This study numerically investigates the spreading of a Newtonian liquid lens over a viscoplastic fluid substrate described by the Herschel–Bulkley–Papanastasiou model. Simulations are performed with the open-source rheoMultiFluidInterFoam solver under the OpenFOAM framework. The droplet spreading process over the viscoplastic fluid is divided into three phases: (i) initial yielding, (ii) main spreading, and (iii) creeping flow. In the first two phases, capillary forces and yield stress govern the droplet dynamics until a quasi-steady state is reached, where all material becomes unyielded according to the Von Mises criterion. The droplet's geometric features in this state differ significantly from those in a purely Newtonian equilibrium. With increasing yield stress, the morphology of droplet resembles that when spreading over a rigid wall. Enhanced shear thinning results in faster yielding and deeper penetration of the lens into the viscoplastic material at early stages. As the spreading parameter is increased, a stronger immersion of the drop into the bottom fluid is observed, and the drop equilibrates to larger aspect ratios. The initial depth of the viscoplastic layer has limited impact on droplet spreading, as the unyielded layer beneath mitigates substrate thickness influence. However, thinner layers cause greater resistance and faster yielding, affecting overall flow dynamics. The viscoplastic behavior, modeled using the regularized Papanastasiou model, allows for finite creeping motion with very high viscosity, leading to a final phase where the material creeps to an equilibrium state similar to that of a Newtonian subphase.
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