adhesion of spherical surfaces" took place on Monday, January 26, 2004. In the run-up to the seminar, Professor Johnson sent me a historical note dated November 18, 2003. In my opinion, this note, which was written in the form of a paper, may be of interest for experts in contact mechanics and tribology. Prof. Johnson did not publish it, so it remained a private communication. For a publication he might have made a revision and would possibly have credited other important contributions. But this we can only guess at, and therefore the note is published below in the form I received it from Kenneth L. Johnson, with only a few misprints corrected. It is interesting as a historical document from Ken Johnson, who played a key role in development of theory of adhesive contacts.

Spatial Mechanism-Environment Contact Geometric Models

This work proposes a theoretical foundation for a general spatial geometric mechanism-environment contact model. In the proposed model the curvature of the environment in the vicinity of the contact is approximated by a number of spherical surfaces with known radii of curvature that constrain/define the movement of the body. We show how the modeled body-environment contact and curvature constraints can be transformed into conditions on spatial velocity and acceleration (i.e. first and second order effects) of certain points of the moving body that can be incorporated in the kinematic task for designing spatial mechanisms. Further, we explore the exact synthesis of a spatial six degrees-of-freedom TPS kinematic chain which end-effector maintains contact with objects in the environment and varies orientation in the vicinity of a contact location. It is discussed how the higher order motion constraints allow for the introduction of kinematic task variations in the vicinity of a contact, resulting in different behaviors of the designed spatial mechanism. The theoretical foundation presented in this paper is crucial in gaining understanding of the constraints in describing mechanism-environment interactions in the vicinity of a contact and is a new contribution.

Bright–Dark Soliton Waves’ Dynamics in Pseudo Spherical Surfaces through the Nonlinear Kaup–Kupershmidt Equation

The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

The Casimir Interaction between Spheres Immersed in Electrolytes

We investigate the Casimir interaction between two dielectric spheres immersed in an electrolyte solution. Since ionized solutions typically correspond to a plasma frequency much smaller than kBT/ℏ at room temperature, only the contribution of the zeroth Matsubara frequency is affected by ionic screening. We follow the electrostatic fluctuational approach and derive the zero-frequency contribution from the linear Poisson-Boltzmann (Debye-Hückel) equation for the geometry of two spherical surfaces of arbitrary radii. We show that a contribution from monopole fluctuations, which is reminiscent of the Kirkwood-Shumaker interaction, arises from the exclusion of ionic charge in the volume occupied by the spheres. Alongside the contribution from dipole fluctuations, such monopolar term provides the leading-order Casimir energy for very small spheres. Finally, we also investigate the large sphere limit and the conditions for validity of the proximity force (Derjaguin) approximation. Altogether, our results represent the first step towards a full scattering approach to the screening of the Casimir interaction between spheres that takes into account the nonlocal response of the electrolyte solution.