## 2 Governing Equations  
The computer simulation of a polymer in the UL 94 scenario is an extremely complex process involving many phenomena, such as fluid flow, heat transfer, material degradation, and flame chemistry. Such simulation requires two different kinds of computational modules: one that deals with the solid phase (i.e., the polymer itself) to calculate the polymer motion, the thermal degradation, and the volatile release (pyrolysis), and a second one that models the combustion process in the flame formed by the mixing of volatiles in the surrounding air. The processes in the polymer (solid) and the gas phase are interdependent, and, thus, the two computational modules must be robustly and efficiently coupled. Next, the mathematical formulation describing the phenomena involved during the burning of a polymer in a fire is presented. Let \(\Omega\subset R^{3}\) be a bounded domain containing the air \(\Omega_{a}\) and the polymer \(\Omega_{p}\) (Figure 1). Note that they are treated as viscous fluids (this assumption becomes satisfactory for higher temperatures and regarding a narrow range of shear rates such as those experienced in the UL 94 test [12]). The problem presented above is governed by the following three-dimensional and unsteady equations written in a compact form as follows:  
\[\frac{\partial\rho\phi}{\partial t}+\
abla_{x}\cdot(\rho\mathbf{v}\phi)= \
abla_{x}\cdot(H_{\phi}\
abla_{x}\phi)+S_{\phi}\quad\text{in }\Omega\times(0,t) \tag{1}\]  
where the variables \(\phi\), \(H_{\phi}\) and \(S_{\phi}\) are defined as follows: \begin{tabular}{c c c c c} \hline \hline
**Transport of** & \(\mathbf{\phi}\) & \(H_{\mathbf{\phi}}\) & \(S_{\mathbf{\phi}}\) & \\ \hline Mass & 1 & 0 & \(\epsilon_{v}\) & (2) \\ Momentum & \(\mathbf{v}\) & \(\mu\) & \(-\
abla_{x}p+\mu\
abla_{x}(\
abla_{x}^{T}\mathbf{v})+\rho\mathbf{f}\) & (3) \\ Energy & \(T\) & \(\kappa/C\) & \(\gamma[w_{T}/C+(\
abla.Q_{R})/C]+(1-\gamma)Q_{v}/C\) & (4) \\ Species & \(Y_{k}\) & \(\kappa/C\) & \(-w_{k}/C\) _for \(k=F\) and \(O\)_ & (5) \\ \hline \hline \end{tabular}   
with \begin{tabular}{c c c} \hline \hline
**Symbol** & \multicolumn{2}{c}{**Parameter**} \\ \hline \(\
abla_{x}=\left\{\partial_{x_{i}}\right\}_{i=1}^{3}\) & \multicolumn{2}{c}{vectorial operator of spatial derivatives} \\ \(\rho\) & \multicolumn{2}{c}{density} \\ \(p\) & \multicolumn{2}{c}{pressure} \\ \(\mu\) & \multicolumn{2}{c}{viscosity} \\ \(\mathbf{f}\) & \multicolumn{2}{c}{gravity force} \\ \(C\) & \multicolumn{2}{c}{capacity} \\ \(\kappa\) & \multicolumn{2}{c}{thermal conductivity} \\ \(w_{T}\) & \multicolumn{2}{c}{rate of production of heat 1  \\ \(Q_{R}\) & \multicolumn{2}{c}{radiative heat flux} \\ \(\epsilon_{v}\) & \multicolumn{2}{c}{mass loss} \\ \(Q_{v}\) & \multicolumn{2}{c}{heat absorbed due to pyrolysis} \\ \(A\) & \multicolumn{2}{c}{pre-exponential function} \\ \(E\) & \multicolumn{2}{c}{activation energy} \\ \(R\) & \multicolumn{2}{c}{universal gas constant} \\ \(\alpha\) & \multicolumn{2}{c}{absorption coefficient} \\ \(\sigma\) & \multicolumn{2}{c}{Stefan-Boltzmann constant} \\ \hline \hline \end{tabular}   
The parameters presented in the table above are considered constant, except for the viscosity of the polymer, which is assumed to be a function of temperature. This is defined in Section 4. The parameter \(\gamma\) is set to 0 in \(\Omega_{p}\) and to 1 in \(\Omega_{a}\). The following table presents the source terms:The incident radiation G adopting the P1 method [13] is computed by solving  
\[-\
abla\cdot\left(\frac{1}{3\alpha}\
abla\mathrm{G}\right)+\alpha G=4\alpha \sigma\mathrm{T}^{4} \tag{12}\]  
which is subject to the Marchak boundary condition [13]. Note that the pyrolysis model used in this work (see Equation (7)) is based on simple, thermally activated, single-step decomposition. Its validation is presented in [12]. Although it is a simple model, the numerical results presented in [6; 7; 12; 14; 15; 16; 17] have revealed that it is good enough to model the pyrolysis during burning.