## 4 Conclusions  
In summary, we have demonstrated in this study the phenomenon of MET convection and its application in microfluidic immunoassay confined within a straight channel embedding a discrete array of 12 metal strips for external dual-frequency AC powering, in terms of both mathematical derivation and numerical calculation. The ideal working frequency of SWET and TWET is first obtained, on the basis of which the MET flow behavior of saline solution experiencing dual-frequency smeared structural polarization under the actuation of hybrid SW and TW voltage waves is revealed. In essence, for MET, the time-averaged synthetic electrothermal body force consists of two force components accounted for, respectively, by SWET and TWET, which are strongly coupled with each other through the action of dual-component Joule heating. The flow profile of active MET streaming commonly behaves as a delicate combination of laterally rotating SWET eddies and horizontally advancing TWET pump motion, giving rise to fully automated simultaneous electrokinetic transport and convective stirring of the incoming antigen sample within the buffer medium subjected to dual-frequency phase-shifted AC forcing. The dependence of the binding performance enhancement on the voltage amplitude and field frequency of the externally imposed SW and TW signal, as well as the reaction time due to the action of MET convection, is studied in detail. It is firmly believed that the detection sensitivity and detection limit of on-chip immunoassay are both greatly improved by implementing the twisted pump streamlines of MET fluid motion on top of an array of conducting metal strips arranged along the channel bottom surface, as long as the SW signal works in the low-frequency limit and the TW counterpart operates at the intermediate relaxation peak for moderate voltage amplitudes and a sufficiently large Da number. The numerical  
Figure 8: Influence of Da number and imposed voltage amplitude on MET-mediated microfluidic immunoassay. (**A**) Da-dependent Be for varying Atw at fixed Asw = 6 V. (**B**) Asw-dependent Be for varying Atw under Da = 1000. (**C**) Da-dependent Be for varying Asw at fixed Atw = 6 V. (**D**) Atw-dependent Be for varying Asw under Da = 1000.  
calculations reported herein prove that the complex electrothermal induced flow under multiple frequency AC actuation serves as a useful tool for accelerating the binding kinetics in inhomogeneous microfluidic bioassays, particularly once the immune reaction of interest is stipulated by a slow mass transfer of the target analyte. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15094748/s1](https://www.mdpi.com/article/10.3390/app15094748/s1)]([https://www.mdpi.com/article/10.3390/app15094748/s1](https://www.mdpi.com/article/10.3390/app15094748/s1)), Video S1: Featured application movie. [PERSON] derived the theory of MET and wrote the paper. [PERSON] wrote the paper and sought the first-order Langmuir adsorption model. [PERSON] conducted the numerical simulation of SWET and revised the paper. [PERSON]. ([PERSON]) conducted the numerical simulation of TWET and revised the paper. [PERSON] conducted the simulation of MET and analyzed the flow behavior. [PERSON]. ([PERSON]) conducted the simulation of the transient binding reaction at different voltage combinations. [PERSON] investigated the effect of signal frequency and Da number on the resulting binding kinetics. [PERSON] outputted the figures and guided the research. [PERSON] proposed the core idea and provided the basic theory of electrokinetics and electrohydrodynamics of conducting electrolytes in microscale systems. All authors have read and agreed to the published version of the manuscript. This research is funded by the National Natural Science Foundation of China (No.