  The spreading of a cap-shaped spherical droplet of non-Newtonian power-law
liquids on a completely wettable spherical substrate is theoretically studied.
Both convex spherical substrates and concave spherical cavities with smooth or
rough surfaces are considered. The droplet on a rough substrate is modeled by
either the Wenzel or the Cassie-Baxter model. The two sources of driving force
of spreading by the surface-tension and the line tension are considered. Also,
the two channels of energy dissipation by the viscous dissipation within the
bulk and the frictional dissipation at the contact line are considered. A
combined theory of spreading on a spherical substrate is constructed by
including those four factors. The spreading process is divided into four
stages, each of which is governed by one of two driving forces and one of two
dissipations. It is found that the dynamic contact angle $\theta$ has a
characteristic time ($t$) dependence at each stage. It does not necessarily
follow the standard power law $\theta \sim t^{-\alpha}$. Instead, the
relaxation can be a power-law with the exponent $\alpha$ different from that on
a flat substrate, or it can be exponential or it can finish within a finite
time. Therefore, various spreading scenarios on a spherical substrate and in a
spherical cavity are predicted.
