Whipping of electrified visco-capillary jets in airflows

Abstract

An electrified visco-capillary jet shows different dynamic behavior, such as cone forming, breakage into droplets, whipping and coiling, depending on the considered parameter regime. The whipping instability that is of fundamental importance for electrospinning has been approached by means of stability analysis in previous papers. In this work we alternatively propose a 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 by Ribe (Proc. Roy. Soc. Lond. A, 2004) describing the jet dynamics with respect to a frame rotating with the a priori unknown whipping frequency that itself becomes part of the solution. The rod model allows for stretching, bending and torsion, taking into account inertia, viscosity, surface tension, electric field and air drag. For the resulting parametric boundary value problem of ordinary differential equations we present a continuation-collocation method. On top of an implicit Runge-Kutta scheme of fifth order, our developed 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 convincing, the whipping effect is qualitatively well characterized.

Figures (11)

• Figure 1.1: From left to right: Sketch of an electrospinning or electrospray device; whipping instability in an electrified jet of glycerine in a bath of hexane (courtesy of A. Gomez-Marin); coiling of an electrified liquid jet (courtesy of G. Riboux)
• Figure 3.1: Jet curve during continuation procedure. From top left to bottom right: Starting solution to $\mathrm{p}_0$, after Part A, after Part B (solution with lay-down end to $\mathrm{\bar{p}}$), after Part C and after Part D (desired solution with stress-free end to $\mathrm{p}^{ref}$). $\mathrm{Re}_\star=1.5$.
• Figure 4.1: Jet curves for variations of single model parameters proceeding from reference $\mathrm{p}^{ref}$, $\mathrm{Re}_\star^{ref}$. The reference jet is visualized in the middle picture and in red in all plots. The jets associated to increased/decreased parameters are given in green/blue.
• Figure 4.2: Elongation of jet end over model parameters in the region $\mathrm{Re}\in [0.4,3.5]$, $\Gamma \in [0,650]$, $\Xi\in [5\cdot 10^3,10^5]$, $\Theta\in [15,170]$, $\epsilon\in [0.02,0.1]$, $\mathrm{M}\in [0,0.1]$, proceeding from $\mathrm{p}^{ref}$.
• Figure 4.3: Jet's throwing range indicating the width of the curve's envelope with respect to the model parameters in the region $\mathrm{Re}\in [0.4,3.5]$, $\Gamma \in [0,650]$, $\Xi\in [5\cdot 10^3,10^5]$, $\Theta\in [15,170]$, $\epsilon\in [0.02,0.1]$, $\mathrm{M}\in [0,0.1]$, proceeding from $\mathrm{p}^{ref}$ (cf. Fig. \ref{['fig:paraElong']}).

Theorems & Definitions (3)

• Remark 1: Stationarity
• Remark 2: Quaternions for rotation
• Remark 3: Selective preconditioning