On Large Deformations of Oldroyd-B Drops in a Steady Electric Field

The deformation of viscoelastic drops under electric fields is central to applications in microfluidics, inkjet printing, and electrohydrodynamic manipulation of complex fluids. This study investigates the dynamics of an Oldroyd-B drop subjected to a uniform electric field using numerical simulations performed with the open-source solver Basilisk. Representative pairs of conductivity ratio ($σ_r$) and permittivity ratio ($ε_r$) are selected from six regions ($PR_A^+$, $PR_B^+$, $PR_A^-$, $PR_B^-$, $OB^+$, and $OB^-$) of the $(σ_r, ε_r)$ phase space. In regions where the first- and second-order deformation coefficients share the same sign ($PR_A^-$, $PR_B^-$, $OB^+$), deviations from Newtonian behavior are negligible. In $PR_A^+$, drops develop multi-lobed shapes above a critical electric capillary number, with elasticity reducing deformation and increasing the critical $Ca_E$ with Deborah number ($De$). In $PR_B^+$, drops form shapes with conical ends above the critical $Ca_E$, while steady-state deformation decreases with $De$ below this threshold, and critical $Ca_E$ shows non-monotonic variation. At high $Ca_E$ and $De$, transient deformation exhibits maxima and minima before reaching steady state, with occasional oscillations between spheroidal and pointed shapes. In $OB^-$, drops deform to oblate shapes and breakup above a critical $Ca_E$, with deformation magnitude increasing and critical $Ca_E$ decreasing with $De$; at low $Ca_E$ and high $De$, dimpling and positional oscillations are observed. These results elucidate viscoelastic-electric interactions and provide guidance for controlling drop behavior in practical applications.

💡 Research Summary

This paper presents a comprehensive numerical investigation of the large‑deformation and breakup behavior of a visco‑elastic drop subjected to a uniform steady electric field, using the Oldroyd‑B constitutive model and the open‑source Basilisk solver. The authors first review the classical leaky‑dielectric (LDM) framework, highlighting its success in predicting small‑deformation prolate or oblate shapes but also its limitations in capturing the strong non‑linearities observed experimentally, especially for non‑Newtonian fluids. They then formulate the governing equations: incompressible Navier‑Stokes coupled with the electric potential equation (∇·(ε∇Φ)=0), surface charge conservation, and the Oldroyd‑B stress evolution (τ + λ ∇τ = η_s γ̇ + η_p γ̇). The interface is tracked with a volume‑of‑fluid (VOF) method, and adaptive mesh refinement (AMR) is employed to resolve the thin interfacial layers and conical tips that appear at high electric capillary numbers (Ca_E).

A systematic parametric study is carried out in the (σ_r, ε_r) plane, where σ_r = σ_in/σ_out and ε_r = ε_in/ε_out. Six representative regions—PR_A^+, PR_B^+, PR_A^−, PR_B^−, OB^+, and OB^−—are identified based on the sign of the first‑ and second‑order deformation coefficients from small‑deformation theory. For each region a pair of conductivity and permittivity ratios is selected, and the electric capillary number Ca_E and Deborah number De = λ/τ_c (τ_c = η_out a/γ) are varied over wide ranges.

Key findings are as follows:

1. Regions with negligible elasticity effects (PR_A^−, PR_B^−, OB^+) – The first‑ and second‑order deformation coefficients share the same sign, leading to behavior essentially identical to that of a Newtonian drop. De has little impact on the steady deformation D or on the critical Ca_E for breakup.

2. PR_A^+ (high conductivity, low permittivity) – Above a critical Ca_E, the drop develops multi‑lobed shapes. Increasing De shifts the critical Ca_E to higher values and reduces the maximum deformation, indicating that elastic stresses oppose the electric normal stresses. The transition is smooth; no oscillations are observed.

3. PR_B^+ (low conductivity, high permittivity) – Drops form conical (pointed) ends once Ca_E exceeds a threshold. For modest De the steady deformation decreases with De, but the critical Ca_E shows a non‑monotonic dependence on De (first increasing, then decreasing). At large Ca_E and De, the transient deformation exhibits overshoots and undershoots, sometimes oscillating between spheroidal and pointed configurations before reaching a steady state.

4. OB^− (moderate conductivity, high permittivity) – The drop is compressed into an oblate shape. The deformation magnitude grows with De, while the critical Ca_E for breakup diminishes. At low Ca_E and high De, dimpling of the interface and positional oscillations of the drop centroid are observed.

The authors attribute these behaviors to the competition between electric normal/tangential stresses (controlled by σ_r and ε_r) and the elastic stress stored in the polymeric network (controlled by De). In PR_A^+ the elastic stress primarily counteracts the normal electric stress, delaying the onset of lobes. In PR_B^+ the elastic stress modifies the balance of tangential stresses, leading to the observed non‑monotonic critical Ca_E and transient oscillations. In OB^− the elastic stress amplifies the compressive electric stress, facilitating earlier breakup.

A grid‑independence study confirms that the observed fine features (conical tips, lobes, dimples) are not numerical artifacts; the minimum cell size used is 2^−10 of the initial drop radius.

The paper concludes that the Oldroyd‑B model, despite its assumption of infinite polymer extensibility, captures the essential visco‑elastic–electric coupling for large deformations. The results provide practical guidelines: increasing De (e.g., by using higher polymer relaxation times) can suppress unwanted multi‑lobed breakup in high‑conductivity systems, while moderate De can be exploited to generate stable conical tips for electrospraying. Future work is suggested to incorporate more complex constitutive models (Giesekus, FENE‑CR) and to validate the predictions experimentally.