# Quantitative Analysis of the Complex Time Evolution of a Camphor Boat
## 3 Results
### Selection of Model Parameters  
The boat motion is a complex phenomenon, and its model involves many parameters. The selection of the majority of their values can be determined in independent experiments. The droplet mass can be measured directly. The diffusion coefficient \(D_{2}\) on the free water surface was estimated by observing tracer particles spreading out after a camphor disk was placed on the surface [7]. The source term \(S_{0}\) was obtained by measuring the mass of the camphor pill as the function of time. Boat friction can also be estimated in a directexperiment [7]. We hope experimental information on the other quantities needed for the model will be available shortly. For example, the maximum surface concentration of camphor \(c_{\text{max}}\) and the functional relationship between the surface camphor concentration and air-water interfacial tension and parameters describing it can be obtained from the quasi-elastic laser scattering technique [21]. However, there are also parameters such as the ratio between surface diffusion below the boat and on the free water surface \(\Delta=D_{1}/D_{2}\), the outflow rate of camphor to the water phase under the boat \(k_{1}\) or the nonlinear contribution to friction \(\eta_{0}\), which seems to be hard to measure directly. Here, we applied the simplest approximation to \(\eta_{0}\) and \(k_{1}\), assuming that both quantities were small and could be regarded as equal to zero. To observe boat motion, the value of \(\Delta\) has to be positive, and here, we fixed its value by the agreement between boat speed observed in experiments and in simulations. The same approach was used to determine \(\gamma(c)\). The results of numerical simulations of boat evolution equations presented in the first paper, where the pulsating mode of boat motion was reported [14], were obtained only for the dimensionless parameters because they were focused on qualitative agreement with experimental results. In [14], as well as in the other papers cited below, the one-dimensional model of the boat motion based on Equations (26) and (27) was used. The results confirmed continuous motion for small \(d\) and pulsating motion when \(d\) is large. The same model but with modified values of model parameters has also been used in our recent publication [15]. After the modification of model parameters, the speed oscillations appeared for intermediate values of \(d\). A step towards the quantitative description of camphor boat motion was reported in [7] and followed by [12], where the values of some model parameters include: \(D_{2}=3940.0\) mm\({}^{2}\) s\({}^{-1}\), \(k_{2}=0.018\) s\({}^{-1}\), \(\mu_{0}=0.0175\) g s\({}^{-1}\) mm\({}^{-1}\) and the rate of supply of camphor to the system \(S_{0}\cdot c_{\text{max}}\) was experimentally measured. In [12], the camphor outflow from disk-shaped pill with 3 mm radius was approximated as 4 mmol s\({}^{-1}\). The boat mass \(m_{0}=0.064\) g corresponds to the 15 mm boat used in our experiments. For the value of \(\Delta=0.0001\), we follow [14]. These values were used in qualitative calculations [15] to match types of boat motion with experimental results. We used two methods to determine the parameter values for a quantitative model. One of them aimed to transform dimensionless parameters used in [15] to the dimensional form. Such a procedure gives:  
\[ \begin{split}& D_{2}=390.0\text{ mm}^{2}\text{ s}^{-1},\quad\Delta=0.0001,\quad k_{2}=1.73\text{ s}^{-1},\quad\mu_{0}=0.0554\text{ g}\,\text{s}^{-1}\,\text{mm}^{-1},\\ &\eta_{0}=0.0\text{ mm}^{-1}\text{ s},\quad S_{0}=0.0346\text{ s}^{-1},\quad m_{0}=0.064\text{ g}.\end{split}  \tag{28}\]  
In another approach, we used the experimental results of [7], the values of \(\Delta\) and \(\eta_{0}\) mentioned above and adjusted the hard-to-measure parameter \(S_{0}\) to obtain the best match with \(v(t)\) measured in experiments. Now, the parameter values are:  
\[