A Theory of Inertia Based on Mach’s Principle

A non-relativistic theory of inertia based on Mach’s principle is presented as has been envisaged, but not achieved, by Ernst Mach in 1872. The central feature is a space-dependent, anisotropic, symmetric inert mass tensor. The contribution of a mass element d m to the inertia of a particle m 0 experiencing an acceleration from rest is proportional to cos 2 α , where α is the angle between the line connecting m 0 and d m and the direction of the acceleration. Apsidal precession for planets circling around a central star is not a consequence of this theory, thereby avoiding the prediction of an apsidal precession with the wrong sign as is done by Mach-like theories with isotropic inert mass.

A Theory of Inertia Based on Mach's Principle

A non-relativistic theory of inertia based on Mach's principle is presented as has been envisaged but not achieved by Ernst Mach in 1872. Central feature is a space-dependent, anisotropic, symmetric inert mass tensor. The contribution of a mass element $dm$ to the inertia of a particle $m_0$ experiencing an acceleration from rest is proportional to $\cos^2\alpha$, where $\alpha$ is the angle between the line connecting $m_0$ and $dm$ and the direction of the acceleration. Apsidal precession for planets circling around a central star is not a consequence of this theory, thereby avoiding the prediction of an apsidal precession with wrong sign as is done by Mach-like theories with isotropic inert mass.

Influence of Added Mass Effect on Rotation of a Drifting Iceberg in Non-Stationary Current

Observations of drifting icebergs by the ice trackers installed at their surface show their rotation relatively vertical axis. Iceberg rotation can influence drift characteristics and potentially dangerous for the iceberg towing. In the present paper 2D motion of floating icebergs in the presence of wind drag and unsteady water flow is considered. The water flow is assumed two dimensional and with uniform vertical profile. The direction and magnitude of the current speed are varying with the time. Temporal variability of the flow is associated with semidiurnal tidal. The inertial properties of an iceberg are characterised by three components of the added mass tensor. The iceberg motion is described by the Kirchoff equations written in the frame of reference sliding with the fluid. The rotation of cylindrical icebergs with elliptic horizontal cross-sections is investigated depending on the wind and fluid flow characteristics.

Axisymmetric dynamics of a bubble near a plane wall

Explicit expressions for the added mass tensor of a bubble in strongly nonlinear deformation and motion near a plane wall are presented. Time evolutions and interconnections of added mass components are derived analytically and analysed. Interface dynamics have been predicted with two methods, assuming that the flow is irrotational, that the fluid is perfect and with the neglect of gravity. The assumptions that gravity and viscosity are negligible are verified by investigating their effects and by quantifying their impact in some cases of strong deformation, and criteria are presented to specify the conditions of their validity. The two methods are an analytical one and the boundary element method, and good agreement is found. It is explained why a strongly deforming bubble is decelerated. The classical Rayleigh–Plesset equation is extended with terms to account for arbitrary, axisymmetric deformation and to account for the proximity of a wall. An expression for the corresponding cycle frequency that is valid in the vicinity of the wall is derived. An equation similar to the Rayleigh–Plesset equation is presented for the most important anisotropic deformation mode. Well-known expressions for the angular frequencies of some periodic solutions without a wall follow easily from the equations presented. A periodically deforming bubble without initial velocity of the centroid and without a dominating isotropic deformation component is eventually always driven towards the wall. A simplified equation of motion of the centre of a deforming bubble is presented. If desired, full deformation computations can be speeded up by selecting an artificially low value of the polytropic constant Cp/Cv.