Analytical expressions of the deformation limit of fluid particles

The deformation of fluid particles is usually directly related to various Lagrangian phenomena in fluids. We show typical analytical expressions under external strain rate fields by defining the deformation limit for small fluid particles, aiming at providing analytical tools for supporting future investigations on the Lagrangian phenomena in fluids. In particular, when particles are driven by a constant strain rate which is greater or equal to the rotation rate, the orientation of fluid elements aligns with strain tensor in short times and then decouples with them in long times, while the fluid particles are flattened in most cases, which are similar to previous observations in turbulent flows. This supports the simplifications in previous Lagrangian models (e.g., Fang et al., 2015 [20]), where the strain rates were assumed to be locally homogeneous.