# Dynamic Pressure Analysis of Hemispherical Shell Vibrating in Unbounded Compressible Fluid
## 2 Control Equation and Solution  
### Formulation for Hemispherical Shell  
Figure 2 shows a mechanical model of a hemispherical shell vibrating in a radial direction in the unbounded fluid. Obviously, structure vibration affected the boundary of flow, while the pressure of flow affected the structure simultaneously. With an inviscid fluid hypothesis, the assumption of \(p=c^{2}\rho\), and the character of axial symmetry, the FSI coupling equation can be simplified as follows [22], where \(m\) represents the areal density of the shell, \(U\) represents the displacement of vibration, \(k\) represents the bending stiffness, \(u\) represents the velocity of fluid in the radius direction, and \(\Delta p\) represents the pressure differential between the upper surface and the bottom surface of the shell.  
\[m\frac{\partial^{2}U}{\partial^{2}t}+k\cdot\
abla^{4}U+\Delta p=0 \tag{1}\]  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho\cdot u}{\partial r}+\frac{ 2\rho\cdot u}{r}=0 \tag{2}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+\frac{1}{\rho} \frac{\partial p}{\partial r}=0 \tag{3}\]  
Equation (1) is the structural control equation of vibration. The first term of Equation (1) presents the inertial force (pressure) derived from the acceleration of motion, and the second term of Equation (1) shows the elastic force contributed from the deformation of the shell. According to [PERSON]'s second law, the combination of inertial force, elastic force, and pressure without damping force are in the condition of balance. The boundary conditions can be written as Equation (4), where the first term condition means the displacement is zero at the initial time, and the second term shows that the initial velocity of the shell is in compliance to the \(u_{s}(0)\) at the initial time.  
\[U|_{t=0}=0,\left.\frac{\partial U}{\partial t}\right|_{t=0}=u_{s}(0). \tag{4}\]  
Equations (2) and (3) are fluid control equations, which are well known as Naiver-Stokes equations (3D axial symmetry), where the density, static pressure, sound speed, and velocity of the fluid is denoted by \(\rho\), \(p\), \(c\), and \(u\), respectively. Equation (2) was derived from the conservation of mass, and Equation (3) was deduced from the conservation of momentum. Let:  
\[\frac{\partial U}{\partial t}=u_{s}(t), \tag{5}\]  
The velocity of fluid boundary vibration is, \(u=u_{s}(t)\). Here, it assumes that \(\mid u_{s}\mid=c\). Therefore, the boundary conditions can be rewritten as Equation (6), where, the first term shows that the boundary of fluid is just the position of the shell, and both the second and third terms present that on the boundary of the far field, the velocity and pressure are approximate to the initial condition 0, \(p_{0}\), respectively.  
\[u|_{r=r_{s}}=u_{s}(t),\left.u\right|_{r=R->\infty}=0,\left.p\right|_{r=R-> \infty}=p_{0} \tag{6}\]According to the physical conditions, the boundary of the fluid is the structural shell itself actually; thus, the initial value of the radius is the radius of the hemispherical shell, \(r_{s}\), therefore:  
\[r_{s}(t)=r_{s}+U(t), \tag{7}\]  
where \(r_{s}(t)\) is the radius of the fluid boundary, which varies with time. Since \(U\) is taken into account in a small value, \(r_{s}(t)\) can be denoted as \(r_{s}\). The initial condition can be expressed by:  
\[\rho|_{t=0}=\rho_{0},\,p|_{t=0}=p_{0} \tag{8}\]