# Sintering of Two Viscoelastic Particles: A Computational Approach
## 3 Governing Equations
### Dimensionless Equations  
To scale the governing equations, we introduce characteristic constant values from the problem parameters. We define the characteristic length as \(x_{\mathrm{c}}=R\), the characteristic velocity as \(u_{\mathrm{c}}=\gamma/\eta_{0}\), the characteristic stress as \(\sigma_{\mathrm{c}}=\gamma/R\), the characteristic pressure as \(p_{\mathrm{c}}=\gamma/R\) and a characteristic time \(t_{\mathrm{c}}\). Herein, \(\eta_{0}\) is the zero-strain rate viscosity and is defined as \(\eta_{0}=\mu\) for Newtonian fluids and \(\eta_{0}=\eta_{\mathrm{s}}+\eta_{\mathrm{p}}\) for viscoelastic fluids, where \(\eta_{\mathrm{p}}=G\lambda\) is the polymer viscosity. A dimensionless variable can be obtained by dividing the original variable by the characteristic value, for example:  
\[x^{*}=\frac{x}{x_{\mathrm{c}}}=\frac{x}{R}. \tag{12}\]  
The dimensionless variable is represented by an asterisk superscript.  
Scaling the boundary condition of Equation (8) leads to:  
\[\frac{dx_{\Gamma}^{*}R}{dt^{*}t_{\mathrm{c}}}=\mathbf{u}^{*}\frac{\eta_{0}}{\gamma}. \tag{13}\]  
Since both sides are assumed to be of the same order of magnitude, a definition for the characteristic time \(t_{\mathrm{c}}=R\eta_{0}/\gamma\) follows, and Equation (13) reduces to:  
\[\frac{dx_{\Gamma}^{*}}{dt^{*}}=\mathbf{u}^{*}. \tag{14}\]  
Scaling the governing Equations (1) and (2) leads to:  
\[\text{La}\ \frac{\text{D}\mathbf{u}^{*}}{\text{D}t^{*}}=\
abla^{*} \cdot\mathbf{\sigma}^{*}+\text{Bo}\,\mathbf{e}_{\mathrm{g}} \quad\text{in}\ \,\Omega, \tag{15}\] \[\
abla^{*}\cdot\mathbf{u}^{*}=0\quad\text{in}\ \,\Omega, \tag{16}\]  
respectively. The two dimensionless numbers defining this flow problem are the Laplace number \(\text{La}=\rho\gamma R/\eta_{0}^{2}\), which is the ratio of the surface tension to the inertial forces, and the Bond number \(\text{Bo}=\rho R^{2}g/\gamma\), which is a measure of the gravity forces versus the surface tension forces. Both of these dimensionless numbers are negligibly small if we use the material parameters of polyamide12 (PA12) powder (Table 1), which is most often used in SLS, i.e., \(\text{La}=\mathcal{O}(10^{-8})\) and \(\text{Bo}=\mathcal{O}(10^{-3})\). Equation (15) reduces to:  
\[-\
abla^{*}\cdot\mathbf{\sigma}^{*}=\mathbf{0}\quad\text{in}\ \Omega. \tag{17}\]  
For the Newtonian case, the scaled Cauchy stress tensor Equation (3) is:  
\[\mathbf{\sigma}^{*}=-p^{*}\mathbf{I}+\left(\
abla^{*}\mathbf{u}^{*}+(\
abla^{*}\mathbf{u}^{*}) ^{\text{T}}\right). \tag{18}\]  
Scaling the Cauchy stress tensor of viscoelastic fluids Equation (4) leads to:  
\[\mathbf{\sigma}^{*}=-p^{*}\mathbf{I}+\beta\left(\
abla^{*}\mathbf{u}^{*}+(\
abla^{*}\mathbf{u} ^{*})^{\text{T}}\right)+\mathbf{\tau}_{\text{P}}^{*}, \tag{19}\]  
where \(\beta=\eta_{\text{s}}/\eta_{0}\). In this work, we choose \(\beta=0.01\). The dimensionless description of Equation (6), the evolution of the conformation tensor \(\mathbf{c}\) for the Giesekus constitutive equation, is:  
\[\text{De}\,\overset{\triangledown}{\mathbf{c}}+\mathbf{c}-\mathbf{I}+\alpha(\mathbf{c}-\mathbf{I} )^{2}=\mathbf{0}. \tag{20}\]  
Besides the material parameter \(\alpha\), we find the Deborah number \(\text{De}=\lambda\gamma/(\eta_{0}R)\) in Equation (20), which is the ratio of the time scale of the fluid response to that of the process. For the XPP model, the dimensionless description of Equation (7) is:  
\[\text{De}\,\overset{\triangledown}{\mathbf{c}}+2\xi\exp[\
u\sqrt{\text{tr}(\mathbf{c} )/3}-1]\left(1-\frac{3}{\text{tr}(\mathbf{c})}\right)\mathbf{c}+\left(\frac{3\mathbf{c}}{ \text{tr}(\mathbf{c})}-\mathbf{I}\right)=\mathbf{0}. \tag{21}\]  
The dimensionless groups in Equation (21) are the Deborah number De as defined before, the ratio between the relaxation time of the backbone orientation and that of the stretch \(\xi=\lambda/\lambda_{\text{s}}\) and \(\
u\), which depends on the number of arms \(q\).  
For the readability of this document, we omit the asterisks in the notation of the dimensionless variables.