# A Validated Thermal Computational Fluid Dynamics Model of Wine Warming in a Glass
## 3 Simulation Model
### Conjugate Heat Transfer  
Describing the non-isothermal flow of air and wine involves conserving mass, momentum, and energy. The differential form of the Navier-Stokes equations reads  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{i})}{ \partial x_{i}} =0\,, \tag{1}\] \[\frac{\partial(\rho v_{i})}{\partial t}+\frac{\partial(\rho v_{i} v_{j})}{\partial x_{j}} =-\frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ij}}{ \partial x_{j}}+\rho g_{i}\,,\] (2) \[\frac{\partial(\rho h)}{\partial t}+\frac{\partial(\rho hv_{i})}{ \partial x_{i}} =\frac{\partial p}{\partial t}+\frac{\partial(\tau_{ij}v_{j})}{ \partial x_{i}}+\rho g_{i}v_{i}-\frac{\partial q_{i}^{\rm S}}{\partial x_{i}}\,, \tag{3}\]  
where \(t\) is time, \(x_{i}\) are cartesian coordinates, \(v_{i}\) are fluid velocities, \(\rho\) represents density, \(p\) is pressure, \(h\) is total enthalpy, and \(g_{i}\) is the gravitational acceleration vector. For surface heat fluxes \(q_{i}^{\rm S}\), [PERSON]'s law in isotropic configuration,  
\[q_{i}^{\rm S}=-\kappa\frac{\partial T}{\partial x_{i}} \tag{4}\]  
is applied, where \(\kappa\) represents the thermal conductivity and \(T\) is temperature. The viscous stress tensor \(\tau_{ij}\) of a Newtonian fluid reads  
\[\tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{ \partial v_{j}}{\partial x_{i}}\right)-\frac{2}{3}\mu\frac{\partial v_{k}}{ \partial x_{k}}\delta_{ij}\,, \tag{5}\]  
where \(\mu\) represents dynamic viscosity and \(\delta_{ij}\) is the Kronecker delta. In Equation (3), viscous heating \(\frac{\partial}{\partial x_{i}}(\tau_{ij}v_{j})\) can be omitted for flows without enormous velocity gradients [22], and work by gravitational forces \(\rho g_{i}v_{i}\) are neglected by the solver [23]. The gravitational body force \(\rho g_{i}\) remains included, as buoyant effects are expected to dominate. All fluid flows are assumed to remain in the laminar regime. In addition to the constitutive material law for the fluid, compared to Equation (5), the thermal and caloric equations of state for density \(\rho=\rho(p,T)\) and specific heat capacity \(c_{p}=c_{p}(p,T)\) close the system of equations describing non-isothermal fluid flow. Boundary conditions and material properties for the given problem are presented in the following sections. For conjugate heat transfer simulations in solids, the total enthalpy equation (Equation (3)) can be simplified with \(v_{i}=0\) and \(\mathrm{d}p=0\) to  
\[\frac{\partial(\rho h)}{\partial t}=\frac{\partial}{\partial x_{i}}\left( \kappa\frac{\partial T}{\partial x_{i}}\right). \tag{6}\]  
Thermal radiation within the space around the glass represents the third mode of heat transfer besides convection and conduction. In the limit case of a non-participating medium, the net radiative heat transfer between surfaces \(q_{\rm rad,net}\) can be included in the boundary condition for Equation (3). It can be derived from the radiative energy conservation at the surface \(k\) of a gray body. Let \(G_{k}\) be the incoming radiative heat flux, also called the irradiation. The radiosity \(J_{k}\), the total radiative heat flux leaving the surface, is  
\[J_{k}=\epsilon_{k}E_{\rm b,\,\,k}+\mathcal{R}_{k}G_{k}\,, \tag{7}\]  
where \(E_{\rm b,\,\,k}=\sigma T^{4}\) is the black body emissive power, \(\epsilon_{k}\) is the emissivity, and \(\mathcal{R}_{k}\) is the fraction of irradiation diffusely reflected by the surface. Taking the difference between radiosity and irradiation yields the net outgoing radiative heat flux  
\[q_{k,{\rm rad,net}}=J_{k}-G_{k}=\epsilon_{k}E_{\rm b,\,\,k}+\mathcal{R}_{k}G_ {k}-G_{k}\,. \tag{8}\]Obeying the energy balance for opaque surfaces, \(\mathcal{A}_{k}+\mathcal{R}_{k}=1\), where \(\mathcal{A}_{k}\) is the body's absorptivity, and applying [PERSON]'s law for gray bodies (\(\mathcal{A}_{k}=\epsilon_{k}\)) [24] yields  
\[q_{k,\text{rad,net}}=\epsilon_{k}E_{\text{b},k}-\mathcal{A}_{k}G_{k}=\epsilon_{ k}(E_{\text{b},k}-G_{k}) \tag{9}\]  
for the additional radiative heat fluxes in the boundary condition for Equation (3).