3.1. Stress singularity versus force singularity: distinct dissipation length scales

To begin, we recognise that even in a power-law fluid, a stress singularity persists at a moving contact line due to a velocity jump between the moving air–liquid interface and the no-slip substrate. This can be seen from the fact that the viscous stress $\tau$ with the power index $n\gt 0$ diverges as the contact line is approached:

(3.2) \begin{align}\tau =\eta \frac{\partial u}{\partial y}=k\left(-\frac{\partial u}{\partial y}\right)^{n}\sim k\left(\frac{U}{h}\right)^{n}\rightarrow \infty\quad\text{as }h\rightarrow 0.\end{align}

However, the viscous braking force F _brake on the contact line does not necessarily diverge at the contact line as can be seen by integrating (3.2) with respect to $x$ . Specifically, with $h\approx \theta _{d}x$ , this force (per unit length) for $n\neq 1$ , F _brake behaves as

(3.3) \begin{align}F_{\textit{brake}}=k\int _{x_{m}}^{L}\left(-\frac{\partial u}{\partial y}\right)^{n}\,\textrm{d}x\sim k\int _{x_{m}}^{L}\left(\frac{U}{\theta _{d}x}\right)^{n}\,\textrm{d}x=\frac{k}{1-n}\left(\frac{U}{\theta _{d}}\right)^{n}\left[L^{1-n}-x_{m}^{1-n}\right]\!,\end{align}

where $L$ is the length of the macroscopic wedge and $x_{m}$ ( $\ll L$ ) is a mesoscopic length bridging the macroscopic and microscopic regions. As evident from (3.3), whether $F_{\textit{brake}}$ is dominated by $L$ or $x_{m}$ critically depends on whether $n\lt 1$ or $n\gt 1$ . The force described by (3.3) is very different than the Newtonian fluid case where the contact line force diverges logarithmically as $x_{m}$ → 0 (De Gennes Reference de Gennes1985):

(3.4) \begin{align} F_{\textit{brake}}\left(n=1\right)\sim \frac{\eta U}{\theta _{d}}\ln \left(\frac{L}{x_{m}}\right) \! . \end{align}

As evident from (3.3), F _brake can still diverge for n > 1 or remain finite for n < 1 as x_m → 0. When n > 1, the viscosity $\eta \sim k (U/h)^{n-1}$ becomes increasingly pronounced as h approaches the contact line, thereby producing a much stronger divergence force of the contact line than that given by (3.4) for the Newtonian case. This causes viscous dissipation to be even more strongly concentrated in the immediate vicinity of the contact line.

In contrast, for n < 1, since the viscosity $\eta \propto h^{1-n}$ vanishes at the contact line (h = 0) and increases toward the bulk, the dissipation must extend beyond the contact line region. Consequently, the force is redistributed over a larger portion of the wedge, leading to a finite contact line force that reflects the integrated dissipation throughout the wedge. Similar conclusions can also be drawn from the viscous energy dissipation rate over the wedge (Carré & Woehl Reference Carré and Woehl2002; Liang et al. Reference Liang, Wang, Duan and Min2012), i.e.

(3.5) \begin{align}T\dot{\Sigma }_{w}\equiv \int _{x_{m}}^{L}\int _{0}^{h}\eta \left(\frac{\partial u}{\partial y}\right)^{2}\,\textrm{d}y\,\textrm{d}x\sim \frac{k}{1-n}\left(\frac{U}{\theta _{d}}\right)^{n+1}\theta_d\left[L^{1-n}-x_{m}^{1-n}\right]\!,\end{align}

from which (3.3) can be readily recovered using $F_{\textit{brake}}\sim T\dot{\Sigma }_{w}/U$ . Both the force and the energy dissipation rate are proportional to the same factor ( $L^{1-n}-x_m^{1-n}$ ), which determines whether viscous effects are localised near the contact line or extend across the wedge. This viscous energy dissipation approach not only reveals that the dominant region of viscous dissipation coincides with that of the contact line force but also provides a picture of how the local stress distribution governs the overall contact line dynamics. In what follows, we discuss the different contact line force characteristics for n < 1 and n > 1.

For shear-thinning fluids ( $n\lt 1$ ), F _brake described by (3.3) is dominated by the macroscopic $L$ term (Carré & Eustache Reference Carré and Eustache2000):

(3.6) \begin{align}F_{\textit{brak}\mathrm{e}}(n\lt 1)\sim \frac{k}{1-n}\left(\frac{U}{\theta _{d}}\right)^{n}L^{1-n}.\end{align}

Thus, although the stress exhibits a singularity at the contact line, it is in fact integrable, causing F _brake to remain finite at the contact line regardless of $x_{m}$ . This integrable singularity feature could be easily overlooked, as illustrated, for instance, by Wang et al. (Reference Wang, Lee, Peng and Lai2007a ) who introduced a microscopic cutoff x_m in their derivation of the dynamic contact angle relationship, in a manner analogous to the Newtonian fluid case. It follows, therefore, that since the viscous dissipation is no longer concentrated at the contact line but grows with L, the local contact line motion can be dissipated through the bulk without needing any microscopic cutoff. This feature is consistent with the key characteristics of shear-thinning fluids identified by King (Reference King2001) and Betelu & Fontelos (Reference Betelu and Fontelos2004), who rigorously solved the governing equations. Here we provide a much simpler reasoning and physical argument to re-establish this feature. As will be seen in § 3.2, this will lead to $ \theta_{d}$ depending on the extent of spreading.

For shear-thickening fluids ( $n\gt 1$ ), on the contrary, the force (3.3) is mainly controlled by the mesoscopic $x_{m}$ term,

(3.7) \begin{align}F_{\textit{brake}}\left(n\gt 1\right)\sim \frac{k}{1-n}\left(\frac{U}{\theta _{d}}\right)^{n}x_{m}^{1-n},\end{align}

indicating that F _brake still diverges as $x_{m}$ → 0. This divergence is of a power-law type and, thus, more severe than the logarithmic divergence (3.4) seen in the Newtonian case. Consequently, a microscopic cutoff is required to alleviate the force singularity at the contact line. The need for such a regularisation at the contact line has also been noted by King (Reference King2001) for the same reason. This necessity also makes the analysis of shear-thickening fluids more challenging and, as a result, less studied.

