## 3 Results and Discussion
### Mechanism of the Secondary Breakup of Single Electrified Al/N-Decane Nanofluid Fuel Droplets
For a perfectly conducting droplet in  
Figure 7: Mechanism of the secondary breakup of an individual electrified nanofluid fuel droplet in uniform electrostatic field, (**a**) production and movement of single droplets, (**b**) initial single spheroidal droplet with uniform charge distribution, (**c**) charge transfer and droplet deformation, (**d**) formation of Taylor cone and secondary breakup. a vacuum, \(\varepsilon_{\text{d}}\)/\(\varepsilon_{\text{g}}\)\(\rightarrow\)\(\infty\), then a conical point forms, as predicted by [PERSON], and the charge and mass may issue from this point [24]. Generally, the conductivity is finite, i.e., there exists an electric field inside the droplet, then the droplet will have to deform. If providing \(\varepsilon_{\text{d}}\)/\(\varepsilon_{\text{g}}\geq 20\), the conical point still forms, although the apex is different from the 'Taylor-Cone' angle of 49.3\({}^{\circ}\). If the permittivity ratio \(\varepsilon_{\text{d}}\)/\(\varepsilon_{\text{g}}<20\), the conical point does not form, and the droplet will elongate along the electric field direction to form a cylindrical shape to accommodate any field strength, then fluid inertia will cause the droplet to oscillate [29; 30]. In this work, although the permittivity ratio of n-decane/air (-2.0) is further lower than the reference value of 20, the secondary breakup of single nanofluid fuel droplets presented the cone-jetting mode and produced uniformly distributive sub droplets (See Figures 2 and 5). This indicates that the Al nanoparticles contribute to the charge distribution and breakup characteristics of the droplets. According to our experimental observation, for pure n-decane, the Taylor cone did not form and the secondary breakup would not occur in air. For low Al nanoparticle concentrations of 1.0-3.0 mg/mL, the breakup probability of single nanofluid fuel droplets is also low. It suggests that, as there are abundant free electrons in the outer layer of aluminum, the addition of a small amount of Al nanoparticles into n-decane increases the free charge density. Adding the induced charges under the action of the external electric field, they together produce an electric field within the droplet and oblige to form the Taylor cone, but it is not strong enough to overcome the surface tension. At high concentrations of 4.0-10.0 mg/mL, the electric field formed by charge transfer within the droplets makes the single droplet break up into finer sub droplets. The electrification of droplets relates to the Al nanoparticle concentration, moreover, their physical properties also strongly depend on the Al nanoparticle concentration, resulting in a corresponding change in electric field force and surface tension, etc., and influencing the secondary breakup performance. For \"large\" nanofluid fuel droplets, their theoretical charge may be defined by the Rayleigh Limit [31; 32] and expressed as  
\[Q_{c}(\varphi)=\sqrt{8\pi^{2}\varepsilon_{0}\gamma_{nf}(\varphi)D_{nf}}^{3} \tag{7}\]  
The surface charges can generate an electric field, normal to the surface [31; 32], and be defined as  
\[Q_{field}=\pi\varepsilon_{0}E_{s}D_{nf}^{2} \tag{8}\]  
where \(E_{s}\) is the electric field strength due to the surface charges. A critical crossover diameter of the droplet where the Rayleigh limit Equation (7) and the breakdown limit Equation (8) should be equal, which can be defined as  
\[D_{nf}=\frac{8\gamma_{nf}(\varphi)}{\varepsilon_{0}E_{s}^{2}} \tag{9}\]  
However, in practice, the Rayleigh limit is scarcely reached since the external electrical or aerodynamic forces deform the drop away from the perfect electrical and physical symmetry. Therefore, the droplet is partly electrified in practice, and its real surface charges of \(q_{s}(\varphi)\) are fractional to \(Q_{C}(\varphi)\)[25]. [PERSON] [33; 34] established the magnitude of charge required to destabilize a given droplet in a given electric field. The fluid motion was driven by the surface electric field, occurring inside dielectric droplets [35].