Advection-solver Steps

[1] Update marker location $(x_m, y_m)$ and spin $w_m$ using a 4th order Runge-Kutta scheme and the current velocity field $v_{x(i,j)_{vx}}$ and $v_{y(i,j)_{vy}}$.

[2] Update marker stress $\sigma_{xx,m}'$ and $\sigma_{xy,m}'$ for rotation using equation eq:stress-rotation.

[3] Update total marker strain $\epsilon_m$ and marker plastic strain rate $\dot{\epsilon}_m^{plastic}$ and plastic strain $\epsilon_m^{plastic}$ (Marker Strain Update Steps).

[4] Correct temperature in sticky air-water markers by setting temperature equal to the temperature at the top of the model domain.

Marker Strain Update Steps

[1] Update total marker strain after copying current marker strain $\epsilon_m$ to $\epsilon_m^o$:

\[\epsilon_m = \epsilon_m^o + \Delta t \sqrt{\dot{\epsilon}_{xx,m}^2 + \dot{\epsilon}_{xy(m)}^2}\]

[2] Update marker plastic strain rate $\dot \epsilon _m^{plastic}$ and plastic strain $\epsilon _{plastic,m}$ using eq:plastic-strain after copying current marker plastic strain to \epsilon _{plastic,m}^o``