# The Variation of Surface Shape in the Gas Jet Forming
## 2 Principles of the Gas Jet Forming and Related Inferences
### Inference of the Relationship between Jet Parameters and Mirror Blank Parameters
\begin{array}{rl}P=\frac{1}{2}\rho_{G}V_{0}^{2}&r<r_{cor}\quad(a)\\ P=\frac{1}{2}\rho_{G}V_{0}^{2}\left(\frac{d}{h}\right)\cdot e^{-c\left(\frac{d }{\rho}\right)^{2}}&r>r_{cor}\quad(b)\end{array} \right. \tag{4}\]  
where \(r_{cor}\) is the core zone radius, \(r_{cor}\propto(1/h)\), and \(\sigma\) is a constant (mm\({}^{2}\)). Equation (4) (a) describes the pressure distribution within the core zone, while Equation (4) (b) describes the pressure distribution outside the core zone to the boundary layer. According to Section 2.1 of this paper, the surface shape of the mirror blank is formed by the combined action of air pressure, surface tension, and the hydrostatic pressure of the liquid. Of these, the air pressure has a positive effect on the expansion of the surface. In contrast, the surface tension and static pressure have a negative effect by inhibiting the expansion of the surface. During the the gas jet forming process, the air pressure is stronger than the combined inhibitory effect of the surface tension and static pressure, causing a tendency for the surface profile to expand outwards until the three forms of action reach equilibrium. For this reason, Equation (4) can be modified to obtain Equation (5). \[P=\frac{\rho V_{0}^{2}d}{2 he^{\left(\frac{hh}{\sigma}\right)^{2}}}, \tag{5}\]  
According to Equation (5), the magnitude of the pressure acting on the surface of the liquid cell is related to the initial jet velocity \(V_{0}\). The greater the \(V_{0}\), the greater the resulting pressure \(P\), resulting in a larger diameter of the surface formed by the gas jet. It can be inferred that the diameter formed by the gas jet increases as the initial gas flow rate \(V_{0}\) increases. In addition to velocity, the pressure is also related to the nozzle's height from the  
Figure 3: Experimental results of the gas jet forming using the self-similarity region. liquid surface. Although the cross-sectional diameter of the flow stream increases as the nozzle's height from the liquid surface increases, the greater the nozzle height, the lower the pressure within the radial cross-section, resulting in a smaller diameter formed by the gas jet. The principal curvature is one of the key parameters describing the profile of an optical mirror, and it is not correlated with the diameter. Therefore, it is also necessary to analyse the law of change in the principal curvature of the surfaces formed in the gas jet forming. Due to the volume reduction in the potential core region and the effect of the hydrostatic pressure of the liquid, the principal curvature is determined by both the pressure in the potential core region and the pressure distribution generated by the flow streams in the vicinity of the core. By taking the derivative of \(P\) with respect to r in Equation (4) (b):  
\[\frac{dP}{dr}=-cr\sigma^{-2}\rho_{G}hdV_{0}^{2}\cdot e^{-c(\frac{th}{r})^{2}}, \tag{6}\]  
The second derivative of the above equation:  
\[\frac{d^{2}P}{dr^{2}}=cr\sigma^{-2}\rho_{G}V_{0}^{2}dh\cdot e^{-c(\frac{th}{r} )^{2}}(2 ch^{2}\sigma^{-2}r-1), \tag{7}\]  
According to the mathematical equation, the curvature \(K\) at each point of the airflow can be obtained from Equation (8). \[K=\frac{\left|\frac{d^{2}P}{dr^{2}}\right|}{\left[1+\left(\frac{dp}{dr}\right) ^{2}\right]^{\frac{3}{2}}}, \tag{8}\]  
Substituting Equations (6) and (7) into Equation (8) yields the curvature equation describing the pressure distribution of the gas jet acting on the fluid surface. \[K=\frac{\left|c\sigma^{-2}\rho_{G}V_{0}^{2}dh\cdot e^{-c(\frac{th}{r})^{2}}(2 ch^{2}\sigma^{-2}r-1)\right|}{\left[1+\left(-cr\sigma^{-2}\rho_{G}hdV_{0}^{2} \cdot e^{-c(\frac{th}{r})^{2}}\right)^{2}\right]^{\frac{3}{2}}}, \tag{9}\]  
The focus of this paper is on the preserving core region and the pressure distribution near the preserving core region (\(r\to 0\)). At this point, the principal curvature equation of the airflow stream is shown in Equation (10).