Characteristic differences between the $n\lt 1$ and $n\gt 1$ cases have been noticed by Liang et al. (Reference Liang, Wang, Duan and Min2012) from an energy dissipation perspective. However, their expression for viscous dissipation energy followed the bulk dissipation mode (3.6) derived by Carré & Eustache (Reference Carré and Eustache2000) for shear-thinning fluids ( $n\lt 1$ ) and was extended to the shear-thickening case ( $n\gt 1$ ) by introducing an additional dissipation factor to account for the effects of the stress singularity. As a result, their formulas for $\theta _{d}$ showed little qualitative difference between these two cases, apart from the dissipation factor. This challenges the notion that their contact line motions are dissipated within different dissipation length scales, as indicated by (3.6) and (3.7). Therefore, their dynamic contact angles must exhibit distinct behaviours, which we demonstrate next.

3.2. Energy dissipation and scaling relationships for ${\theta }_{{d}}$

Having obtained the distinct viscous force characteristics (3.6) and (3.7), we now balance each with the surface tension force to derive the corresponding scaling relationships for $\theta _{d}$ . This will enable us to reveal essential features of how $\theta _{d}$ varies with $U$ and the relevant length scales.

The surface tension force that drives the contact line is often expressed as a pulling tension force at the dynamic state relative to the static state with the equilibrium contact angle $\theta _{e}(\lt \theta _{d})$ (Carré & Eustache Reference Carré and Eustache2000):

(3.8) \begin{align}F_{s}=\gamma (\cos \theta _{e}-\cos \theta _{d}).\end{align}

For small $\theta _{d}$ , as in the case of complete wetting, (3.8) simplifies to (De Gennes Reference de Gennes1985)

(3.9) \begin{align}F_{s}\approx (\gamma /2)\theta _{d}^{2}+S.\end{align}

Here $S=\gamma _{SV}-\gamma _{LS}=\gamma (\cos \theta _{e}-1)\approx -(\gamma /2)\theta _{e}^{2}$ is the spreading coefficient, which measures the excess interfacial free energy per unit area. The terms $\gamma _{SV}$ and $\gamma _{LS}$ represent the solid–vapour and the liquid–solid interfacial tensions, respectively.

As indicated by (3.9), this surface tension force includes an additional contribution from $S$ to account for the wettability of a substrate. In the case of complete wetting, it is well established that for Newtonian fluids, the spreading kinetics described by Tanner’s law remains universal, independent of the value of $S$ (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). This implies that $S$ does not enter the dynamic contact angle relationship $\theta _{d}(U)$ , which also explains the universality of the Cox–Voinov law (2.3).

De Gennes (Reference de Gennes1985) has shown for the Newtonian case that the excess interfacial free energy $S$ is virtually consumed by the precursor film ahead of the macroscopic wedge. Here we establish that this also holds for power-law fluids. As shown analytically in Appendix B, the viscous dissipation energy (per unit time) within the film $T\dot{\Sigma }_{f}$ is nearly equal to the work (per unit time) done by $S$ , i.e.

(3.10) \begin{align}T\dot{\Sigma }_{f}\equiv \int _{-d}^{0}\int _{0}^{h}\eta \left(\frac{\partial u}{\partial y}\right)^{2}\,\textrm{d}y\,\textrm{d}x\approx SU,\end{align}

where $d$ is the width of the ‘truncated’ film with thickness given by $h_{e}=a(3\gamma /2S)^{1/2}$ , with $a$ being the molecular length associated with disjoining pressure (De Gennes Reference de Gennes1985). The complete burnout of $S$ by the precursor film also indicates that the effects of substrate wettability or $\theta _{e}$ will only enter the contact line microstructure without altering how the surface tension force drives the macroscopic contact line. Previous studies (Dandapat & Singh Reference Dandapat and Singh2010; Liang et al. Reference Liang, Wang, Duan and Min2012) used the full tension force (3.8) to derive dynamic contact angle relationships for small $\theta _{d}$ , often neglecting $S$ in (3.9) under the assumption that $\theta _{e}\ll \theta _{d}$ . However, if $S\approx -(\gamma /2)\theta _{e}^{2}$ is included, this can lead to an overestimation of $\theta _{d}$ or even fundamentally change its dependence on $U$ .

Thus, only the dynamic component $(\gamma /2)\theta _{d}^{2}$ in (3.9) will be balanced to the viscous braking force over the wedge. In the following, we demonstrate that the distinct dissipation length scales in (3.6) and (3.7) for the $n\lt 1$ and $n\gt 1$ cases will result in fundamentally different characteristics of the corresponding dynamic contact angles.

For shear-thinning fluids ( $n\lt 1$ ), balancing $(\gamma /2)\theta _{d}^{2}$ to (3.6) gives

(3.11a) \begin{align}\theta _{d}^{n+2}\sim \frac{k}{\gamma }U^{n}L^{1-n},\end{align}

or expressed in terms of the local wedge height $h\,\sim L\theta _{d}$ :

(3.11b) \begin{align}\theta _{d}^{3}\sim \frac{k}{\gamma }U^{n}h^{1-n}.\end{align}

Thus, $\theta _{d}$ increases with the macroscopic scale $L$ or $h$ over which the contact motion is mainly dissipated, as indicated by (3.6). In drop spreading problems, $L$ corresponds to the spreading radius $R$ , making $\theta _{d}$ dependent on the extent of spreading. Given $R=({\Omega} /\theta _{d})^{1/3}$ from the constant volume constraint $R^{3}\theta _{d}={\Omega}$ , this changes the dependence of $\theta _{d}$ on $U$ in (3.11a ) to

(3.12) \begin{align}\theta _{d}\propto U^{3n/(2n+7)}.\end{align}

It is worth pointing out that while the mesoscopic length $x_{m}$ does not enter the apparent contact angle $\theta _{d}$ at the macroscopic level, it may still affect the inner portion of the contact line as $L$ approaches $x_{m}$ or $h$ approaches $h_{m}$ in (3.11). In other words, although a shear-thinning fluid does not require an additional microscopic cutoff to avoid the stress singularity in the macroscopic contact line motion, it may still admit a meso/micro contact line structure. This situation is similar to that of a strongly slipping moving contact line, where the dynamic contact angle is found to be proportional to $L^{1/2}$ or $h^{1/3}$ (Wei et al. Reference Wei, Tsao and Chu2019), corresponding to the strong shear-thinning $n\rightarrow 0$ limit in (3.11).

