Whipping of electrified visco-capillary jets in airflows
An electrified visco-capillary jet shows different dynamic behavior, such as cone forming, breakage into droplets, and whipping and coiling, depending on the considered parameter regime. The whipping instability, which is of fundamental importance for electrospinning, has been approached by means of stability analysis in previous papers. In this work, we propose an alternative model framework in which the instability can be computed straightforwardly as the stable stationary solution of an asymptotic Cosserat rod description. For this purpose, we adopt a procedure from Ribe [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2051 (2004), pp. 3223--3239] by describing the jet dynamics with respect to a frame rotating with the a priori unknown whipping frequency, which itself becomes part of the solution. The rod model allows for stretching, bending, and torsion and takes inertia, viscosity, surface tension, electric field, and air drag into account. For the resulting parametric boundary value problem of ordinary differential equations we present a continuation-collocation method. On top of an implicit sixth order Runge--Kutta scheme, which leads to a fifth order collocation scheme, our continuation procedure makes the efficient and robust simulation and navigation through a high-dimensional parameter space possible. Despite the simplicity of the employed electric force model, the numerical results are very convincing, and the whipping effect is qualitatively well characterized.