Effect of viscoelasticity on electrohydrodynamic drop deformation

Abstract

The impact of viscoelasticity on drop deformation in the presence of an electric field is investigated using both analytical and numerical methods. The study focuses on two configurations: a viscoelastic drop suspended in a Newtonian fluid and a Newtonian drop suspended in a viscoelastic medium. Oldroyd-B constitutive equation is employed to model constant viscosity viscoelasticity. Effect of Deborah number (ratio of polymer relaxation time to convective time scale) on drop deformation is studied and explained by examining the electric, elastic and viscous stresses at the interface. For small deformations, we apply the method of domain perturbations, and show that the viscoelastic properties of the drop significantly influence its deformation more than when the surrounding fluid is viscoelastic. Numerical computations are performed using a finite volume framework for larger drop deformations. The transient dynamics of the drops show distinct oscillatory patterns before eventually stabilizing at a steady deformation value. We observe a trend of decreased deformation in both configurations as the Deborah number increases. Relative magnitude of normal and tangential stresses plays a crucial role in drop deformation.

Figures (12)

• Figure 1: Schematic representation of a drop of radius $a$ in the presence of an imposed uniform electric field $\boldsymbol{E}_\infty$. A spherical coordinate system ($r, \theta, \varphi$) is considered which is attached to the centroid of the drop.
• Figure 2: (a) Variation of drop deformation with $R$ for $Q=10$. The other dimensionless parameters considered are $C=0.2$, $De=0.2$, $\beta=0.1$, $Re=1$, and $\mu_r=1$. Markers show the $O(C) + O(C^2)$ deformation from Ajayi1978. (b) Deviation from Newtonian behavior is higher for lower values of $R$.
• Figure 3: Streamlines for $V_dN_m$ configuration obtained from numerical simulations. $\beta=0.1, \mu_r=1, C=0.2, De=0.2$. The left side shows the contributions for the prolate case, $R=10, Q=0.1$, and the right side for the oblate case, $R=0.5, Q=2$. (a) $O(1)$ to prolate. (b) $O(1)$ to oblate. (c) $O(C)$ to prolate. (d) $O(C)$ to oblate. (e) $O(De)$ to prolate. (f) $O(De)$ to oblate. (g) Final streamlines for prolate. (h) Final streamlines for oblate
• Figure 4: Schematic representation (not to scale) of the initial configuration and boundary conditions in an axisymmetric coordinate system ($r,z$). The size of the domain is $16a \times 16a$, where $a$ is the radius of the drop. Electric field $(\boldsymbol{E}_\infty)$ is applied along the negative $z$ axis.
• Figure 5: Temporal evolution of drop deformation ($D$) for different levels of grid refinement. The inset shows the zoomed view for better convergence analysis. The dimensionless parameters for the computations are $R=0.5$, $Q=2$, $De=1$, $C=0.2$, $Re=1$, $\beta=0.1$, $Re_E=0.01$ and $\mu_r=1$.