## 3 Results and Discussion
### Numerical Characterization of Electrothermal Induced Convection
#### 3.1.2 Traveling-Wave Induced Electrothermal Flow  
A traveling-wave field is the electric field of spatial phase variations generated by a linear electrode track with a given phase difference (e.g., 90\({}^{\circ}\)) between neighboring electrodes. The complex amplitude method is used here to analyze the phase-shifted electric field in the simulation of traveling-wave induced electroconvection, and the complex electric field is decomposed into real and imaginary parts, as shown in Figures 3A and 3B, respectively. The same electrodes differ by 90\({}^{\circ}\) in the distribution of the real and imaginary parts of the sinusoidal voltage, namely, they are equally separated by an electrode gap. Figure 3C shows the temperature field generated by TWET. Referring to the temperature field generated by SWET (Figure 2B), it can be found that the difference between the temperature field profile and maximum temperature elevation in TWET and SWET is not large at the same voltage amplitude. A low temperature region is located right on the surface of the discrete electrode array, which corresponds to a smaller electrical conductivity of liquid molecules, and the temperature value increases gently with a larger vertical distance from the electrode surface on the channel bottom surface. Figure 3D-I shows the pumping effect generated by traveling-wave signals of different oscillating frequencies, and it can be found that the electrokinetic transport direction of the liquid medium is always in the direction of the traveling wave. The reason behind this is the formation of a positive conductivity gradient pointing from the perfectly polarizable surface of metal electrodes to the channel top wall, so the current TWET device serves as an attraction-type pump, wherein the TWET pumping force is in the same direction as the propagation direction of the traveling wave. When f = 1 kHz, as shown in Figure 3D, no obvious pumping effect is found in the bulk fluid, and instead, a series of small electrothermal vortices are formed on top of the electrode array, mainly due to the in-phase induced polarization. Increasing the TW frequency to 1 MHz, as shown in Figure 3E, the pumping effect is enhanced within the working fluid, but the vortex shedding is still significant. When f is increased to 10 MHz, as shown in Figure 3F, the pumping behavior of the liquid medium is significant, and the vortex flow obviously disappears, but there are still electrothermal vortices induced near the left channel inlet. Continue to raise the TW signal frequency to f MW = 22.5 MHz, as shown in Figure 3G. At this time, the pumping motion due to out-of-phase electrothermal induced polarization reaches the ideal working state, the horizontal pump flow rate is high, and no obvious vortex is found in the transversal direction. When f is raised to 1 GHz, as shown in Figure 3H, the flow rate decreases severely, and vortices begin to appear at both the entrance and exit of the discrete electrode array. For f = 10 GHz, as shown in Figure 3I, the liquid pump motion tends to vanish, and two huge vortices of opposite flow direction with geometric symmetry are formed at the channel inlet and outlet. Through the numerical analysis of TWET fluid motion, the out-of-phase TWET pumping force achieves unidirectional delivery of the incoming analyte. If biomolecules are added into the buffer solution, rapid convective transport of the incoming particle samples can be achieved by the TWET pumping effect, and therefore avoiding the slow thermal diffusion in a concentration gradient, greatly improving immune adsorption dynamics on functionalized interfaces in microfluidic channels. To find the ideal working frequency, a COMSOL parametric sweep is conducted to arrive at the conclusion. Figure 4A,B show the horizontal pump flow velocity, overall flow rate, and rotating vorticity of the TWET fluid motion within a broad frequency range.