BlinkingRotlet¶
Systems · Continuous · Climate geophysics
Blinking-rotlet flow — a model of chaotic advection in Stokes mixing.
Dimension: 3 · Reference: Meleshko & Aref (1996), Phys. Fluids 8, 3215-3217
Equations¶
\[
\begin{aligned}
\dot{r} &= \sigma \left(- b \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(- \frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right)}{\left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} + \frac{1}{b^{2} + 2 b r \cos{\left(\theta \right)} + r^{2}} - \frac{1}{a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}}\right) \sin{\left(\theta \right)} + b \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \left(- \frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right)}{\left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} + \frac{1}{b^{2} - 2 b r \cos{\left(\theta \right)} + r^{2}} - \frac{1}{a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}}\right) \sin{\left(\theta \right)}\right) \\
\dot{theta} &= \frac{\sigma \left(\left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(\frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right) \left(b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}\right)}{\left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} - \frac{b \cos{\left(\theta \right)} + r}{b^{2} + 2 b r \cos{\left(\theta \right)} + r^{2}} + \frac{b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}}{a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}} + \frac{bc r \left(a^{2} + b^{2} - \frac{2 b^{2} r^{2}}{a^{2}}\right)}{a^{2} \left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)}\right) + \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \left(\frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right) \left(- b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}\right)}{\left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} - \frac{- b \cos{\left(\theta \right)} + r}{b^{2} - 2 b r \cos{\left(\theta \right)} + r^{2}} + \frac{- b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}}{a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}} + \frac{bc r \left(a^{2} + b^{2} - \frac{2 b^{2} r^{2}}{a^{2}}\right)}{a^{2} \left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)}\right)\right)}{r} \\
\dot{t} &= 1
\end{aligned}
\]
Parameters¶
| parameter | default |
|---|---|
a |
1.0 |
b |
0.5298833894399929 |
bc |
1.0 |
sigma |
-1.0 |
tau |
3.0 |
Variables: r, theta, t
