## 2 Numerical Simulation  
### Mathematical Model  
#### 2.1.1 Liquid Phase Model  
During running, the drilling fluid flows through the differential motion assembly. The continuity equations and momentum equations are as follows [39]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{u }\right)=0 \tag{1}\]  
Figure 1: (**a**) Schematic diagram of the IPP-Coring system. (**b**) Schematic diagram of the working principle of the IPP-Coring system and the differential motion assembly. (**c**) Photographs of the IPP-Coring system and the differential motion assembly. \[\frac{\partial}{\partial t}\bigg{(}\vec{u}\bigg{)}+\
abla\cdot\bigg{(}\rho\vec{u} \vec{u}\bigg{)}=-\
abla P+\
abla\cdot\bigg{\{}\mu\bigg{[}\bigg{(}\
abla\vec{u}+ \
abla\vec{u}^{\top}\bigg{)}-\frac{2}{3}\
abla\cdot\vec{u}\,I\bigg{]}\bigg{\}}+ \rho\vec{g}+\vec{s}_{M} \tag{2}\]  
where \(\rho\) is the liquid phase density, \(\vec{u}\) represents the instantaneous velocity vector, \(P\) is pressure, \(\mu\) is the molecular viscosity, \(I\) is the unit tensor, \(\rho\vec{g}\) is the gravitational body force, and \(S_{M}\) is the added momentum caused by the discrete phase. The standard \(k\)-\(\varepsilon\) model is very popular in industrial flow and heat transfer simulations because of its robustness and economy, and it has shown reasonable accuracy for fully developed turbulent flow conditions [40]. For the differential motion assembly, during the IPP-Coring process for deep oil and gas exploration, the drilling fluid, which is characterized by a high velocity (pump displacement can reach 20-40 L/s), high pressure, and high solid particle content, circulates through it. Due to the high velocity and the resulting high Reynolds number, the flow in the differential motion assembly is usually turbulent. Thus, the standard \(k\)-\(\varepsilon\) model, which has reasonable accuracy for a wide range of turbulent flows and a fast convergence rate [40], was employed in this study to gain an insight into the flow characteristics in the differential motion assembly. The equations are as follows [39; 41; 42]:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{i }}=\frac{\partial}{\partial x_{i}}\bigg{[}\bigg{(}\mu+\frac{\mu_{T}}{\sigma_{ k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\bigg{]}+P_{k}-\rho\varepsilon+S_{k} \tag{3}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\varepsilon u _{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+ \frac{\mu_{T}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{ \partial x_{j}}\bigg{]}+C_{\varepsilon 1}\frac{\varepsilon}{P}P_{k}-C_{ \varepsilon 2}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{4}\]  
\[P_{k}=\mu_{T}\
abla\vec{u}\cdot\bigg{(}\
abla\vec{u}+\
abla\vec{u}^{\top} \bigg{)}-\frac{2}{3}\
abla\vec{u}\bigg{(}3\mu_{T}\
abla\cdot\vec{u}+\rho k \bigg{)} \tag{5}\]  
\[\mu_{T}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{6}\]  
where \(\mu_{T}\) is the turbulent eddy viscosity; \(k\) and \(\varepsilon\) are the turbulent kinetic energy and the turbulent dissipation rate, respectively; \(u_{i}\) denotes the velocity component in the \(i\)-direction; \(x_{i}\) and \(x_{j}\) are the coordinates in space; \(C_{\varepsilon 1},C_{\varepsilon 2},\) and \(C_{\mu}\) are model constants; \(S_{k}\) and \(S_{\varepsilon}\) represent the source terms defined by the discrete phase caused by the particle damping and turbulence eddies; \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) denote the turbulent Prandtl numbers; and \(P_{k}\) is the kinetic energy of the turbulence caused by the mean velocity gradients. In the standard \(k\)-\(\varepsilon\) model, \(C_{\mu}=0.99,C_{\varepsilon 1}=1.44,C_{\varepsilon 2}=1.92,\sigma_{k}=1.0,\) and \(\sigma_{\varepsilon}=1.3\)[40; 43]. After the velocity and pressure fields in the fluid domain were solved by the momentum equation, the \(k\)-\(\varepsilon\) model was used to obtain the turbulent kinetic energy and its dissipation rate.