# Controlled Release Using Gas Detonation in Needle-Free Liquid Jet Injections for Drug Delivery
## 3 Results and Discussion
It is worth noting, that unlike the water hammer effect, which describes pressure variations in a pipeline of which the pressure wave dynamics and damped oscillatory behavior can be accurately obtained using the method of characteristics [44; 45], the present liquid jet injection phenomenon also involves a detailed analysis of complete system dynamics, i.e., the fluid-structure interaction between the rapidly moving piston, water column and the flow behind the reflected shock, after the detonation impact. Typically, the water hammer effect is a result of a rapid closing of valves in a flow stream, causing a pressure wave to propagate upstream in the pipe. For such a situation, the numerical solutions  
Figure 6: Stagnation pressure evolution from the analytical model with combustible initial pressures of (a) 30 kPa; (b) 40 kPa and (c) 50 kPa. to the water-hammer equations governing the propagation of the pressure surge can predict the wave velocity and damping of the pressure oscillations. It is worth mentioning that an equivalent analysis has been considered by [PERSON] and [PERSON] [10], referred to as \"wave analysis\". This study illustrates that the wave analysis results were only valid over a very short time span, i.e., the short duration over which the first pressure spike occurs and when piston movement is negligible and assumed to be zero. Unlike the water-hammer effect, the present phenomenon involves piston acceleration to a high velocity which is no longer negligible, and the water-hammer equations are not sufficient to describe the full dynamics of the injection pressure profile evolution. The continuum analysis approach detailed in [PERSON] and [PERSON] has become a standard model with continuous improvement for different types of needle-free liquid injection devices driven by a high-velocity plunger, e.g., [15; 18; 28; 32] and thus, is also used in this work. It is important to note that the oscillatory behavior, i.e., both the amplitude and damping of the jet pressure variation are not simply wave dynamics within the liquid column, moreover, they do not solely depend on the liquid acoustic and thermodynamic properties. The oscillatory behavior is a result of system dynamics, which must be modeled considering piston movement caused by the driving force and subsequently countered by the frictional and fluid forces which arise due to the piston movement [10]. All these aforementioned effects are taken into consideration in the continuum analysis, although more accurate quantitative sub-models, e.g., O-ring seals and piston driving force by the detonation wave, are needed to precisely capture the damping of the jet pressure oscillation. Despite the simplicity, the model does capture the two main jet properties, namely, the peak and average stagnation pressure values, and the period of oscillations correlate well to experimental observations. Qualitatively, in our previous work, the effects that strongly influence damping are identified [18]. The friction from sealing is found to be dominant and the oscillations are caused primarily by the piston dynamics. Nevertheless, the liquid viscosity, as can be seen in this study, is also another damping parameter which affects the oscillatory behavior [19; 46]. In order to further improve the continuum analysis and obtain more accurate predictions of the jet pressure oscillation, future work will implement an improved quantitative description of the arising frictional force due to the O-ring seals and the detonation reflection process interacting with an accelerating piston. The pressure traces from both the experimental measurement and analytical results depict both the peak and average jet pressures for different acetylene-oxygen gas mixtures and initial pressures, as can be seen in Figure 7. The solid line represents the analytical model results. For each initial pressure condition, at least five experimental shots were performed.