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Fixed Surfaces

The initial conditions that we use for all the spherical and ellipsoid simulations are set to

\begin{align*} u(0,\mathbf x)&=u_0(\mathbf x)=\frac{1}{2}(\tanh(30y)+1)\\ v(0,\mathbf x)&=v_0(\mathbf x)=\frac{1}{2}0.375(\tanh(30(-x+0.01))+1)\end{align*}

which is a smooth version of

\begin{align*} u(0,\mathbf{x})&=u_0(\mathbf x)=\mathbb{I}_{y>0.01}\\ v(0,\mathbf{x})&=v_0(\mathbf x)=\frac{3}{8}\mathbb{I}_{x<0}.\end{align*}

Static Sphere

The first attempt to simulate a heart-like structure is implemented by simulating spiral waves on the unit sphere. The constants are set to

\begin{equation*} \epsilon =0.02 \; \; \; a= {1 \over {179.049}} \; \; \; b=0.75 \; \; \; c=0.01 . \end{equation*}

The results are summarised below.

Static Ellipsoid

The next step is to deform the sphere to an ellipsoid as this is a more realistic model of the heart. The sphere is deformed to an ellipsoid by streching by factor of 1.5 along the \(y\)-axis; all other parameters are the same as above. The results are summarised below.

Note that the results are qualitatively similar to those on the static sphere.