## 3 Results and Discussion
### Mechanism of the Secondary Breakup of Single Electrified Al/N-Decane Nanofluid Fuel Droplets
The droplet stability condition is still the Rayleigh Limit, and at the droplet surface, the surface charge density can be given as  
\[\frac{q_{s}(\varphi)}{Q_{c}(\varphi)}=\left(\frac{\varepsilon_{g}}{2\varepsilon _{g}-1}\right)\left(1-\frac{3}{2}E\sqrt{\frac{D_{nf}\varepsilon_{0}}{2 \varepsilon_{g}\gamma_{nf}(\varphi)}}\left(\frac{\varepsilon_{nf}(\varphi)- \varepsilon_{g}}{\varepsilon_{nf}(\varphi)+2\varepsilon_{g}}\right)\right) \tag{10}\]  
In these cases, the charges depend on the Al nanoparticle concentration, but further less than the Rayleigh limit. According to our estimation, the maximum surface charge density was 24.0% at 6.0 mg/mL, and the minimum value was only 20.7% at 8.0 mg/mL. Figure 6 demonstrated that, at a corresponding concentration of 6.0 mg/mL and 8.0 mg/mL, the average diameters of secondary breakup sub droplets almost reached minimum and maximum. It suggests that the larger the surface charge density, the finer the average diameters breakup sub droplets are. According to the Weber and [PERSON] number correlations published by [PERSON] [36], [PERSON] [37] extended the Rayleigh limit to non-stationary droplets. It demonstrated that the surface charge may reduce the effective surface tension. For spheroidal deformation of non-stationary charged droplets, its effect will be magnified by spheroidal eccentricity at the poles of a prolate ellipsoid. Assuming an ellipsoid of the major and minor axis \(a\) and \(b\) and eccentricity of \(e\), the ellipticity parameter \(I_{1}=\frac{1}{2e}\ln\left(\frac{1+e}{1-e}\right)\). The behavior of charged non-stationary droplet could be assessed:  
\[\left(1-\left(\frac{a}{b}\right)^{1/3}\right)\left(1-\left(\frac{q_{\mathrm{s} }^{2}(\varphi)}{8\pi^{2}\varepsilon_{0}\gamma_{nf}(\varphi)D_{nf}^{3}}\right) \left(\frac{NI_{1}}{4}\right)^{2}\right)=K\frac{\rho_{nf}(\varphi)u_{nf}^{2}D_ {nf}}{\gamma_{nf}(\varphi)} \tag{11}\]  
where \(N^{2}=\frac{2\delta^{2/3}(2-\delta^{1/2}-\delta^{3/2})}{1-\delta}\) and \(\delta=1-e^{2}\); \(K=0.0850\) for \(Oh\leq 1\), \(K=0.0475\) for \(Oh\geq 1\)[37]. In this work, \(Oh\leq 1\), thus \(K=0.0850\). Therefore, the dimensionless effective electrical Reynolds number (\(Re_{\mathrm{E}}(\varphi)\)), Weber number (\(We_{\mathrm{E}}(\varphi)\)), and Ohnesorge number (\(Oh_{\mathrm{E}}(\varphi)\)) can be more effective to evaluate the charge contribution for the secondary breakup of droplet. For most of the cases in this work, only one Taylor cone formed and towards the grounding electrode, and thus one cone jet produced (See Figure 5). Occasionally, two Taylor cones formed and faced the two electrodes respectively, illustrating two jets. Figure 8 demonstrates the producing process of two Taylor cones and two jets along the direction of the electric field at Al nanoparticle concentration of 8.0 mg/mL. In the current case, the two jets were produced in a sequence, i.e., the jet at the left side sprayed from the single droplet is slightly earlier than that on the right. It means that the charge polarity and numbers at the surface were a little asymmetrical to the droplet center. The charges of sub droplets on the left presented positive, while negative on the right. The average diameter of secondary breakup sub droplets on the left (100.22 \(\upmu\)m) is slightly larger than that on the right (87.08 \(\upmu\)m). Figure 8: The secondary breakup of both sides of single electrified Al/n-decane nanofluid fuel droplet at the concentration of 8.0 mg/mL. [PERSON] et al. (2020) demonstrated similar phenomena. For conducting droplet breakup like water/alcohol immersed in a neutral density oil, large droplets with a diameter of 10-16 mm elongated, and at a critical tip field strength, the surface deformed identically at both ends. Jet filaments were produced and broke up into fine droplets. As the droplet was displaced nearer to one electrode and jet formation only occurred at one end. At the other end, part of the droplet broke off to form a small satellite droplet.