On the Rheology of Two-Dimensional Dilute Emulsions

The single droplet under shear is a foundational problem in fluid mechanics. In computational fluid dynamics, the two-dimensional (2D) formulation offers advantages in both computational efficiency and relevance, yet its theoretical treatment remains relatively underdeveloped. In this brief note, we present an analytical treatment of this problem, beginning with a derivation of the Lamb solution for 2D Stokes flows, which in turn is used to obtain the flow fields around a droplet in a purely extensional flow. Using these flow fields, expressions are obtained for the apparent viscosity, $\mu^*$, of a dilute emulsion as well as a small deformation theory. We show that $\mu^* = \mu( 1 + f(\lambda) \phi) + \mathcal{O}(\phi^2)$ with $f(\lambda) = (2\lambda + 1)/(\lambda + 1)$ where $\lambda$ is the ratio of the droplet viscosity to the matrix viscosity and $\phi$ is the area fraction covered by the suspended phase. Also the steady state value of the Taylor deformation parameter $D_T^\infty$, in the capillarity-dominated regime, obeys $D_T^\infty = g(\lambda)\,\text{Ca}$, where Ca is the capillary number and $g(\lambda) = 1$. This contrasts with the 3D case, where $g(\lambda)$ depends on $\lambda$. These results are then validated through direct numerical simulations across a wide range of viscosity ratios ($0.01 < \lambda < 100$). Our results provide a basic theoretical framework for interpreting 2D droplet simulations and provide clear benchmarks for computational fluid dynamics.