The single droplet under shear is a foundational problem in fluid mechanics. In computational fluid dynamics (CFD), the two-dimensional (2D) problem offers advantages in both computational efficiency and relevance, yet it lacks a comprehensive theoretical treatment equivalent to that of the three-dimensional (3D) case. In this brief note, we present an analytical treatment of this problem beginning with a derivation of the Lamb solution for 2D Stokes flows and then applying it to obtain the flow field around a droplet in a purely extensional flow. Using these flow fields, expressions are obtained for the apparent viscosity, ${\mu }^{*}$, of a dilute 2D emulsion as well as a small deformation theory. We show that in the limit of both zero Reynolds number ($\text{Re}\to 0$) in which inertia is negligible and zero capillary number ($\text{Ca}\to 0$) in which surface tension prevents any droplet deformations, ${\mu }^{*}=\mu (1+f\left(\lambda \right)\varphi )+\mathcal{O}\left({\varphi }^2\right)$ with $f\left(\lambda \right)=(2\lambda +1)/(\lambda +1)$ where $\lambda$ is the ratio of the droplet viscosity to the matrix viscosity and $\varphi$ is the area fraction covered by the suspended phase. In addition, we show that in the capillarity-dominated regime ($\text{Ca}\ll 1$), the steady-state value, $D_T^{\infty }$, of the Taylor deformation parameter obeys $D_T^{\infty }=g\left(\lambda \right)\text{Ca}$, where Ca is the capillary number and $g\left(\lambda \right)=1$. This contrasts with the 3D case, where $g\left(\lambda \right)$ depends on $\lambda$. These results are then validated through direct numerical simulations (DNSs) of a single droplet at the center of a periodic square box undergoing simple shear. The DNS and theory are compared across a wide range of viscosity ratios ($0.01\le \lambda \le 100$) and area fractions ($0.002\le \varphi \le 0.125$) and they are found to agree asymptotically in the limit $\varphi \to 0$ for $\text{Ca}\ll 1/(\lambda +1)$. Our results provide a basic theoretical framework for interpreting 2D droplet simulations and provide clear benchmarks for CFD codes in 2D.

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• Rheology
• 2-dimensional systems
• Drops & bubbles
• Emulsions