Deformation and breakup behaviors of a Giesekus viscoelastic droplet in Newtonian shear flow

The deformation and breakup of a Giesekus viscoelastic droplet suspended in a Newtonian matrix under simple shear flow is investigated through a lattice Boltzmann method, which uses the color-gradient model for immiscible two-phase flows while coupling a lattice advection-diffusion scheme for the Giesekus constitutive equation. The influence of the Deborah number (De) and shear-thinning mobility parameter (α) of the droplet is first explored for relatively low capillary numbers (Ca). The transient overshoot is observed before the deformation reaches the steady state, and the overshoot generally increases with De but decreases with α. With the increase of De, the steady-state deformation parameter first decreases and then increases, which is attributed to two opposite effects on the elastic force, namely the promoting effect from the increased polymer extension and the inhibiting effect caused by the deviation of the location where the maximum elastic force occurs from the droplet tips. The droplet inclination angle varies little with α but increases with De, where the latter can be explained as an increased vertical component of viscous and elastic forces in the tip regions. Upon increasing α, the steady-state deformation parameter becomes higher due to the reduced elastic force at the droplet tips. In addition, the phase diagrams of Ca-De and Ca-α are presented in which the critical capillary number Ca_cr characterizing the droplet breakup can be identified. It is found that Ca_cr varies non-monotonically with De, while Ca_cr is not very sensitive to α.