# Evolution of Flow and Streaming in Exponential Variable Cross-Section Resonators
## 5 Conclusions  
In this study, the flow and streaming motions in an exponential-type resonator were studied, using the gas-kinetic BGK model. The geometry and driving strength were found to be the main factors affecting the streaming patterns and the performance of the system. The observations are as follows:  
1. An optimal shape parameter, i.e., \(m\) = 2.2, for the exponential-shaped resonator, can be used to obtain the largest oscillatory pressure amplitude. Therefore, some numerical experiments are carried out for an \(m\) = 2.2 exponential-shaped resonator under five different driving strengths, which lead the acoustic fields to change from quasi-linear to highly non-linear. 2. Vortex formation and evolution are closely linked to the waveform of pressure, velocity, and temperature. The third type of DC streaming is found under quasi-linear and moderate non-linear conditions. Driven by the asymmetry of fluid flow and temperature, the DC flow is formed as a result of the generation, evolution, and shedding of vortices, and it accompanies a periodic energy storage and release. 3. The flow and streaming patterns are found to be very sensitive to the driving strength. With increasing driving strength, a very large pressure and temperature asymmetry will break the vortex and disorder the flow direction and waveforms. In particular, a turbulent flow with irregular and unsteady direction and secondary flow may occur if the driving amplitude increases beyond a critical value. 4. The largest growth ratio for peak-to-peak pressure amplitude and axial flow velocity occurs when the driving strength varies from 300 to 400 \(\upmu\)m, while the smallest occurs when the driving strength varies from 400 to 500 \(\upmu\)m. Therefore, to obtain a larger output power and avoid \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\Delta P\) (Pa) & \begin{tabular}{c} **Growth** \\ **Ratio (Pa)** \\ \end{tabular} & \(|u_{\text{max}}|\) (m/s) & \
\begin{tabular}{c} **Growth** \\ **Ratio (m/s)** \\ \end{tabular} & \(\Delta T\) (K) &
\begin{tabular}{c} **Growth** \\ **Ratio (K)** \\ \end{tabular} \\ \hline A\(\rightarrow\)B & 3556 & - & 2.75 & - & 1.05 & - \\ B\(\rightarrow\)C & 3150 & \(-\)406 & 2.80 & 0.05 & 1.19 & 0.14 \\ C\(\rightarrow\)D & 6119 & 2969 & 2.91 & 0.11 & 1.42 & 0.23 \\ D\(\rightarrow\)E & 1195 & \(-\)4924 & 1.66 & \(-\)1.25 & 3.33 & 1.91 \\ \hline \hline \end{tabular}
\end{table} 
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