As for shear-thickening fluids ( $n\gt 1$ ), by balancing the tension force $(\gamma /2)\theta _{d}^{2}$ to the viscous braking force (3.7), we obtain the scaling for $\theta _{d}$ , i.e.

(3.13a) \begin{align}\theta _{d}^{n+2}\sim \frac{k}{\gamma }U^{n}x_{m}^{1-n},\end{align}

or, written in terms of the mesoscopic meniscus height $h_{m}\sim x_{m}\theta _{d}$ ,

(3.13b) \begin{align}\theta _{d}^{3}\sim \frac{k}{\gamma }U^{n}h_{m}^{1-n}.\end{align}

In contrast to (3.11), (3.13) now involves the mesoscopic length $x_{m}$ or height $h_{m}$ , which helps alleviate the contact line singularity due to (3.7). It is important to note that these microscopic length scales are generally not fixed but vary with $U$ , depending on the detailed contact line microstructure (De Gennes Reference de Gennes1985; Eggers Reference Eggers2005; Wei et al. Reference Wei, Tsao and Chu2019). Typically, $h_{m}\propto U^{-\alpha }$ where $\alpha \gt 0$ , with its value determined by the nature of the contact line microstructure (Wang & Zhu Reference Wang and Zhu2014). This alters the dependence of $\theta _{d}^{3}$ on $U$ in (3.13b ) to

(3.14) \begin{align}\theta _{d}^{3}\propto U^{n+\alpha (n-1)},\end{align}

implying that the dynamics of the macroscopic contact line are strongly influenced by the microscopic structure of the contact line.

Note that the above results are derived for the complete wetting scenario where the contribution from the spreading coefficient $S\approx -(\gamma /2)\theta _{e}^{2}$ is not included and is entirely dissipated within the precursor film according to (3.10).

However, in the case of a partial wetting situation, this contribution becomes significant and can no longer be ignored. By incorporating this term into (3.9) and balancing it with (3.7), (3.13a ) is modified to

(3.15) \begin{align}\theta _{d}^{2+n}\sim \theta _{d}^{n}\theta _{e}^{2}+\frac{k}{\gamma }U^{n}x_{m}^{1-n}.\end{align}

Unlike the complete wetting case, the required microscopic cutoff $x_{m}$ becomes a fixed quantity, typically dependent on the equilibrium contact angle $\theta _{e}$ and the equilibrium film thickness $h_{e}=a(3\gamma /2S)^{1/2}$ (De Gennes Reference de Gennes1985). During the early stages of spreading where $\theta _{d}\gg \theta _{e}$ and the contact line speed $U$ is high, $\theta _{d}$ can still be described by (3.13a ), which can be deduced from (3.15). As spreading progresses and $\theta _{d}$ approaches $\theta _{e}$ with a significant reduction in $U$ at later stages, the behaviour of $\theta _{d}$ transitions to that described by (3.15).

It is worth mentioning that Wang et al. (Reference Wang, Lee, Peng and Lai2007a ) also derived a dynamic contact angle relationship for complete wetting shear-thickening fluids, which is essentially (3.13a ). However, in their expression, the microscopic cutoff $x_{m}$ was treated as fixed, which is not consistent with the assumed complete wetting situation unless a microscopic slip length exists to provide a physically meaningful fixed cutoff (Eggers Reference Eggers2004).

4.1. Assumptions

Prior to performing detailed analysis, we explicitly state some of the key assumptions underlying our theoretical framework. This serves to establish the scope of applicability of the theory, identifies the physical regimes most relevant to the present problem and clarifies the validity and generality of the results derived in the subsequent sections.

As our primary focus is on dynamic wetting and spreading, we consider the motion of an advancing contact line. We assume that the apparent dynamic contact angle $\theta _{d}$ remains small throughout the contact line motion, typically $\theta _{d}\lt$ 30°, as also adopted in the preceding scaling analysis. This corresponds to complete wetting scenarios where surface tension forces dominate, thereby justifying the use of the standard lubrication approximation, under which tan $ \theta_{d}$ ≈ $ \theta_{d}$ . This simplification allows a tractable description of the interface geometry and the stress balance near the contact line. Importantly, within this regime, the dynamic contact angle $\theta _{d}$ is determined purely by hydrodynamic effects and is effectively independent of substrate wettability. For larger contact angles, as encountered in partial wetting conditions, the present analytical framework would need to be extended to account for finite wedge geometry and the influence of substrate wettability (Cox Reference Cox1986; Snoeijer Reference Snoeijer2006; Chan et al. Reference Chan, kamal, snoeijer, spritles and eggers2020).

We further assume that the interface slope remains small, or equivalently, that the interface varies slowly along the horizontal direction x, i.e. $|\partial h/\partial x|\ll 1$ everywhere except possibly within a narrow transition region near the contact line, where the slope may become large due to the loss of scale separation. This implies that the characteristic horizontal length scale L – typically a macroscopic scale such as the drop radius – greatly exceeds the vertical scale H such that $ \epsilon = H/L\ll 1 $ . Under this small-slope assumption, inertial effects are often negligible compared with viscous effects, since their ratio ε ² Re is typically very small, where $\textit{Re} = \rho UL/ \eta_{\textit{char}} $ is the Reynolds number with $ \rho $ the fluid density, U the contact line speed and $\eta_{\textit{char}} $ a characteristic viscosity. This condition justifies the lubrication approximation, allowing us to simplify the momentum equations by neglecting terms of O(ε ²) and higher. Moreover, with inertia neglected, the flow field adjusts instantaneously to the contact line motion, enabling the steady-state formulation that leads to the governing equation (4.1), presented in § 4.2 next.

