# Thermal Accelerometer Simulation by the R-Functions Method
## 5 R-Functions Method and Transfinite Interpolation
This justification is based on variational principles [43] and is appropriate for the wide class of bases composed of both spectral and compactly supported functions. It is convenient to take normalized functions due to the fact that with them it is easier to evaluate some important differential characteristics of the solution, in particular, the Nusselt number, Nu. The local Nusselt number at a point of boundary \(\partial\Omega\) is expressed as:  
\[\mathrm{Nu}|_{\partial\Omega}=-\frac{\partial\theta}{\partial\mathbf{n}}\bigg{|} _{\partial\Omega}. \tag{28}\]  
Taking into account Equations (3) and (11) and properties of the normalized function \(\omega\), we can write that:  
\[\mathrm{Nu}|_{\partial\Omega}=-\frac{\partial}{\partial\mathbf{n}}\bigg{(} \omega\sum\limits_{n=0}^{N}c_{n}^{(k)}\chi_{n}+\Phi\bigg{)}\bigg{|}_{\partial \Omega}=-\bigg{(}\frac{\partial\omega}{\partial\mathbf{n}}\sum\limits_{n=0}^{ N}c_{n}^{(k)}\chi_{n}+\omega\sum\limits_{n=0}^{N}c_{n}^{(k)}\frac{\partial\chi_{n}}{ \partial\mathbf{n}}+\frac{\partial\Phi}{\partial\mathbf{n}}\bigg{)}\bigg{|}_{ \partial\Omega}=\bigg{(}\sum\limits_{n=0}^{N}c_{n}^{(k)}\chi_{n}-\frac{ \partial\Phi}{\partial\mathbf{n}}\bigg{)}\bigg{|}_{\partial\Omega}. \tag{29}\]  
Substituting Equation (26) into Equation (29) and taking into account properties of the normalized functions \(\omega_{C},\omega_{H}\), we get:  
\[\frac{\partial\Phi}{\partial\mathbf{n}}\bigg{|}_{\partial\Omega}=\left.\frac{ \partial}{\partial\mathbf{n}}\bigg{(}\frac{\theta_{H}\omega_{C}+\theta_{C} \omega_{H}}{\omega_{C}+\omega_{H}}\bigg{)}\right|_{\partial\Omega}=\left.\frac {1}{\omega_{C}+\omega_{H}}\right[\left(\theta_{H}\frac{\partial\omega_{C}}{ \partial\mathbf{n}}+\theta_{C}\frac{\partial\omega_{H}}{\partial\mathbf{n}} \right)-\Phi\left(\frac{\partial\omega_{C}}{\partial\mathbf{n}}+\frac{ \partial\omega_{H}}{\partial\mathbf{n}}\right)\right]\bigg{|}_{\partial\Omega}.\]  
Therefore on each part of the boundary \(\partial\Omega\) we have the simple expressions:  
\[\frac{\partial\Phi}{\partial\mathbf{n}}\bigg{|}_{\partial\Omega_{C}}=\left( \theta_{C}-\theta_{H}\right)\frac{1}{\omega_{H}}\bigg{|}_{\partial\Omega_{C} },\;\frac{\partial\Phi}{\partial\mathbf{n}}\bigg{|}_{\partial\Omega_{H}}= \left(\theta_{H}-\theta_{C}\right)\frac{1}{\omega_{C}}\bigg{|}_{\partial\Omega _{H}}, \tag{30}\]  
and  
\[\mathrm{Nu}|_{\partial\Omega_{C}}=\sum\limits_{n=0}^{N}c_{n}^{(k)}\left.\chi_{ n}\right|_{\partial\Omega_{C}}+\left(\theta_{H}-\theta_{C}\right)\frac{1}{ \omega_{H}}\bigg{|}_{\partial\Omega_{C}},\;\mathrm{Nu}|_{\partial\Omega_{H}}= \sum\limits_{n=0}^{N}c_{n}^{(k)}\left.\chi_{n}\right|_{\partial\Omega_{C}}+ \left(\theta_{C}-\theta_{H}\right)\frac{1}{\omega_{C}}\bigg{|}_{\partial\Omega _{H}}. \tag{31}\]  
Average Nusselt numbers on boundaries \(\partial\Omega_{C},\;\partial\Omega_{H}\) are found by integration of local Nusselt numbers along corresponding contours:  
\[\overline{\mathrm{Nu}}\big{|}_{\partial\Omega_{C}}=\frac{1}{|\partial\Omega_{C }|}\int\limits_{\partial\Omega_{C}}\mathrm{Nud}s,\;\overline{\mathrm{Nu}} \big{|}_{\partial\Omega_{H}}=\frac{1}{|\partial\Omega_{H}|}\int\limits_{ \partial\Omega_{H}}\mathrm{Nud}s. \tag{32}\]