force-free convected 2-tensors at a moving spheroid
We present a comparison of solutions of force-free transported surface 2-tensors q, i.e. D_t(q) = 0, where D_t is the upper-upper, lower-lower, upper-lower and lower-upper convected time derivative. The moving surface is a constantly stretching spheroid along one euclidean axis. The solutions are depicted in an app-like CDF-file, i.e. the Wolfram CDF player is required. This player is free and can be downloaded at https://www.wolfram.com/cdf-player/index.html for all common operating systems. We recommend version 12.0.0.. The solutions are represented in parallelogram-shaped tensor glyphs, where the diagonals representing the eigenvectors with lengths equal eigenvalues, see initial condition at left bottom. To demonstrate that the transported tensors are not aligned with the corresponding transported eigensystem generally, we present also the upper and lower convected eigenvectors times eigenvalue with same tensor glyphs wrt. initial condition. This app is interactive, where we can control the rotational symmetric initial conditions by setting eigenvectors and real-valued eigenvalues, which can be restricted to symmetric and/or trace-free initial tensors. Moreover we can control the time of evaluation, the maximum time and latitude of evaluation. The depiction of the tensor glyphs can be taken place with or without scaling wrt. eigenvalues. To validate the rotational invariance of convected derivatives it is also possible to add a rigid rotation to the spheroid. Note that the initial settings are the same as in the tensor field example in our paper "Observer-invariant time derivatives on moving surfaces".
Steps to reproduce
open document with Wolfram CDF Player (version 12.0.0)