In addition, the temperature is assumed to remain uniform throughout the fluid, and thermal effects – including Marangoni flows due to temperature gradients – are considered negligible. This assumption is appropriate for experiments conducted at controlled ambient temperature, where heat generation from viscous dissipation is minimal.

Lastly, we assume that the fluid viscosity follows a simple power-law constitutive relation, which serves as an idealisation valid over a finite range of shear rates. Many complex fluids exhibit more intricate rheological behaviour, such as yield stress, thixotropy or viscoelasticity, which would require more sophisticated constitutive models. Nevertheless, the power-law model captures the essential shear-rate dependence of viscosity and provides a tractable yet physically meaningful framework for analysing the fundamental hydrodynamic mechanisms governing dynamic wetting in non-Newtonian fluids.

As shown below, under these assumptions, the present framework provides a minimum set of ingredients required to capture the essential physics of contact line motion in power-law fluids under conditions commonly encountered in experimental studies.

4.2. Modified Tanner equation and characteristic dissipation length

With the above assumptions, in this section we now apply the standard lubrication theory to derive the equation governing the contact line motion for power-law fluids (see Appendix A for a detailed derivation). This leads to the modified Tanner equation

(4.1) \begin{align}h^{\prime\prime\prime}=-K\frac{U^{n}}{h^{n+1}},\end{align}

where $K=(k/\gamma )(2+1/n)^{n}$ . The interface must satisfy the condition that its curvature vanishes at infinity,

(4.2) \begin{align}h^{\prime\prime}(x\rightarrow \infty )=0.\end{align}

Rearranging (4.1) into the form analogous to (2.1) for the Newtonian case, we express it as

(4.3) \begin{align} h^{\prime \prime \prime}=- \frac{1}{h^{2}}\left(\frac{h^{*}}{h}\right)^{n-1} , \end{align}

where

(4.4) \begin{align} h^{*} = \left(\textit{KU}^{n}\right)^{ \frac{1}{n-1}} \end{align}

represents an inherent length scale. The factor $ (h^{*}/h)^{n-1} $ , with the exponent (n−1) originating from the power-law viscosity (3.1), quantifies the contact line resistance relative to the Newtonian case’s $ 1/h^{2} $ . Alternatively, $ (h^{*}/h)^{n-1} $ in (4.3) can be interpreted as a local capillary number,

(4.5) \begin{align} Ca_{\textit{local}}=\left(\frac{h^{*}}{h}\right)^{n-1} , \end{align}

measuring the ratio of the local viscous to surface tension effects near the contact line. A small $ \textit{Ca}_{\textit{local}} $ indicates that surface tension dominates, enabling smooth wetting with a small dynamic contact angle, whereas a large $ \textit{Ca}_{\textit{local}} $ reflects enhanced viscous resistance that tends to steepen the interface due to weakening wetting.

As shown below, the requirement that this local capillary number remains small for smooth wetting will define the extent of $ h^{*} $ . Thus, $ h^{*} $ can be regarded as a characteristic dissipation length that determines the extent of viscous dissipation, depending on the values of n. It is important to note that for the Newtonian case (n = 1), (4.4) formally yields $ h^{*} $ → ∞, indicating that $ h^{*} $ is not a meaningful characteristic length scale for Newtonian fluids. In this case, the local capillary number (4.5) reduces to $ \textit{Ca}_{\textit{local}} $ equal to a constant (independent of h), reflecting the absence of a characteristic dissipation length in Newtonian fluids. This renders the Newtonian fluid a special case, exhibiting the scale-invariant solution form (2.2), which in turn leads to the logarithmic form of the Cox–Voinov law (2.3) for the dynamic contact angle. In contrast, for power-law fluids (n ≠ 1), $ \textit{Ca}_{\textit{local}} $ varies spatially with h, delineating regions where viscous dissipation or surface tension dominates, with the transition occurring at h ∼ $ h^{*} $ where $ \textit{Ca}_{\textit{local}} $ ∼ 1.

To examine how $ h^{*} $ influences the contact line structure, we let the local wedge length x scale as L. Then, the interface slope $ h^{\prime} = \text{tan}\,\theta \sim h/L $ . With $ h^{\prime\prime\prime}h^{2} \sim (h/L)^{3} $ , (4.3) yields the following scaling for the local interface angle θ, similar to the Newtonian case but with a locally varying capillary number:

(4.6) \begin{align} \tan^{3}\theta\sim \left(h/L\right)^{3}\sim Ca_{\textit{local}}. \end{align}

For wetting to be favourable, $ \textit{Ca}_{\textit{local}} $ is necessarily small with surface tension effects dominating over viscous effects when h is close to the contact line or when θ approaches the contact angle $ \theta_{d}$ , which is also necessarily small. This provides the necessary condition for wetting:

(4.7) \begin{align}\textit{Ca}_{\textit{local}} \ll 1 \quad\text{ as }\quad h\rightarrow 0\quad\text{ or }\quad \theta \rightarrow \theta_{d} \ \left(\ll 1\right).\end{align}

We now examine how different values of n control the extent of dissipation. It is evident that n = 1 reduces (4.3) to the usual Newtonian case (2.1), where dissipation is independent of length scales.

Figure 2.

Schematic illustrations of how the characteristic dissipation length $ h^{*} $ influences the contact line structures for (a) shear-thinning fluids (n < 1) with large $ h^{*} $ extending into the bulk, and (b) shear-thickening fluids (n > 1) with small $ h^{*} $ near the contact line.

