A universal rescaling law for the maximum spreading factor of non-Newtonian droplets with power-law fluids

The maximum spreading diameter of non-Newtonian fluid droplets impacting on the solid surface is a key concern in a variety of industrial and medical applications. In this work, we focus on the effect of the shear-thinning, one of the most important non-Newtonian properties, on the spreading dynamics of impacting droplets. A finite element scheme combined with a phase field method and dynamic contact angle model has been employed to perform extensive studies on the spreading process of power-law fluid droplets on solid surfaces with various rheological parameters, impact conditions and surface wettability. The simulation results show that on both hydrophilic and hydrophobic surfaces, impacting droplets exhibit two typical morphologies at the maximum spreading state: a spherical cap in the low-Weber-number range (capillary regime) and a thin-film form in the high-Weber-number range (viscous regime). The maximum spreading factor ${\beta }_{max}$, of droplets with different degrees of shear-thinning converges to the equilibrium spreading state for a droplet with $U_0=0$ at the low-Weber-number limit. Furthermore, a theoretical relationship of ${\beta }_{max}\sim W{e}^{1/2}$ has been derived in the capillary regime. In contrast, the effect of the shear-thinning property becomes significant in the high-Weber-number regime. We discussed the influence of the power-law coefficients $K$ and $n$ on the spreading process and ${\beta }_{max}$ independently. Specifically, as the power-law index $n$ decreases, the morphology of the shear-thinning droplet at the maximum spreading state tends to change from a spherical cap to a thin-film form. Considering the non-uniform distribution of shear rates in the spreading shear-thinning droplet, a new scaling relationship of ${\beta }_{max}\sim ln\left(R{e}_n^{1/\left(2n+3\right)}\right)$ has been proposed based on theoretical derivation and numerical simulations. By introducing an interpolation function on the scaling relationships between the capillary and viscous regimes, we obtained a universal rescaling model that agrees well with numerical and experimental results of non-Newtonian droplets with shear-thinning fluid over a wide range of $We$ numbers, surface wettability and rheological parameters.