A Robust Truncated-Domain Approach for Cone--Jet Simulations in Electrospinning and Electrospraying

Abstract

Direct numerical simulations of electrospinning and electrospraying are computationally demanding due to large-scale separation between the needle and the tip-to-collector distance. The cone-jet mode that occurs in the vicinity of the needle arises from a delicate balance between surface tension, viscous stresses, inertia, and electric stresses. This mode has a central role in determining the subsequent instabilities of the jet and the eventual outcomes on the collector. Truncated-domain simulations offer a viable alternative but depend critically on the accuracy of far-field electrostatic boundary conditions. Existing truncated-domain approaches based on analytical expressions for the electric potential systematically underestimate the electric field near the needle tip and require empirical tuning informed by prior experiments or full-domain simulations, thereby limiting their predictive capability. Here, we present a general truncated-domain framework for electrohydrodynamic (EHD) simulations of the cone-jet mode that avoids these limitations. Our approach exploits inexpensive full-domain electrostatic simulations to obtain the exact electric field and potential distributions near the needle, which are then imposed as boundary conditions in an EHD simulation carried out on a truncated domain. Comparisons with full-domain EHD simulations and experimental data demonstrate that the proposed approach accurately reproduces cone-jet shapes as well as key physical quantities, including electric currents, charge distributions, velocity fields, and Maxwell stresses, while converging at substantially smaller domain sizes. The formulation eliminates tunable parameters, does not require prior knowledge of the cone-jet configuration, and significantly reduces computational cost, providing a reliable and predictive framework for studying electrohydrodynamic cone-jet flows.

Figures (9)

• Figure 1: (a) Schematic of the experimental configuration used in electrospinning and electrospraying, where $L$ denotes the distance from the axis of symmetry of the needle to the lateral confinement wall and $H'$ represents the axial distance from the needle tip to the collector. (b) Electric field distribution obtained from purely electrostatic simulations with a lateral boundary distance $L = H'/2$, where $H'$ is the tip-to-collector distance. Electric field lines are shown in black with arrows, while equipotential lines are perpendicular to them. (c) Electric field distribution for a larger lateral boundary distance $L = H'$, with the collector plate size kept identical to that in panel (b). The red box indicates the computational domain size corresponding to $L = H'/2$. All cases correspond to purely electrostatic simulations with an applied field of $E_\infty = 1.6 \times 10^{6}\,\mathrm{V/m}$.
• Figure 2: Schematic representation of the two computational domains: the full domain (solid black), corresponding to the experimental geometry, and the truncated domain (red dashed). The figure is not to scale. Full-domain simulations (FDS) employ standard boundary conditions, whereas truncated-domain simulations (TDS) require appropriate boundary conditions on the red dashed boundaries. The local mesh structure is illustrated in the inset.
• Figure 3: Grid-independence study comparing interface shapes for different mesh resolutions (a) and for varying interpolation factor $f$ (b). The study was conducted for an applied voltage of $2000~\mathrm{V}$ and a flow rate of $1~\mathrm{mL\,h^{-1}}$. The geometrical parameters are $R_o/R_i = 1.1$, $H'/R_i = 10$, and $L/H' = 1.5$. Convergence is achieved for $f = 5$ and a minimum grid spacing of $\Delta x = 2~\mu\mathrm{m}$.
• Figure 4: Variation of the electric field magnitude along the needle axis for different lateral boundary positions ($L/H'$), with $H'/R_i = 10$ and $R_o/R_i = 1.1$. The needle length is $L_n/R_i = 1.77$, and convergence of the electric field is observed for $L/H' = 1.5$. All cases correspond to purely electrostatic simulations with an applied field of $E_\infty = 1.6 \times 10^6~\mathrm{V/m}$.
• Figure 5: (a) Variation of the electric field magnitude along the needle axis for truncated-domain electrostatic simulations (TDS-ES-JT) with three domain sizes: $7R_i \times 7R_i$ (red), $10R_i \times 10R_i$ (green), and $10R_i \times 15R_i$ (blue). The results are compared with those from the full-domain electrostatic simulation (FDS-ES, black), for which the domain size was determined from the electrostatic convergence study. (b) Electric potential contours for the corresponding cases, with the colour scheme consistent with that in panel (a). In all cases, the needle length is $L_n/R_i = 1.77$ and the outer-to-inner radius ratio is $R_o/R_i = 1.1$. The applied electric field is $E_\infty = 1.6 \times 10^6~\mathrm{V/m}$. (c) Interface evolution for the truncated-domain electrohydrodynamic simulation (TDS-EHD-JT) with domain size $10R_i \times 15R_i$, showing an unstable cone–jet transitioning to the dripping mode. The sequence is shown at four time instants: $t = 3.0$, $3.2$, $3.4$, and $3.6~\mathrm{ms}$. (d) Interface evolution for the remaining truncated-domain cases, $7R_i \times 7R_i$ (red) and $10R_i \times 10R_i$ (green), together with the full-domain electrohydrodynamic simulation (black). The principal non-dimensional parameters used in the electrohydrodynamic simulations are listed in Table \ref{['table:Simulation_summary']}. The cases shown correspond to \ref{['case:3(a)']} and \ref{['case:2(c)']}.