For $ n \lt 1 $ , corresponding to shear-thinning fluids, since $ \text{tan}\,\theta \sim {\textit{Ca}_{\textit{local}} }^{1/3} \propto h^{(1-n)/3} $ in (4.6) together with (4.5), the interface slope increases with distance from the contact line, forming a concave profile as illustrated in figure 2(a). The wetting condition (4.7) requires $ h^{*} \gg h $ according to (4.5) with $n\lt1$ , indicating that viscous dissipation extends far beyond the immediate contact line region. Because $ h^{*} \propto U^{n/(n-1)} $ from (4.4), $ h^{*} $ diverges as U→0, suggesting that the dissipation progressively expands into the bulk as the contact line advances and its speed decreases. This outer bulk dissipation region – far away from the contact line – has h > $ h^{*} $ or $ \textit{Ca}_{\textit{local}} \gt 1 $ , yielding $ h \sim L $ according to (4.6) and, hence, $ h^{*} \sim L $ . Conversely, the inner wetting region near the contact line satisfies h < $ h^{*} $ or $ \textit{Ca}_{\textit{local}} \lt 1 $ . The lateral length scale $ L^{*} $ corresponding to $ h^{*} $ , which separates these two regions, thus occurs at h ∼ $ h^{*} $ or $ \textit{Ca}_{\textit{local}} $ ∼ 1, giving $ L^{*} \sim h^{*} $ . It should be noted that in the extreme shear-thinning limit (n → 0), (4.3) reduces to the equation governing the motion of a strongly slipping contact line (Wei et al. Reference Wei, Tsao and Chu2019), where viscous dissipation is primarily confined within a fixed region of size $ h^{*} $ proportional to the slip length λ. With $ \text{tan}\,\theta \sim \theta $ when θ is small under (4.6) and (4.7), the dynamic contact angle is

(4.8) \begin{align} \theta_{d}\sim (h/h^{*})^{(1-n)/3}, \end{align}

which is consistent with (3.11b ), showing that the wetting can be enhanced with $ \theta \rightarrow 0 $ as $ h \rightarrow 0 $ due to the drag reduction effect of shear thinning.

Conversely, when $ n \gt 1 $ for shear-thickening fluids, the situation is markedly different. In this case, the viscosity $\eta \sim k(U/h)^{n-1}$ increases sharply near the contact line. According to (4.6) together with (4.5), $ \text{tan}\,\theta \propto 1/h^{(n-1)/3} $ , so the interface slope decreases with distance from the contact line, yielding a convex profile as illustrated in figure 2(b). The wetting condition (4.7) requires that the characteristic dissipation length satisfies $ h^{*}\ll h $ . This implies that the viscous dissipation is highly concentrated in a narrow region very close to the contact line. Because $ h^{*} \propto U^{n/(n-1)} $ from (4.4), $ h^{*} $ decreases as U→0, indicating that the dissipation becomes increasingly localised near the contact line as the wetting progresses. When n is large, the rate of shrinkage of $ h^{*} $ approaches its maximum, with $ h^{*} \propto U $ , corresponding to extremely shear-thickening fluids (Starov et al. Reference Starov, Tyatyushkin, Velarde and Zhdanov2003).

As also illustrated in figure 2(b), the contact line behaviours in the inner and outer regions for shear-thickening fluids are markedly different from those for shear-thinning fluids. The inner region with $ h \lt h^{*} $ (or $ \textit{Ca}_{\textit{local}} \gt 1 $ ) is now dominated by strong viscous dissipation and confined within the immediate vicinity of the contact line. The outer region with $ h\gt h^{*} $ (or $ \textit{Ca}_{\textit{local}} \lt 1 $ ) corresponds to the wetting zone, where surface tension prevails and interface relaxation takes place. The lateral dimension associated with $ h^{*} $ separates these two regions and again scales as $ L^{*} $ ∼ $ h^{*} \ll L $ , corresponding to the location where $ \textit{Ca}_{\textit{local}} \sim 1 $ .

However, to determine the apparent dynamic contact angle $ \theta_{d}$ in the outer wetting zone with $ h \gt h^{*} $ , certain precautions are needed. Since (4.6) together with (4.5) indicate that θ →π/2 as $ h \rightarrow 0 $ , this suggests a complete suppression of wetting. In fact, under this condition, the lubrication equation (4.3) breaks down near the contact line where the interface steepens significantly due to the mounting viscous resistance caused by the more severe contact line singularity. To sustain wetting with $ \theta \ll 1 $ and to alleviate the interface’s steepening and resistance, a microscopic mechanism must operate at a scale h _m , much larger than the dissipation length $ h^{*} $ in the vicinity of the contact line. Consequently, when wetting occurs under (4.7), a well-defined apparent dynamic contact angle can exist only if the interface satisfies $h^{*}/h_{m} \ll 1$ in (4.6) with (4.5) evaluated at $h=h_m$ , leading to

(4.9) \begin{align} \theta_{d}\sim \left(h^{*}/h_{m}\right)^{\left(n-1\right)/3}, \end{align}

which is consistent with (3.13b ).

4.3. Review of solutions to ( 4.1 ) obtained by previous studies

Regarding the actual solution to (4.1), it is worth revisiting what has been done previously. Wang et al. (Reference Wang, Lee, Peng and Lai2007a ) analysed spreading power-law fluid drops and reported that (4.1) admits an analytical solution in a simple power-law form, i.e.

(4.10) \begin{align}h=\textit{bU}^{n/(n+2)}x^{3/(n+2)},\end{align}

with $b \gt 0$ being the coefficient required to satisfy (4.1). A similar solution form was also identified by Carré & Eustache (Reference Carré and Eustache2000) for shear-thinning fluids ( $n\lt 1$ ). In fact, this solution was first obtained by Boatto et al. (Reference Boatto, Kadanoff and Olla1993) who investigated various classes of travelling-wave solutions of the evolution equation $h_{t}+(h^{n+1}h_{xxx})_{x}=0$ . However, as pointed out by Bertozzi & Pugh (Reference Bertozzi and Pugh1996), (4.10) is valid only for $n\lt 1$ but not for $n\geq 1$ . Indeed, it can be readily verified that (4.10) does not satisfy (4.1) for $n\gt 1$ , since the left-hand side of (4.1) remains positive while the right-hand side is negative. Moreover, even if (4.10) were assumed in this case, it would result in $h^{\prime}$ diverging at the contact line ( $x=0$ ), yielding an apparent contact angle $\theta _{d}=90^{\mathrm{o}}$ . This is clearly unphysical, as the contact line would not advance at all, not to mention that this contradicts the small-slope assumption under which (4.1) is derived.

