On the deformation of a shear thinning viscoelastic drop in a steady electric field

Abstract

The deformation of viscoelastic drops under electric fields plays a crucial role in applications such as microfluidics, inkjet printing, and electrohydrodynamic manipulation of complex fluids. This study examines the deformation and breakup dynamics of a linear Phan-Thien-Tanner (LPTT) drop subjected to a uniform electric field using numerical simulations performed with the open-source solver Basilisk. Representative combinations of conductivity ratio ($σ_r$) and permittivity ratio ($ε_r$) are chosen from six characteristic regions of the ($σ_r$, $ε_r$) phase space, $PR_A^+$, $PR_B^+$, $PR_A^-$, $PR_B^-$, $OB^+$, and $OB^-$. In regions where the first- and second-order deformation coefficients have the same sign ($PR_A^-$, $PR_B^-$, $OB^+$), the LPTT drops exhibit deformation dynamics that negligibley deviate from the Newtonian behavior. In the $PR_A^+$ region, drops deform into prolate spheroidal shapes below a critical electric capillary number and transition to stable multi-lobed shapes or breakup beyond this threshold. Increasing elasticity of drop opposes the deformation, thereby reducing deformation and increasing critical $Ca_E$ with the Deborah number ($De$). In the $PR_B^+$ region, drops form prolate shapes below critical $Ca_E$ and develop conical ends above it. The steady-state deformation exhibits a non-monotonic dependence on $De$, increasing at low $De$ and decreasing at higher values. A similar non-monotonic variation is also observed in critical $Ca_E$. In the $OB^-$ region, LPTT drops attain oblate shapes below critical $Ca_E$ and undergo breakup beyond it. The deformation magnitude shows a non-monotonic variation with $De$, increasing initially and decreasing at higher elasticity.

Figures (26)

• Figure 1: Schematic of the problem. LPTT drop of radius $R$ is subjected to an external electric field, $\bm{E}_\infty$, aligned along the axis of symmetry. The domain size is set to $32R \times 32R$ to minimize the boundary effects.
• Figure 2: Deformation vs. time for various sizes of the simulation domain. Simulation parameters are $\mu_r=1$, $\rho_r=1$, $Re=1$, $\beta_i = 1/9$, $\sigma_r = 10$, $\epsilon_r=1.37$, $De=5$, $Ca_E=0.4$, $\varepsilon=0.25$. $R/\Delta x_{min}$ is taken as 256.
• Figure 3: (a) Deformation parameter variation with time at various refinements of the grid. (b) Steady state deformed interface of the drop for various grid refinements. Simulation parameters: $Re=1$, $Ca_E=0.4$, $De=5$, $\sigma_r=10$, $\epsilon_r=1.37$, $\beta=1/9$, $\varepsilon=0.25$
• Figure 4: $(\sigma_r, \epsilon_r)$ pairs selected for the study are enumerated and marked (red circle markers) on $(\sigma_r, \epsilon_r)$ phase plot on log-log scale.
• Figure 5: Deformation versus $Ca_E$ for various $De$, for $\rho_r=1$, $\mu_r=1$, $Re=1$, $\beta_i = 1/9$, $\varepsilon=0.25$. (a) $\sigma_r=0.1$, $\epsilon_r=0.04$$(PR_A^-)$ (b) $\sigma_r=0.01$, $\epsilon_r=0.1$$(PR_B^-)$ (c) $\sigma_r=2$, $\epsilon_r=20$$(OB^+)$