## 3 Results and Discussion
### Numerical Characterization of Electrothermal Induced Convection
#### 3.1.1 Standing-Wave Electrothermal Fluid Motion  
A standing wave has a phase difference of 180\({}^{\circ}\) in the applied harmonic voltage between neighboring electrodes of opposite polarity. Every electrode can be approximated as a point charge under this condition, so the in-phase electric field is quite similar to the electric field generated by a static point charge. From the space distribution of the standing-wave electric field shown in Figure 1A, we can treat the voltage gradient generated by the standing-wave signal for the zero-order electric field with uniform conductivity, since there is no evident deflection of equipotential lines. When the standing-wave voltage signal is applied to the discrete electrode array, a large amount of Joule heat is generated in the bulk electrolyte and then diffused into the liquid solution and solid substrate due to the resistive effect. The temperature field induced by electric heat generation in the liquid solution is shown in Figure 2B, with the electrode surface behaving as the low temperature region. Figure 2: Numerical analysis of SWET flow field in a straight PDMS microchannel embedding a three-period four-phase discrete microelectrode array situated on a silicon substrate, with a given SW voltage amplitude V\({}_{\text{SW}}\) = 6 V. (**A**) A surface plot of zeroth-order electrostatic potential and an arrow plot of zeroth-order electric field vector independent of the SW field frequency. (**B**) An arrow and surface plot of the time-averaged temperature field caused by Joule medium heating. (**C–E**) A streamline and surface plot of electrothermal flow field due to SWET forcing at distinct signal frequencies, (**C**) f = 100 kHz in the low frequency limit, (**D**) f = \(\sqrt{11}\)f\({}_{\text{MW}}\) = 77.83 MHz at the critical frequency point for flow reversal, (**E**) f = 10 GHz in the high frequency range. (**F**) Frequency-dependent electrothermal flow velocity driven by SW voltage. (**G**) Frequency-dependent vorticity magnitude of SWET fluid motion. The fluid is subjected to time-averaged Joule heating to produce an internal temperature gradient, and thus a continuous conductivity gradient across the channel depth direction. Due to the objective constraints of the full electric current continuity condition, the interaction of the electrical property gradient with the imposed standing-wave electric field inevitably leads to the in-phase smeared structural polarization, generating an induced space charge cloud, which imparts an electrostatic force density to the fluid bulk and drives the SWET flow field in the form of a series of counter-rotating micro-vortices. As shown in Figure 2C-E, the distribution of time-averaged SWET flow field in space is analyzed for three different field frequencies. In the low frequency limit of f = 100 kHz, as dominated by the Coulomb force, fluid of higher conductivity is transported by the SWET forcing to occupy the region of larger electric field strength. This is in good agreement with Figure 2C, which shows the fluid is dragged from the top of the electrode gap of the largest temperature elevation to the center of the electrode surface, and finally forms a series of closed eddies in counter-rotating directions. However, different from the repeating flow pattern actuated by periodic electrode structures, the SWET flow field on top of the finite electrode array presents a larger vortex on its two sides, mainly due to the electric field end effects on electrothermal induced bulk polarization.