It is worth noting that due to nonlinearity, (4.1) can exhibit a variety of solution characteristics, depending on the value of $n$ (Boatto et al. Reference Boatto, Kadanoff and Olla1993). While an exact solution can sometimes exist (Abdel Kader et al. Reference Abdel Kader, Abdel and Nour2013), it is often not easily converted into a form that facilitates a clear understanding of the underlying physics. Furthermore, even with a simple solution form like (4.10), it is difficult to identify how the dynamic contact angle $\theta _{d}=h^{\prime}$ is influenced by the length scales involved in connection to the nature of the contact line singularity. Therefore, an approximate yet more physically oriented approach is needed to solve (4.1), which is presented next.

4.4. New dynamic contact angle relationships and Cox–Voinov generalisation

Under the assumption that $h^{\prime}$ is small, we employ the slowly varying slope approach to solve equations (4.1) or (4.3) approximately. This approach was first introduced by Voinov (Reference Voinov1976) in his derivation of (2.3) and later extended by Snoeijer (Reference Snoeijer2006) for the large-slope scenario. The essence of this approach is that, by assuming the shape of the macroscopic wedge to vary slowly with respect to the distance to the contact line, the dynamic contact angle formula can be derived through a straightforward integration of the equation. This approach is able to recover the Cox–Voinov law (2.3) (Wei Reference Wei2018), and its validity is supported by the fact that the numerical error due to this approximation, when compared with the full formula derived by Cox (Reference Cox1986), does not exceed 1 % for $\theta _{d}\lt 3\pi /4$ (Voinov Reference Voinov1976; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). Moreover, the same framework has been successfully applied to model surfactant superspreaders (Wei Reference Wei2018) and to develop a theory of slipping moving contact lines (Wei et al. Reference Wei, Tsao and Chu2019), demonstrating the robustness of this approach.

In the derivation below, following the approach of Voinov (Reference Voinov1976), we use $h$ as the variable to integrate (4.1) under the assumption of slowly varying $h^{\prime}$ . We first rewrite $h^{\prime\prime\prime}=h^{\prime}\textrm{d}h^{\prime\prime}/\textrm{d}h$ in (4.1) and integrate it along with the vanishing curvature condition (4.2). This yields

(4.11) \begin{align}h^{\prime\prime}=-\textit{KU}^{n}\int _{h_{ \infty }}^{h}\left(\frac{1}{h^{\prime}}\right)\,\frac{\textrm{d}x}{h^{n+1}},\end{align}

where $h_{ \infty }\equiv h(x\rightarrow \infty )$ . Since $h^{\prime}$ varies slowly, the integral on the right-hand side of (4.11) with $n\gt 0$ can be approximated as

(4.12) \begin{align}\left(\frac{1}{h^{\prime}}\right)\int _{h_{ \infty }}^{h}\frac{\textrm{d}x}{h^{n+1}}\approx -\left(\frac{1}{h^{\prime}}\right)\frac{1}{n}h^{-n},\end{align}

wherein we neglect $h_{ \infty }^{-n}$ in the final result because $h_{ \infty }\gg h$ . Substituting (4.12) into (4.11) and rewriting $h^{\prime\prime}=h^{\prime}\textrm{d}h^{\prime}/\textrm{d}h$ , we arrive at

(4.13) \begin{align}(h^{\prime})^{2}\frac{\textrm{d}h^{\prime}}{\textrm{d}h}=\left(\frac{K}{n}\right)U^{n}h^{-n}.\end{align}

As anticipated, for $n=1$ , integrating (4.13) recovers the Cox–Voinov law (2.3) for the Newtonian case. For power-law fluids with $n\neq 1$ , integrating (4.13) yields a new law for the apparent dynamic contact angle $\theta _{d}\equiv h^{\prime}(x)$ at $x\gg x_{m}$ :

(4.14) \begin{align}(h^{\prime})^{3}=\left({h_m}^{\prime}\right)^{3}+\left[\frac{3K}{n(1-n)}\right]U^{n}\left[h^{1-n}-h_{m}^{1-n}\right]\!,\end{align}

where $h_{m}$ ( $\ll h)$ and ${h_m}^{\prime}$ ( $\ll h^{\prime}$ ) are evaluated at a mesoscopic length $x_{m}$ .

As indicated by (4.14), the behaviour of $h^{\prime}$ can be dominated by either the macroscopic $h^{1-n}$ term or the mesoscopic $h_{m}^{1-n}$ term, depending on whether $n\lt 1$ or $n\gt 1$ . These macroscopic and mesoscopic terms correspond to distinct dissipation length scales $L$ and $x_{m}$ for $n\lt 1$ and $n\gt 1$ , respectively, as also described by the braking force behaviour in (3.3).

4.4.1. The shear-thinning case

For shear-thinning fluids (n < 1), (4.14) is dominated by the $h^{1-n}$ term. By neglecting the contributions from $h_{m}$ , (4.14) reduces to

(4.15) \begin{align}(h^{\prime})^{3}=c \textit{KU}^{n}h^{1-n},\end{align}

where $ c = 3/n(1-n) $ . This confirms the scaling result (3.11b ). When expressed in terms of the dissipation length $ h^{*} $ = (KU ⁿ )^1/(n−1) given by (4.4), (4.15) reduces to the familiar Cox–Voinov form, written in terms of the local capillary number defined by (4.5), consistent with (4.8):

(4.16) \begin{align}(h^{\prime})^{3}= c \textit{Ca}_{\textit{local}} = c (h / h^{*})^{1-n} .\end{align}

The corresponding interface profile is then

(4.17) \begin{align} h=\left[\frac{n+2}{3}\left(\textit{cK}\right)^{1/3}\right]^{3/\left(n+2\right)}U^{n/\left(n+2\right)} x^{3/\left(n+2\right)}, \end{align}

which recovers the exact solution reported previously (Boatto et al. Reference Boatto, Kadanoff and Olla1993; Carré & Eustache Reference Carré and Eustache2000; Wang et al. Reference Wang, Lee, Peng and Lai2007a ). Re-expressing (4.15) in terms of x using (4.17), we find that the apparent dynamic contact angle $\theta _{d}\equiv h^{\prime}(x)$ varies as

