Moving contact lines of power-law fluids: a Cox-Voinov generalization revealing how nonlinear fluid rheology alters stress singularity and wetting dynamics

Rate-dependent viscosity in power-law fluids significantly affects contact line stress singularities and moving contact line behaviour. Contact line forces show more severe divergence for shear-thickening fluids ( $n\gt 1$ ) or remain finite for shear-thinning fluids ( $n\lt 1$ ). Complementing earlier self-similar derivations of spreading laws by Starov et al. ( J. Colloid Interface Sci. vol. 257, 2003, pp. 284–290) for shear-thinning drops, we extend the classical Cox-Voinov theory to power-law fluids and obtain explicit dynamic contact angle relationships – results that are more fundamental than previously reported spreading laws. This development provides a unified yet fundamentally distinct description of advancing contact line behaviour across the full range of shear-thinning and shear-thickening rheologies. We show that the apparent dynamic contact angle $\theta _{d}$ depends critically on the characteristic dissipation length $h^{*}\propto U^{n/(n-1)}$ , fundamentally altering its dependence on contact line speed $U$ . For shear-thinning fluids ( n < 1) with less diverging contact line stresses, this length scale yields $\theta _{d}\sim C{a_{\textit{local}}}^{1/3}$ in the familiar Cox–Voinov form in terms of the local capillary number $ \textit{Ca}_{\textit{local}} = (h/h^{*})^{1-n}$ , with the contact line motion dissipated within $h^{*}$ extending beyond local wedge height $h$ , thereby eliminating the need for a microscopic cutoff. This feature also renders $\theta _{d}$ size dependent and varying with the spreading radius $R$ , recovering $R\propto t^{n/(3n+7)}$ and $\theta _{d}\propto U^{3n/(2n+7)}$ as previously derived by Starov et al. (2003). For shear-thickening fluids ( n > 1) that exhibit more strongly diverging contact line stresses, by contrast, the contact line motion is dissipated within a much narrow region $h^{*}$ that is much smaller than the required microscopic cutoff h m . A complete precursor theory is also developed, showing $ h_{m} \propto U^{-n/(4-n)}$ . This leads to $\theta_{d} \propto U^{n/(4-n)}$ , making the global spreading behaviour highly sensitive to the contact line microstructure. Importantly, regardless of the microscopic mechanisms, the apparent dynamic contact angle relationship can always be expressed in the analogous Cox–Voinov form $\theta _{d}\sim {\textit{Ca}_{\textit{eff}}}^{1/3}$ in terms of the effective capillary number