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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.

No Deformation

Deformation with \( \alpha = 0.1 \)

Deformation with \( \alpha = 0.2 \)

Deformation with \( \alpha = 0.3 \)

Deformation with \( \alpha = 0.4 \)

Deformation with \( \alpha = 0.5 \)