Spatiotemporal Viscosity Gradients Shape Breakup and Relaxation in Non-Newtonian Drops

Unlike elongated Newtonian drops, whose relaxation to a sphere is approximately exponential and governed by a single characteristic timescale, particle-filled drops can exhibit complex, multistage relaxation and may fail to recover a spherical shape. These dynamical features are important because they determine the morphology of particle-filled immiscible blends, which in turn controls their macroscopic properties. Our prior experiments demonstrated that, at sufficiently high particle loading, the initially rapid relaxation eventually slows down and appears to arrest at a finite, non-spherical deformation. However, the experiments could not resolve the spatiotemporal evolution of internal quantities such as viscosity, velocity gradients, and stress fields. Here, we investigate whether the retardation and subsequent apparent arrest of initially elongated drops can arise solely from strong shear thinning, without invoking fluid elasticity or a true yield stress. Motivated by experiments on particle-filled polyisobutylene drops, we perform axisymmetric, two-phase volume-of-fluid simulations in which the drop phase follows a Carreau–Yasuda constitutive law and the suspending fluid is Newtonian. The numerical method is first validated against classical predictions for Newtonian drop relaxation. We find that Carreau–Yasuda drops exhibit a two-stage response comprising rapid initial relaxation followed by pronounced late-time retardation. The onset, magnitude and rate of this slowing are governed by a dimensionless parameter S that compares the material shear-thinning rate scale with the characteristic capillary relaxation rate. Spatially resolved fields reveal that geometrically induced low-strain-rate regions near the retracting poles enter the high-viscosity regime before the remainder of the drop. These localized regions resist the flow required for continued shape recovery and act as viscous anchors, producing a self-reinforcing positive coupling between decreasing strain rate, increasing viscosity, and suppressed relaxation. Because the Carreau–Yasuda fluid never becomes truly unyielded, we introduce an operational stress-based criterion for apparent arrest using the characteristic stress η0/λ where η0 is the zero-shear viscosity of the Carreau–Yasuda fluid, and λ is the Carreau–Yasuda relaxation time. We demonstrate and rationalize via a stress balance argument that the deformation at apparent-arrest Darr ∝ S in this weak-to-moderate deformation regime. Taken together, spatiotemporal variations in the viscosity, strain-rate, and normalized-stress fields identify the microscopic origin of the arrested relaxation observed experimentally: a geometrically inherited low-strain-rate region near the drop ends, amplified by the shear-thinning constitutive response into a high-viscosity anchor, which couples back through positive feedback and stabilizes the drop at a finite arrested deformation.