## 2 Theoretical Model  
The THM coupling problem involves the interaction between the temperature, flow, and solid deformation fields. The coupling effect between fields works through interface forces. The coupling solution is achieved through one-way coupling using two field-cross iterations [12].  
Continuity equation  
The conservation of mass in fluid flow is expressed through continuity equations:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{x})}{\partial x}+\frac {\partial(\rho u_{y})}{\partial y}+\frac{\partial(\rho u_{z})}{\partial z}=0 \tag{1}\]  
where \(u_{x}\), \(u_{y}\) and \(u_{z}\) represent the velocity components in three directions, \(t\) is time, and \(\rho\) is the density.  
Momentum equation  
For a given fluid element, the momentum change rate over time is equal to the sum of various external forces acting on the element.  
\[\frac{\partial(\rho u_{x})}{\partial t}+\
abla\bullet(\rho u_{x}\overset{ \rightarrow}{u})=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{xx}}{ \partial z}+\rho f_{x} \tag{2}\]  
\[\frac{\partial(\rho u_{y})}{\partial t}+\
abla\bullet(\rho u_{y}\overset{ \rightarrow}{u})=-\frac{\partial p}{\partial y}+\frac{\partial\tau_{xy}}{ \partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{xy}}{ \partial z}+\rho f_{y} \tag{3}\]  
\[\frac{\partial(\rho u_{z})}{\partial t}+\
abla\bullet(\rho u_{z}\overset{ \rightarrow}{u})=-\frac{\partial p}{\partial z}+\frac{\partial\tau_{xz}}{ \partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{ \partial z}+\rho f_{z} \tag{4}\]  
where \(\rho\) is the pressure on the fluid element, \(\tau_{xx}\), \(\tau_{xy}\) and \(\tau_{xz}\) are the components of viscous stress on the surface of a microelement caused by molecular viscosity, and \(f_{x}\), \(f_{y}\) and \(f_{z}\) are the unit mass forces in three directions.  
Energy equation  
\[\frac{\partial(\rho E)}{\partial t}+\
abla\bullet[\overset{\rightarrow}{u}( \rho E+p)]=\
abla\bullet[k_{eff}\
abla T-\sum_{j}h_{j}]_{j}+(\tau_{eff} \bullet\overset{\rightarrow}{u})]+S_{h} \tag{5}\]  
\[E=h-p/\rho+u^{2}/2 \tag{6}\]  
where \(E\) is the total energy of the fluid micro cluster, \(h\) is enthalpy, \(h_{eff}\) is the enthalpy of component \(j\), \(k_{eff}=k+k_{t}\), \(k_{t}\) is the effective thermal conductivity coefficient, \(J_{j}\) is the diffusion flux of component \(j\), and \(S_{h}\) includes chemical reaction heat and other user-defined volumetric heat source terms.  
Discrete solid structure's equation of motion The solid structure motion equation can be described as follows:  
\[\mathbf{M}_{\mathrm{s}}\ddot{r}+\mathbf{C}_{\mathrm{s}}r+\mathbf{K}_{\mathrm{s}}r +\mathbf{f}_{\mathrm{p}}+\mathbf{f}_{0}=0 \tag{7}\]  
where \(\mathbf{M}_{\mathrm{s}}\) is the displacement mass matrix of the structure, \(\mathbf{C}_{\mathrm{s}}\) is the damping matrix of the structure, \(\mathbf{K}_{\mathrm{s}}\) is the stiffness matrix of the structure, \(\mathbf{f}_{\mathrm{p}}\) is the nodal vector of fluid at the fluid-structure interface of the strike plate, and \(\mathbf{f}_{0}\) is the vector of the external motivator.  
Heat transfer control equation  
\[Q=kA\Delta t_{m} \tag{8}\]  
where \(k\) is the heat transfer coefficient, \(A\) is the heat transfer area, and \(\Delta t_{m}\) is the average temperature difference for heat transfer [13].  
Numerical calculations are conducted in the fluid domain of the gas regulator. The temperature load and flow field of the fluid-solid boundary calculated in the fluid domain are loaded onto the corresponding fluid-solid interface of the gas regulator. The temperature and heat flux fields on the inner wall of the gas regulator are also calculated. Then, the temperature field and thermal stress in the solid domain are calculated, and the final coupling analysis of the calculation results in the fluid and solid domains is conducted to obtain the deformation and stress situation of the overall structure of the gas pressure regulator.