(4.18) \begin{align}\theta _{d}(x)=C_{1}U^{n/(n+2)}x^{(1-n)/(n+2)},\end{align}

where C ₁ = [(n + 2)/3]^{(1−n)/(n + 2)} (cK)^{1/(n + 2)}. Note that the above results hold only for $x$ sufficiently far from the microscopic position $x_{m}$ , as the contributions from $h_{m}$ at that position are neglected in (4.14). Suppose that $\theta _{d}$ is evaluated at a fixed relative position $r=R-x=(1-\alpha )R$ with respect to the spreading radius $R$ with $\alpha$ being a constant ( $\lt 1$ ). Then, $\theta _{d}$ described by (4.18) not only varies with the extent of the spreading but also is constrained by the drop volume $V=(\pi /4)R^{3}\theta _{d}$ . Substitution of $x=\alpha R$ and $R=(4V/\pi \theta _{d})^{1/3}$ into (4.18) leads $\theta _{d}$ to vary with $U$ as

(4.19) \begin{align} \theta _{d}=\left[\left(\frac{4V\alpha ^{3}}{\pi }\right)^{1-n}C_{1}^{3\left(n+2\right)}\right]^{1/\left(2n+7\right)}U^{3n/\left(2n+7\right)} . \end{align}

Equation (4.19) is consistent with the $ \theta_{d}$ -U relationship obtained by Starov et al. (Reference Starov, Tyatyushkin, Velarde and Zhdanov2003) using self-similar analysis. In their work, the spreading law R(t) was derived first by solving the full lubrication equation across the entire drop, and the spreading speed U was subsequently expressed in term of $ \theta_{d}$ via the constant drop volume constraint. In contrast, the present derivation of (4.19) is achieved by resolving the local contact line behaviour directly. Furthermore, unlike the Newtonian case where the $ \theta_{d}$ -U relationship is independent of the drop volume V, as indicated by the Cox–Voinov law (2.3), (4.19) exhibits an explicit dependence on V, as also noted by Starov et al. (Reference Starov, Tyatyushkin, Velarde and Zhdanov2003). This arises because the dissipation length $ h^{*} $ given by (4.4) now extends across the entire drop body, i.e. $ h^{*} $ ∼R. For this reason, the actual value of α– which quantifies the extent of dissipation – must be determined by solving the full equation governing the interfacial evolution of a spreading drop, as carried out by Starov et al. (Reference Starov, Tyatyushkin, Velarde and Zhdanov2003).

Writing $U\sim R/t$ and using $R^{3}\theta _{d}=$ constant again, we can use (4.19) to determine how $\theta _{d}$ and $R$ vary with time $t$ :

(4.20a) \begin{align}\theta _{d}\propto t^{-3n/(3n+7)}, \\[-28pt] \nonumber\end{align}

(4.20b) \begin{align}R\propto t^{n/(3n+7)}. \\[-2pt] \nonumber\end{align}

Equation (4.20b ) reproduces the spreading law derived by Starov et al. (Reference Starov, Tyatyushkin, Velarde and Zhdanov2003) and Betelu & Fontelos (Reference Betelu and Fontelos2004) with complete prefactors using self-similar analysis and is also consistent with the solution form proposed by King (Reference King2001). Furthermore, (4.20a ) for the dynamic contact angle evolution agrees with that derived by Betelu & Fontelos (Reference Betelu and Fontelos2004). This consistency confirms that the present formulation – based on the original contact angle relation (4.15) derived directly from the governing equation (4.1) under the volume constraint – faithfully reproduces the spreading dynamics for shear-thinning fluids. While (4.20a ) and (4.20b ) naturally arise from the self-similar analysis, our formulation underscores that these power-law dependences are not merely mathematical consequences but are intrinsically governed by the local dynamic contact angle behaviour encapsulated in (4.15). This also reinforces that the dynamic contact angle (4.15) serves a more fundamental description from which macroscopic spreading laws emerge.

In the strong shear-thinning $n\rightarrow 0$ limit, the approximation used in (4.12) for the right-hand side of (4.1) is no longer valid, and thus, the preceding results do not apply. In this limit, (4.1) simplifies to

(4.21) \begin{align}h^{\prime \prime \prime}=-\frac{K_{0}}{h} .\end{align}

Here $K_{0}=k/\gamma$ and its inverse is precisely the dissipation length described by (4.4). The same equation form also describes the motion of a strongly slipping contact line with $K_{0}$ being inversely proportional to the slip length (Wei et al. Reference Wei, Tsao and Chu2019). Following a similar approach as in (4.11) and (4.12), the dynamic contact angle relationship is found to be

(4.22) \begin{align}(h^{\prime})^{3}=3K_{0}h \ln (h_{ \infty }/h).\end{align}

So ( $h^{\prime})^{3}$ is roughly proportional to $h$ , similar to (4.15) with small $n$ , albeit it weakly depends on the macroscopic height $h_{ \infty }$ through the logarithmic term. The corresponding interface profile is

(4.23) \begin{align}h\approx \left(3/2\right)^{3/2}\left[3K_{0}\ln (h_{ \infty }/h)\right]^{1/2}x^{3/2},\end{align}

which approximately follows $x^{3/2}$ , resembling (4.17) when $n$ is small. Both (4.22) and (4.23) have also been observed in strongly slipping moving contact lines (Wei et al. Reference Wei, Tsao and Chu2019). These results indicate that the contact line behaviour in the strong shear-thinning $n\rightarrow 0$ limit resembles that in the strong slip limit because of strong drag reduction due to a substantial decrease in the viscosity.

It should also be noted that the viscosity $\eta$ of a shear-thinning fluid does not necessarily vanish as the shear rate $\dot{s}$ increases indefinitely, as suggested by (3.1), especially near the contact line. Instead, $\eta$ may approach a finite constant value, known as the infinite-shear viscosity $\eta _{ \infty }$ , as $\dot{s}$ → ∞, as described by the Carreau model (Bird, Armstrong & Hassager et al. Reference Bird, Armstrong and Hassager1987; Carreau, Kee & Daroux Reference Carreau, Kee and Daroux1979). In this case, the contact line structure can be conceptually divided into three subregions: an inner Newtonian-like region characterised by viscosity $\eta _{ \infty }$ , an outer region exhibiting shear-thinning behaviour with a higher apparent viscosity and an intermediate transition region connecting the two (Charitatos et al. Reference Charitatos, Suszynski, Carvalho and Kumar2020).

