# Controlled Release Using Gas Detonation in Needle-Free Liquid Jet Injections for Drug Delivery
## 2 Fundamentals and Methods
The detonation propagates at \(D_{CJ}\) into the unburned reactants and impinges upon the plunger of the injector module at \(x=L\). The detonation wave reflection results in an even higher pressure on the injector's piston. The resulting maximum pressure occurring at the moment of reflection can be estimated using a simple gas dynamic analytical model based on the Rankine-Hugoniot equations for a constant-\(\gamma\) ideal gas [38; 39], i.e. :  
\[\frac{P_{R0}}{P_{CJ}}=\frac{5\gamma+1+\sqrt{17\gamma^{2}+2\gamma+1}}{4\gamma} \tag{1}\]  
where \(P_{CJ}\) is the CJ detonation pressure, \(P_{R0}\) the immediate reflected-detonation shock pressure, \(\gamma\) the ratio of specific heats. Taking an average \(\gamma=1.275\) at the detonation CJ state, \(P_{R0}\approx 2.54\ P_{CJ}\). The CJ pressure and reflected pressure versus initial pressure of the combustible are plotted in Figure 3 shown by dotted and solid lines, respectively. Due to the solid boundary at \(x=0\), a non-steady expansion wave--also referred to as the Taylor-Zel'dovich wave--follows behind the detonation lowering the pressure and temperature to match the boundary conditions. As shown in [40; 41] and also recently in [42], the immediate reflected pressure \(P_{R0}\) will decay exponentially toward the final expansion pressure, i.e.,:  
\[P_{R}(t)=\Big{(}P_{R0}-P_{f}\Big{)}exp\Big{[}-\frac{t}{\tau}\Big{]}+P_{f} \tag{2}\]  
Figure 3: Incident Chapman–Juguet (CJ) detonation pressure, reflected pressure and expansion pressure for stoichiometric C\({}_{2}\)H\({}_{2}\)/O\({}_{2}\) mixture at various initial pressures. Figure 2: A sketch showing different gas dynamic states of the detonation reflection process. where \(\tau\) is a time decay constant and \(P_{R}(t)\) asymptotes to \(P_{f}\) within the typical injection period. \(P_{f}\) is the pressure behind the Taylor-Zel'dovich wave, which can be calculated using the isentropic relationship across the expansion:  
\[P_{f}=P_{C}\!\left(\frac{c_{f}}{c_{C}}\right)^{2\gamma/(\gamma-1)} \tag{3}\]  
where the sound speed \(c_{f}\) can be obtained by noting \(u_{f}=0\) at \(x=0\) end wall and using the Riemann invariants along the C\({}^{-}\) characteristics for the detonation:  
\[\Gamma_{-}=u_{CJ}-\frac{2c_{CJ}}{\gamma-1}=-\frac{2c_{f}}{\gamma-1} \tag{4}\]  
where \(u_{CJ}\) is the flow velocity immediately behind the detonation. According to the Chapman-Jouguet criterion, \(u_{CJ}\) is equal to the detonation velocity \(D_{CJ}\) minus the sound speed at the CJ state, \(c_{CJ}\). Hence:  
\[c_{f}=\frac{\gamma+1}{2}c_{CJ}-\frac{\gamma-1}{2}D_{CJ} \tag{5}\]  
The expansion pressure \(P_{f}\) is also plotted in Figure 3 given by the dashed line. The initial reflected shock pressure \(P_{R0}\) provides a sufficiently large driving force to punch the skin and generate the injection jet with high inertia and pressure after the expansion process \(P_{f}\) for the rate constant drug delivery. To model the jet evolution and obtain its flow properties, a model was developed by [PERSON] and [PERSON] [10] by performing a mass balance and force analysis on the injection device. Assuming that the water is incompressible, the jet pressure can be described by integrating the following expression:  
\[\frac{dP_{jet}}{dt}=\frac{\left(B+P_{jet}\right)\frac{dx_{p}}{dt}-\frac{BA_{a}} {A_{p}}\frac{\sqrt{2P_{jet}}}{\rho_{o}}}{L-x_{p}} \tag{6}\]  
where the piston acceleration driven by the detonation wave reflection is given by the following equation of motion derived from a force balance:  
Figure 1:0  
It consists of the driving force generated by the reflected shock pressure (Equation (2)), the fluid pressurization, as well as frictional losses due to the O-ring sealing in the plunger, \(F_{\text{O-rings}}(t)\). The latter term is difficult to model because the frictional forces due to O-ring sealing consist of a complex phenomenon as there are many factors in play that have reciprocal influence [18; 19].