# 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  
In this section, we describe approximations that allow us to reduce the model of the boat motion to a set of differential equations in which the most complex part is a reaction-diffusion equation for a one-dimensional spatial variable. Table 1 lists symbols used and model parameters. Rigorous numerical simulations of time evolution for a self-propelled object on the water surface are a hard numerical problem because the model should describe a few coupled phenomena. The driving force from the boat motion comes from the continuous flux of camphor molecules from the source through the surface layer to the environment. Camphor molecules decrease water surface tension depending on their local concentration. The local differences in surface tension generate Marangoni flows in the water phase and can produce forces and torques acting on the object and changing the position and orientation of the camphor source. The appearing flows influence camphor transport on the water surface. As a result, the description of object motion requires coupled equations for flows in all water volume (a vector function in three-dimensional space), the surface concentration of camphor represented by a scalar function of two spatial variables and a few scalar functions that describe object location and orientation [17; 18]. Of course, solving these equations is computationally complex. As a workable approximation to numerical simulations of the time evolution for a camphor-propelled object, the reaction-diffusion model was proposed in [8; 10]. Within this approach, the flow is not directly included in the model, and the surface transport of camphor is approximated by a reaction-diffusion equation with an \"effective\" diffusion coefficient [7; 19]. Therefore, the system is described by the surface concentration of camphor \(c(x,y,t)\) and all parameters that define the location of the considered object \(p_{i}(t)\). Here, \(x\) and \(y\) are coordinates defining a point on the surface. Figure 1: (**a**) An example of the setup for our experiments with a rotating camphor boat. The diameter of the Petri dish was 20 cm. The camphor boat was located at the end of a 7 cm-long arm and can rotate around the vertical axis located at the dish center. A typical water level was 5 mm. (**b**) The boat was 10 mm long and 4 mm wide, as seen below. It was cut from an aluminum U-shaped angle profile. The mass of the cuboid camphor pill (4 mm \(\times\) 4 mm \(\times\) 1 mm) was 0.008 g. The mass of the aluminum 15 mm boat profile was 0.056 g. The mass of the arm was 0.020 g, but we neglected it in calculations of inertia. The square marks a pill of camphor at the distance \(d\) from the stern [16]. The evolution equation for \(c(x,y,t)\) has the form:  
\[\frac{\partial c}{\partial t}= \
abla(D(x,y,t)\
abla c)-\{\text{camphor dissipation}\}\] \[+\{\text{camphor inflow from the source}\}. \tag{1}\]  
The first term of the equation denotes the diffusive transport of the camphor molecules on the water surface [7; 19]. Here, \(D(x,y,t)\) is the effective diffusion coefficient of the molecules at the water surface selected and introduced to approximate the flow-induced transport. As we discuss later, \(D(x,y,t)\) can depend both on the position and time because the transport of camphor molecules below the object can be different from those on the free surface. Let us assume that the time-dependent shape of the object (in the following a camphor boat) is given by the function \(\Omega_{\text{B}}(x,y,t)\). The diffusion coefficient \(D\) can be represented by a two-valued function:  
\[D(x,y,t)=\left\{ \begin{array}{ll}D_{1},&(x,y)\in\Omega_{\text{B}}(x,y,t)\\ D_{2},&(x,y)\
otin\Omega_{\text{B}}(x,y,t)\end{array} \right., \tag{2}\]  
where \(D_{1}\) and \(D_{2}\) are the diffusion coefficients below the boat and on the free water surface off the boat, respectively.