标题： 二维稀释乳液的流变学 显示英文标题

标题： On the Rheology of Two-Dimensional Dilute Emulsions

摘要： 在剪切作用下的单个液滴是流体力学中的一个基础问题。 在计算流体力学中，二维（2D）公式在计算效率和相关性方面具有优势，但其理论处理仍然相对不发达。 在本简短的注释中，我们对该问题进行了分析处理，首先推导了二维斯托克斯流动的兰姆解，进而用于获得纯拉伸流动中液滴周围的流动场。 利用这些流动场，得到了稀释乳液的表观粘度$\mu^*$以及小变形理论的表达式。 我们证明了$\mu^* = \mu( 1 + f(\lambda) \phi) + \mathcal{O}(\phi^2)$在$f(\lambda) = (2\lambda + 1)/(\lambda + 1)$的情况下成立，其中$\lambda$是液滴粘度与基质粘度的比值，$\phi$是悬浮相所覆盖的面积分数。 同时，在毛细管主导的条件下，泰勒变形参数$D_T^\infty$的稳态值服从$D_T^\infty = g(\lambda)\,\text{Ca}$，其中 Ca 是毛细管数，$g(\lambda) = 1$。这与三维情况形成对比，其中$g(\lambda)$依赖于$\lambda$。这些结果通过在广泛粘度比范围内的直接数值模拟得到了验证（$0.01 < \lambda < 100$）。我们的结果为解释二维液滴模拟提供了一个基本的理论框架，并为计算流体力学提供了明确的基准。 显示英文摘要

摘要： 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.