# Numerical Infeasibilities of Nanofibrous Mats Process Design
## 2 Theoretical Support  
The control of the fiber diameter is an important element of the electrospinning process [24]. There are several attempts to model the electrically charged jet behavior described in the literature [25; 26]. A basic electrospinningning setup consists of an infusion pump with a polymer solution, a high voltage power supply, and a collector [27]. The process involves applying a high voltage to electrodes, which are a syringe needle and the collector [28]. In other words, a strong electrical field is applied to the polymer solution. This operation induces a charge on the surface of the fluid. As the electrical field strength increases, a conical shape called the Taylor cone begins to be created. A straight jet is then formed, which undergoes a bending instability phenomenon, as it travels towards the collector [29; 30]. This process causes the fiber diameter to decrease (by orders of magnitude) relative to the diameter of the syringe needle [31]. Basically, there are four zones in the ES: the Taylor cone formation (T); the stable part of the jet (J); the unstable, whipping part (W); and the area where fibers are stopped and collected (C) [32]. A schematic view of the basic ES equipment with the four separate states is depicted in Figure 1.  
[PERSON] and co-authors [20] in 2003 presented the model of a charged Newtonian fluid jet applicable to the whipping instability; the model allowed the prediction of a terminal fiber diameter with the known values of an electric current, flow rate and surface tension. The model neglected elastic effects and solvent evaporation. The model of [PERSON] et al. [20] has been widely discussed and modified [33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. [PERSON], in [40], has noticed that the model presented in [20], due to the omission of important dependencies (i.e., viscosity, solution evaporation, polymer contribution), leads to a significant exaggeration in the importance of the surface tension. [PERSON] et al. [21; 22] proposed the combined version of the stable and unstable jet based on works [43; 44] and [20]. The model enabled the prediction of the final fiber diameter regarding the volumetric flow rate, the applied electrical field, and the polymer concentration. Moreover, the authors of the work [38] described the approach where the exponential model was proposed. The model related the stable jet length and the terminal fiber radius. The authorsshowed that for fully charged fibers, the scaling exponent was \"\(-2/3\)\" and that this becomes smaller for partially charged fibers. The parameter \(\chi\) is a very interesting part of the work described above. The literature definitions of \(\chi\) can be presented as follows: \"\(\chi^{-1}\) is the local aspect ratio, which is assumed to be small\" [43], \"\(\chi\sim R/h\) is the dimensionless wavelength of the instability responsible for the normal displacements\" [20], \(\chi\) is \" a characteristic length parameter (...) influenced by the first stage of electrospinning, which is the stable jet\" [21], and \"\(\chi\) is the aspect ratio of the jet\" [38]. Thus, we can also state:  
\[\chi\sim\frac{\lambda}{r_{sj,end}}, \tag{1}\]  
where \(\lambda\) denotes the wavelength and \(r_{sj,end}\) is the critical radius of the stable jet (at the end of the straight jet region), and  
\[\lambda=\frac{2\pi}{k}, \tag{2}\]  
that can be estimated by the radial instability wavelength from the fastest growing mode [38]. The definition of \(\chi_{bi}\) described as \"the ratio of the characteristic axial length scale\" can also be found in the work of [21]. It is also worth exploring the \(\chi\) range and determining which values lead to nanofibers at certain diameters. [PERSON] et al. [20] in all computations used \(\chi=100\); the authors assumed also that the \(\chi\) value between 10 and 1000 was reasonable. The estimation method of the \(\chi\) parameter and the \(\mathrm{F_{D}}(\chi)\) relation was presented, analyzed and discussed in detail in [23].