# Influence of the Polarity of the Electric Field on Electrorheometry
## 3 Results and Discussion  
### Newtonian Fluid  
Extra virgin olive oil was the suspending fluid, and it exhibited a Newtonian behavior. It was measured both under shear and extensional flows, with and without the application of the electric field, obtaining different results. Figure 4 shows the steady-state deformation with the stress of the olive oil measured in the rotational rheometer at 20 \({}^{\circ}\)C with and without the influence of an electric field. It can be observed that all the curves overlapped, which was clear evidence for the presence of the electric field, and its intensity did not affect the rheological behavior of the olive oil under shear flow. As mentioned above, the electrorheological cell in the rotational rheometer did not allow changing the polarity in the electric field, which was perpendicularly oriented to the flow field. In the absence of particles, the electric field had no possibility to build any structure in the olive oil, and the viscosity did not change with \(E_{0}\). Figure 5 shows the time evolution of the minimum filament diameter once the fluid bridge reached the final position \(h_{f}=6\) mm. It can be observed that the presence of the imposed electric field \(E_{0}\) slightly modified the curve measured without the presence of the electric field. The influence of the polarity imposed on that change could also be appreciated; the filament lived longer when the electric field was pointing upward (inverse polarity), while its breakup time became shorter when the polarity was oriented with the direction of gravity (direct polarity). Additionally, at \(h_{0}\), once the field was applied, a slight vibration of the liquid-air interface (Video S1 in the Supplementary Material) was observed, which was modified. Figure 6a shows that the polarity of the electric field had an effect on the minimum diameter of the liquid bridge. Figure 6b shows the silhouette of the liquid bridges before applying the voltage and after 8.5 s of the application of the voltage. It can be observed that the inverse polarity helped to keep the shape of the filament unchanged (\(\Delta D_{0}^{Inv}=\frac{\Delta(R/R_{0})}{\Delta t}=0.0124\)), while the direct polarity led to a more intense decrease in the diameter (\(\Delta D_{0}^{Dir}=\frac{\Delta(R/R_{0})}{\Delta t}=0.0308\)). Moreover, it can be observed that in the case of the direct polarity, the vertical position of the minimum diameter moved down. These results confirmed that the direct polarity had a similar effect to having a larger Bond number, which would lead to shorter break up times in the filament. According to [PERSON] et al. [26], the electrical currents are very small in a highly insulating liquid bridge, and then, the electrical forces (\(F_{V,\epsilon}\)) participating in the problem would be the following ones:  
* Coulombian-type forces (\(\rho_{e}\bar{E}\)), which are due to the presence of the free charge density (\(\rho_{e}\)) that probably comes from the dissociation of impurities or of the liquid itself, apart from the inevitable electrochemical reactions at the metal-electrode/liquid interface. The dynamics of the liquid will be strongly affected by this type of force in a DC electric field. * Dielectrophoretic force (\(\frac{1}{2}E^{2}\
abla e\)), which results from the inhomogeneity in the electrical permittivity (\(\epsilon\)) when the fluid undergoes electrical stress, acts at the air-liquid interface and perpendicularly to it. Therefore, it is not a volume force. Figure 4: Viscosity curve of the olive oil measured in the rotational rheometer at 20 \({}^{\circ}\)C with and without the influence of an electric field. Figure 5: Time evolution of the normalized diameter for the Newtonian fluid. Influence of the presence of the electric field and its polarity. * Electrostrictive force (\(\frac{1}{2}\
abla\left[\rho\left(\frac{dc}{d\rho}\right)_{T}E^{2}\right]\)), which acts both at the interface and on the liquid bulk, is irrelevant to the dynamics of the liquid, and the usual convention neglects its contribution.