## 2 Materials and Methods
### Theory of MET in Electrically Conductive Fluid  
#### 2.2.1 Quasi-Electrostatics  
For alternating electric fields \(\phi(t)=\text{Re}(\widetilde{\phi}e^{j\omega t})\), the complex electric field satisfies the electric current continuity condition [31]:  
\[\
abla\cdot\left[(\sigma+j\omega\varepsilon)\widetilde{\mathbf{E}}\right]=0 \tag{1}\]  
where \(\widetilde{\mathbf{E}}=-\
abla\widetilde{\phi}\) is the phasor amplitude of the complex electric field, \(\varepsilon\) is the dielectric permittivity of the electrolyte, \(\sigma\) is the solution conductivity, f is the field frequency, and \(\omega=2\pi\)f is the angular frequency. If the liquid's dielectric properties vary spatially, the divergence of the electric field can be derived from the previous equation [32]:  
\[\
abla\cdot\widetilde{\mathbf{E}}=-\frac{\
abla\sigma+j\omega\
abla\varepsilon }{\sigma+j\omega\varepsilon}\cdot\widetilde{\mathbf{E}} \tag{2}\]  
As the local temperature of the buffer medium \(T\) changes, the electrical properties of the liquid would change linearly according to the known values of electric conductivity and permittivity at the reference temperature T\({}_{0}\)[33]:  
\[\varepsilon(T)=\varepsilon(T_{0})(1+\alpha(T-T_{0})) \tag{3}\]  
\[\sigma(T)=\sigma(T_{0})(1+\beta(T-T_{0})) \tag{4}\]  
where the two thermal diffusivity coefficients \(\alpha=\frac{\partial\varepsilon}{\partial T}/\varepsilon=-0004\) K\({}^{-1}\) and \(\beta=\frac{\partial\sigma}{\partial T}/\sigma=0.022\) K\({}^{-1}\) show that even though permittivity decreases as temperature increases, liquid electric conductivity rises at an even faster rate. By integrating Equation (2) with the Gauss law equation, we derive a precise mathematical formula for the phasor amplitude of the induced spatial free charge density:  
\[\widetilde{\rho}_{f}=\
abla\cdot\left(\varepsilon\widetilde{\mathbf{E}} \right)=\frac{(\sigma\
abla\varepsilon-\varepsilon\
abla\sigma)}{\sigma+j \omega\varepsilon}\cdot\widetilde{\mathbf{E}} \tag{5}\]  
Observing that the electrical property varies with the temperature field, the first-order space free charge density can then be explicitly given by:  
\[\widetilde{\rho}_{f}=\frac{\sigma\
abla\varepsilon-\varepsilon\
abla\sigma}{ \sigma+j\omega\varepsilon}\cdot\widetilde{\mathbf{E}}=\frac{\varepsilon( \alpha-\beta)}{1+j\omega\tau}\
abla T\cdot\widetilde{\mathbf{E}} \tag{6}\]  
where \(\tau=\frac{\varepsilon}{\sigma}\) is the Debye relaxation time of the induced charge in the liquid bulk. Under the combined SW/TW electric signal, the temporary electrothermal body force density acting on each water element is essentially a sum of the Coulomb force from the induced free charge and the dielectric force from the polarized bound charge:  
\[ \begin{array}{l}f_{\text{ET}}(t)=\left(\text{Re}\big{(}\widetilde{\rho}_{1 }e^{j\omega_{1}t}\big{)}+\text{Re}\big{(}\widetilde{\rho}_{2}e^{j\omega_{2}t} \big{)}\right)\left(\text{Re}\big{(}\widetilde{\mathbf{E}}_{1}e^{j\omega_{1} t}\big{)}+\text{Re}\big{(}\widetilde{\mathbf{E}}_{2}e^{j\omega_{2}t}\big{)} \right)\\ -\frac{1}{2}\text{Re}\Big{(}\big{(}\widetilde{\mathbf{E}}_{1}e^{j\omega_{1}t} \big{)}+\big{(}\widetilde{\mathbf{E}}_{2}e^{j\omega_{2}t}\big{)}\Big{)}\text{ Re}\Big{(}\big{(}\widetilde{\mathbf{E}}_{1}e^{j\omega_{1}t}\big{)}+\text{Re}\big{(} \widetilde{\mathbf{E}}_{2}e^{j\omega_{2}t}\big{)}\Big{)}\
abla\varepsilon\end{array}  \tag{7}\]  
where subscripts 1 and 2 represent the SW and TW components, respectively. For the purpose of simplification, we need to expand Equation (7) as follows:\[