# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 2 Physical and Mathematical Models  
The subject of the study was a multilayer elliptical system, as shown in Figure 1, where \(\eta\), \(\psi\) are the elliptical-cylindrical coordinates, whereas \(a_{1}\), \(a_{2}\)... \(a_{N}\) and \(b_{1}\), \(b_{2}\)... \(b_{N}\) are, respectively, the successive semi-major and semi-minor axes of ellipses. All ellipses making up the model have common focuses (\(\pm c\),0), and their individual perimeters are described with the equations \(\eta=\eta_{i}=const_{i}\) for \(i=1\), \(2\)... \(N\). In each layer of the system (Figure 1), there is a uniform heat source of efficiency \(g_{i}\) for \(i=1\), \(2\)... \(N\). It is assumed that the length \(l\) of the system under consideration in the \(z\) axis direction is much greater than the length of the external ellipse axis (\(l>>2a_{\rm N}\)). This means that the thermal field is two-dimensional and depends on the coordinates \(\eta\), \(\psi\). The mathematical model of the thermal field was formulated with respect to increments \(v_{i}(\eta,\,\psi)\) related to ambient temperature \(T_{a}\)  
\[v_{i}(\eta,\,\psi)=T_{i}(\eta,\,\psi)-T_{a}, \tag{1}\]  
where \(T_{i}(\eta,\,\psi)\) is the two-dimensional stationary distribution of the thermal field in the \(i\)-th layer of the system. The stationary temperature increment in the model's \(i\)-th layer is described by the heat equation (Poisson's) [3; 32; 33; 34], which, in the elliptical-cylindrical system and assuming \(\lambda_{i}=const_{i}\), has the following form  
\[\frac{1}{c^{2}(\cosh^{2}\eta-\cos^{2}\psi)}\bigg{(}\frac{\partial^{2}v_{i}( \eta,\,\psi)}{\partial\eta^{2}}+\frac{\partial^{2}v_{i}(\eta,\,\psi)}{\partial \psi^{2}}\bigg{)}=-\frac{g_{i}}{\lambda_{i}},\text{for}\eta_{i-1}\leq\ \eta\ \leq\ \eta_{i},\ 0\leq\psi\ \leq 2\pi,i=1,\ 2,\ \ldots\ N \tag{2}\]where \(\eta_{0}=0\), \(\lambda_{i}\) is the thermal conductivity of the \(i\)-th layer. It is assumed that on the system's external surface for \(\eta=\eta_{N}\), the exchange of heat with the environment takes place according to [PERSON]'s law [3]. The above-mentioned energy exchange is described by the Hankel boundary condition [3; 33], which in the elliptical-cylindrical coordinates has the following form  
\[\frac{1}{c\sqrt{(\cosh^{2}\eta-\cos^{2}\psi)}}\left.\frac{\partial v_{N}(\eta, \psi)}{\partial\eta}\right|_{\eta=\eta_{N}}=-\frac{\alpha}{\lambda_{N}}v_{N}( \eta=\eta_{N},\psi),\ \text{for}0\leq\psi\ \leq 2\pi, \tag{3}\]  
where \(\alpha\) is the total heat transfer coefficient (the sum of the convective and radiative coefficients [34]). The proposed method also enables the analysis of a system with the boundary condition of the first kind (Dirichlet's). For this purpose, it is necessary to substitute \(\alpha\)\(\rightarrow\)\(\infty\) in Equation (3). After making the respective transformation, this will lead to the relationship \(T_{N}(\eta=\eta_{N},\psi)=T_{a}\). In physical interpretation, the latter equation corresponds to a system with constant temperature on the system's outer perimeter, \(\eta=\eta_{N}\). Individual elliptic layers (Figure 1) adhere closely to each other. Therefore, the conditions of the continuity of increment in temperature and heat flux at the interface \(\eta=\eta_{i}\) between the layers \(i\) and \(i+1\) are satisfied  
\[v_{i}(\eta_{i},\psi)=v_{i+1}(\eta_{i},\psi)\ \ \text{for}\ \ 0\leq\psi\ \leq 2\pi,\ i=1,\ 2,\ \ldots\ N-1, \tag{4}\]  
\[\lambda_{i}\left.\frac{\partial v_{i}(\eta,\psi)}{\partial\eta}\right|_{\eta= \eta_{i}}=\lambda_{i+1}\left.\frac{\partial v_{i+1}(\eta,\psi)}{\partial\eta }\right|_{\eta=\eta_{i}}\text{for}\ 0\leq\psi\ \leq 2\pi,\ i=1,\ 2,\ \ldots\ N-1. \tag{5}\]  
Formulas (1)-(5) form a mathematical model of the temperature field in the system. Figure 1: Layered elliptical structure. The solution of Equation (2) is the sum of its particular and general integrals [29]. The particular integral has been determined by the method of variation of parameters [29], while the general integral has been determined using the method of separation of variables [29; 32].