The spreading of a cap-shaped spherical droplet of non-Newtonian power-lawliquids on a flat and a spherical rough and textured substrate is theoreticallystudied in the capillary-controlled spreading regime. A droplet whose scale ismuch larger than that of the roughness of substrate is considered. Theequilibrium contact angle on a rough substrate is modeled by the Wenzel and theCassie-Baxter model. Only the viscous energy dissipation within the dropletvolume is considered, and that within the texture of substrate by imbibition isneglected. Then, the energy balance approach is adopted to derive the evolutionequation of the contact angle. When the equilibrium contact angle vanishes, therelaxation of dynamic contact angle $\theta$ of a droplet obeys a power lawdecay $\theta \sim t^{-\alpha}$ except for the Newtonian and the non-Newtonianshear-thinning liquid of the Wenzel model on a spherical substrate. Thespreading exponent $\alpha$ of the non-Newtonian shear-thickening liquid of theWenzel model on a spherical substrate is larger than others. The relaxation ofthe Newtonian liquid of the Wenzel model on a spherical substrate is evenfaster showing the exponential relaxation. The relaxation of the non-Newtonianshear-thinning liquid of Wenzel model on a spherical substrate is fastest andfinishes within a finite time. Thus, the topography (roughness) and thetopology (flat to spherical) of substrate accelerate the spreading of droplet.
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