## 2 Theoretical Assumptions in the Ansys Fluent Program [17; 18; 19; 20; 21; 22]
### Straight-Line Flows  
(a) Principle of conservation of mass;  
The mass conservation equation, also known as the continuity equation, can be written as follows:  
\[\frac{\partial\mathbf{p}}{\partial t}+\mathbf{\
abla}\times\left(\rho\overset{\to}{ \mathbf{v}}\right)=S_{\mathbf{m}} \tag{1}\]  
Equation (1) is the general form of the mass conservation equation and is valid for incompressible and compressible flows. The source is the mass \(S_{\mathbf{m}}\) added to the continuous phase from the dispersed second phase (e.g., due to evaporation of liquid droplets) and any user-defined sources. For 2D axisymmetric geometries, the continuity equation is given by Equation (2),  
\[\frac{\partial\mathbf{p}}{\partial t}+\frac{\partial}{\partial\mathbf{x}}(\rho\mathbf{v} _{\mathbf{x}})+\frac{\partial}{\partial\mathbf{r}}(\rho\mathbf{v}_{\mathbf{r}})+\frac{ \partial\mathbf{v}_{\mathbf{r}}}{\mathbf{r}}=S_{\mathbf{m}} \tag{2}\]  
where \(\mathbf{x}\) is the axial coordinate, \(\mathbf{r}\) is the radial coordinate, \(\mathbf{v}_{\mathbf{x}}\) is the axial velocity and \(\mathbf{v}_{\mathbf{r}}\) is the radial velocity. 2. The principle of conservation of momentum; The conservation of momentum in an inertial (non-accelerating) reference frame is described by Equation (3) \[\frac{\partial}{\partial t}\Big{(}\rho\overset{\rightarrow}{\overline{v}}\Big{)}+ \mathbf{\
abla}*\left(\rho\overset{\rightarrow}{\overline{v}}\overset{\rightarrow} {\overline{v}}\right)=-\mathbf{\
abla}\mathbf{p}+\mathbf{\
abla}*\left(\overline{\overline {\mathbf{r}}}\right)+\rho\overset{\rightarrow}{\overline{g}}+\overset{ \rightarrow}{\overline{\mathbf{F}}}\] (3) where \(\rho\) is the static pressure, \(\overline{\overline{\mathbf{r}}}\) is the stress tensor (described below), and \(\rho\overset{\rightarrow}{\overline{g}}\) and \(\overset{\rightarrow}{\overline{\mathbf{F}}}\) are the gravitational force and external body forces (for example, due to interactions with the dispersed phase), respectively. \(\overset{\rightarrow}{\overline{\mathbf{F}}}\) also includes other model-dependent source terms, such as porous media and user-defined sources. The stress tensor \(\overline{\overline{\mathbf{\tau}}}\) is given by \[\overline{\overline{\mathbf{\tau}}}=\mathbf{\mu}\bigg{[}\bigg{(}\mathbf{\
abla}\overset{ \rightarrow}{\overline{v}}+\mathbf{\
abla}\overset{\rightarrow}{\overline{v}} ^{T}\bigg{)}-\frac{2}{3}\mathbf{\
abla}*\overset{\rightarrow}{\overline{v}}\bm {I}\bigg{]}\] (4) where \(\mathbf{\mu}\) is the molecular viscosity, \(\mathbf{I}\) is the unit tensor, and the second term on the right is the volume dilation effect. 3. User-defined scalar transport equations (UDS); Ansys Fluent can solve the transport equation for any user-defined scalar (UDS) in the same way that it solves the transport equation for a scalar, such as the mass fraction of a species. In some types of combustion applications or, for example, modeling plasma-enhanced surface reactions, additional scalar transport equations may be needed. There are two basic subsections for UDS: single-phase and multi-phase flows. * Single-phase flow For any scalar \(\mathbf{\Phi}_{k}\), Ansys Fluent solves the equation \[\frac{\partial p\mathbf{\Phi}_{k}}{\partial t}+\frac{\partial}{\partial x_{i}} \bigg{(}\rho\mathbf{u}_{i}\mathbf{\Phi}_{k}-\mathbf{\Gamma}_{k}\frac{\partial\mathbf{\Phi}_{k} }{\partial x_{i}}\bigg{)}=S_{\mathbf{\omega}_{k}}\mathbf{k}=\mathbf{1},\ldots,\mathbf{N}\] (5) where \(\mathbf{\Gamma}_{k}\) and \(S_{\mathbf{\omega}_{k}}\) are the diffusion coefficient and source term given for each of the \(\mathbf{N}\) scalar equations. Note that \(\mathbf{\Gamma}_{k}\) is defined as a tensor in the case of anisotropic diffusivity. The diffusion term is, therefore, \(\mathbf{\
abla}*\left(\mathbf{\Gamma}_{k}*\mathbf{\Phi}_{k}\right)\). For isotropic diffusivity, \(\mathbf{\Gamma}_{k}\) can be written as \(\mathbf{\Gamma}_{k}\mathbf{I}\) where \(\mathbf{I}\) is the identity matrix.