Moving Surfaces

Now consider a object that periodically oscillates between a unit sphere and an ellipsoid. To do this multiply the $$y$$-axis of the unit sphere by a factor of

\begin{equation*} 1+ \alpha \sin \left( 2 \pi \beta t \right) \end{equation*}

where $$\alpha$$ is the deformation and $$\beta$$ controls the period.

The deformation must be chosen carefully: if it is too small there will be no significant difference between this case and the case of the static sphere. Obviously the deformation $$\alpha$$ cannot be greater than 1 otherwise the sphere collapses. We tested with the deformations $$\alpha=0.1$$, $$0.2$$, $$0.3$$, $$0.4$$ and $$\alpha=0.5$$.

The choice of $$\beta$$ is also important: if it is too small, the period is very long and the simulation must be run for an unfeasible time to obtain results. However if $$\beta$$ is too large, the period is very short and the sphere oscillates rapidly meaning a very high level of time refinement is needed to avoid numerical instability. The results given below are for $$\beta=0.1$$.

It can be seen that all these oscillating spheres contain spiral waves, however comparison between the different magnitudes of deformation show different behaviour. The two most notable effects are

• the wave-form is deformed; at the maximum vertical oscillation the wave-form becomes much wider and at the maximum horizontal oscillation the wave-form becomes much thinner,
• the centre of the spiral moves appearing to move in the direction of the oscillation.

Both these effects are emphasised more with larger oscillation.