Pulse

Motivation

Our motivation here is to show that the equations we have chosen can, in certain circumstances, give a pulse which travels from one side of our object to the other. This is what happens in normal heart behaviour: there is a pulse that occurs periodically at one side of the heart, travels across the heart and dissipates, and then the process is repeated.

The Model

The surface

The surface we use for this simulation is the pulsating ellipsoid with deformation $$\alpha =0.1$$ and period $$\beta =0.1$$ (see the section on moving surfaces for the definitions).

The Initial Conditions

We now change the initial conditions for both $$u$$ and $$v$$ to
\begin{align*} u_0 (x,y) &= 0 \\ v_0 ( x,y ) &=0. \end{align*}

The New Equations

We add an extra term to the right hand side of our $$u$$ equation:

\begin{equation*} (1-u) \mathbb I _{y >0.95} \mathbb I _ {u <0.99} \mathbb I _ { \mathbb Z < t < \mathbb Z +0.05} \end{equation*}

This term has the effect of creating an impulse in the value of $$u$$. We do not want multiple waves on our surface so we only allow the impulse if $$u$$ is decreasing ($$u<0.99$$) . We also confine the impulse to a small circular region in the upper hemisphere ($$y>0.95$$ ). The impulse occurs every 1 time unit and lasts for $$0.05$$ time units

\begin{align*} \dot u + u \nabla_\Gamma \cdot \mathbf{v} - a \Delta_\Gamma u &= f(u,v) + (1-u) \mathbb I _{y >0.3} \mathbb I _ {u <0.99} \mathbb I _ { \mathbb Z < t < \mathbb Z +0.05} \\ \dot v + v \nabla _ \Gamma \cdot \mathbf{v} &= g(u,v) \end{align*}