# Quantitative Analysis of the Complex Time Evolution of a Camphor Boat
## 2 Reduction of the Two-Dimensional Reaction-Diffusion Model to the One-Dimensional One
We assume that the values of \(D_{1}\) and \(D_{2}\) (\(D_{1}\ll D_{2}\)) can differ by a few orders of magnitude [14]. The dissipation term combines camphor sublimation from the water surface with its dissolution in the bulk water. This rate of camphor dissipation from the surface at the point \((x,y)\) is proportional to \(c(x,y,t)\) thus:  
\[\{\text{camphor dissipation}\}\propto c(x,y,t)=k(x,y,t)\cdot c(x,y,t) \tag{3}\]  
The proportionality constant \(k\) can depend on the position and time because dissipation from the free water surface is different from that in the area covered by the object. It is common to assume that:  
\[k(x,y,t)=\left\{ \begin{array}{ll}k_{1},&(x,y)\in\Omega_{\text{B}}(x,y,t)\\ k_{2},&(x,y)\
otin\Omega_{\text{B}}(x,y,t)\end{array} \right., \tag{4}\]  
where \(k_{1}\) is the constant outflow rate to the water phase under the boat, and \(k_{2}\) is the constant sublimation rate beyond the boat to the air phase. In the numerical experiments in this paper, we followed the assumption \(k_{1}=0\) in accordance with the paper [14]. The source term describes the inflow of camphor and it also depends on position and time via the location of camphor source. Let us assume that the function \(\Omega_{\text{S}}(x,y,t)\) describes the location of the source as the function of time. Thus: COMMENT  
\[\{\text{camphor inflow from the source}\} =S(x,y,t)\] \[=\left\{ \begin{array}{ll}S_{0}\cdot(c_{\text{max}}-c(x,y,t)),& (x,y)\in\Omega_{\text{S}}(x,y,t)\\ 0,&(x,y)\
otin\Omega_{\text{S}}(x,y,t)\end{array} \right., \tag{5}\]  
where \(c_{\text{max}}\) is the maximum surface concentration of camphor at the specific conditions, and the factor \((c_{\text{max}}-c(x,y,t))\) takes into account the decrease in camphor dissipation if its concentration at the source is close to \(c_{\text{max}}\). Combining the terms discussed above, we obtain:  
\[\frac{\partial c}{\partial t}(x,y,t)= \
abla(D(x,y,t)\
abla c(x,y,t))-k(x,y,t)\cdot c(x,y,t)\] \[+X_{\Omega_{\text{S}}}(x,y,t)S_{0}\cdot(c_{\text{max}}-c(x,y,t)). \tag{6}\]  
where \(X_{\Omega_{\text{S}}}(x,y,t)\) is the characteristic function of camphor source location. Equations describing the time evolution of parameters defining the location of the object (\(p_{i}\), including cartesian coordinates and angles) are of the Newton type and include forces and torques generated by gradients of the surface tension \(\gamma(x,y)\). They should also include friction resulting from the water phase. Formally, they can be formulated as:  
\[\frac{d^{2}p_{i}}{dt^{2}}\sim\{\text{forces and torques depending on }\gamma(x,y)\}-\{\text{friction terms}\}. \tag{7}\]  
The force acting on the object is generated by a difference in the surface tension around its boundary \(\delta\Omega_{B}(x,y,t)\) and can be calculated as:  
\[\{\text{force}\}=\int_{\delta\Omega_{B}(x,y,t)}\gamma(c(x,y,t))\;\;d\mathbf{s}, \tag{8}\]  
where \(d\mathbf{s}\) is a normal outward vector element of the object boundary. Similarly, a torque acting on the object is calculated with respect to an arbitrarily selected axis O (cf. Figure 2) as:  
\[\{\text{torque}\}=\int_{\delta\Omega_{B}(x,y,t)}\gamma(c(\mathbf{r},t))\;\; \mathbf{r}\times d\mathbf{s}. \tag{9}\]  
where \(\mathbf{r}=(x,y)\) is a vector pointing towards the surface element. Friction terms are standard and they can include translational and angular velocities and nonlinear combinations. Figure 2: A section of the Petri dish (the black arc) illustrating the geometry of the experimental system. The camphor boat (blue rectangle, cf. Figure 3) is located at the end of an arm (green line, radius \(r_{0}=70\) mm) and can rotate around the vertical axis at point O. The trajectory of the boat center is illustrated with a red arc. The boat width is \(2a=4\) mm. The dashed blue rings show trajectories of two points located below the arm at the opposite edges of the boat.