Determination of the bifurcation parameter (), as a function of time in the electrospinning process using Bratu equation

Polymeric nanofibers derived from electrospinning hold potential applications in various fields such as fluid filtration, biomedicine, catalyst supports, drug delivery, tissue engineering, nanowires, and more. However, theoretical modeling of the electrospinning process remains a problem to be solved, as ignorance of the physical nature of the process severely hinders the improvement of the quality of the fibers as well as the efficiency of the process. In mathematical terms, a nonlinear system is one whose evolution equations are nonlinear. For these types of systems, a slight variation in a parameter can trigger a sudden and dramatic change (or bifurcation) in a system's quantitative and qualitative properties, which can lead the arrangement to behave chaotically. This study establishes a mathematical model to obtain an approximation of the axial velocity of the exit of the electrospun polymeric jet, which was simulated in MATLAB, only using the tip-to-collector distance (10 and 20 cm) as a unique variable in the mathematical model, this does not mean that the other parameters such as solution viscosity, molecular weight, among other, are not necessary, However, this study screened the tip-to-collector distance as the initial variable, followed by an experimental evaluation to examine the relationship between axial velocity and the resulting morphology and fiber's diameter. The electric force was evaluated as the dominant parameter over the different forces and the Bratu equation, which can explain the instability (bifurcation) in electrospinning. The time-space differential equation result where the axial velocity enters chaos when the potential overcomes the surface tension of the polymer droplet is observed for values of bifurcation lambda = 0.5. In conclusion, controlling distance between 10 and 20 cm does not represent an essential variable in searching for fiber diameter's reproducibility using the one-dimensional mathematical model.