In fact, the behaviour of such Carreau shear-thinning fluids is somewhat similar to polymeric liquids where viscosity is significantly reduced near a substrate due to the depletion layer (De Gennes Reference de Gennes1985). This results in an apparent slip length $\lambda$ near the contact line where viscosity decreases from the bulk value η _b to $\eta_{\infty}$ . As the shear stress near the contact line behaves as

(4.24) \begin{align}\tau =\eta \frac{\partial u}{\partial y}\sim \eta _{b}\frac{U}{h+\lambda }\sim \eta _{ \infty }\frac{U}{h} ,\end{align}

the slip length λ can be related to the viscosity reduction ratio through

(4.25) \begin{align}\frac{\lambda }{h}\sim \frac{\eta _{b}}{\eta _{ \infty }}-1,\end{align}

establishing a connection between shear-thinning effects and slip effects. When this viscosity-reduction-induced slip effect becomes pronounced, i.e. $ \lambda/h \gg 1$ , the apparent dynamic contact angle can be shown to behave as $ \theta_{d}$ ∼ Ca _b ^1/2 in terms of the capillary number $ \textit{Ca}_{b} = \eta_{b} U/\gamma $ based on the bulk viscosityη _b (Wei et al. Reference Wei, Tsao and Chu2019). This scaling deviates significantly from the classical Cox–Voinov law $ \theta_{d}$ ∼ Ca _b ^1/3 given by (2.3). Recently, Yancheshme et al. (Reference Yancheshme, Yoon, Palmese and Alvarez2024) performed drop spreading experiments for a variety of shear-thinning fluids and found that the measured dynamic contact angles approximately collapse according to $ \theta_{d}$ ∼ Ca _b ^0.51. The excellent agreement with this prediction strongly supports the notion that shear-induced viscosity reduction near the contact line effectively enhances slip, thereby altering the local dissipation mechanism and leading to a stronger dependence of the dynamic contact angle on the capillary number.

Another aspect worth noting is that as the shear rate $\dot{s}$ → 0, the viscosity η should asymptotically approach the zero-shear viscosity η ₀, which corresponds to the bulk viscosity η _b mentioned above and cannot be captured by the simple power-law model (3.1). It might happen that in the outer region sufficiently far away from the contact line, the local shear rate becomes very small, η is therefore dominated by the zero-shear viscosity plateau η ₀. This might accompany the inner contact line region where $\dot{s}$ is large but not yet high enough to reach the infinite-shear viscosity plateau $\eta_{\infty}$ , its viscosity may still follow the power-law dependence described by (3.1). Under such circumstances, η can be more accurately represented by the Ellis model, which incorporates only a zero-shear viscosity plateau, or by the Carreau model, which accounts for both the zero-shear and infinite-shear viscosity plateaus (Bird et al. Reference Bird, Armstrong and Hassager1987). In either model, the transition from the outer Newtonian (low $\dot{s}$ ) regime to the inner shear-thinning (high $\dot{s}$ ) regime occurs at a critical shear rate, $\dot{s}_{c}$ , below or above which the contact line behaviour is respectively Newtonian or shear-thinning. This also defines the corresponding critical liquid height $ h_{c} \sim U/\dot{s}_{c} $ below or above which the contact line dynamics exhibit shear-thinning or Newtonian characteristics. When $\dot{s}_{c}$ is large, h _c becomes small and, thus, is easily exceeded, making the contact line behaviour predominantly Newtonian. This height may effectively act as a velocity-dependent slip length λ to alleviate the contact stress singularity in Newtonian fluids. In that case, the apparent dynamic angle can be described by the classical Cox–Voinov law with $\ell = 0.54 \lambda \textit{Ca}^{-1/3}$ in (2.3) (Eggers & Stone Reference Eggers and Stone2004), where $ \textit{Ca} = \eta_{0} U/\gamma $ is the capillary number based on the zero-shear viscosity η ₀. When $\dot{s}_{c}$ is small, however, the contact line region becomes shear thinning and the local viscosity decreases markedly with increasing shear rate. In this regime, an effectively large slip length may emerge due to the substantial viscosity reduction according to (4.25). This in turn results in a weak stress singularity, making the dynamic contact angle deviated from the classical Cox–Voinov law as indicted by (4.19).

4.4.2. The shear-thickening case

For shear-thickening fluids with $n\gt 1$ , we rewrite (4.14) in terms of the characteristic dissipation length $h^{*}$ defined by (4.4),

(4.26) \begin{align}(h^{\prime})^{3}=(h_{m}')^{3}+c\left[\left(\frac{h^{*}}{h_{m}}\right)^{n-1}-\left(\frac{h^{*}}{h}\right)^{n-1}\right]\! ,\end{align}

with $ c = 3/n (n-1)$ . Equation (4.26) shows that the slope $h^{\prime}(x)$ increases monotonically with $h$ as $h$ grows toward the bulk. As $h\rightarrow \infty $ when $x\rightarrow \infty $ , $h'$ approaches a constant value, representing the apparent dynamic contact angle $\theta _{d}\equiv \tan ^{-1}(h^{\prime}(x\rightarrow \infty ))$ , leading (4.26) (neglecting $(h_{m}^{\prime})^{3}$ ) to

(4.27) \begin{align} {\theta_{d}}^{3}=cCa_{\textit{local}}\left(h_{m}\right)= (h^{*}/ h_{m})^{n-1}, \end{align}

which is essentially (4.9) with the local capillary number $ \textit{Ca}_{\textit{local}} $ evaluated at h = h _m . Equation (4.27) indicates that wetting under small values of $\theta _{d}$ would become more favourable when $h^{*}\lt h_{m}$ (see figure 3 a). Since $h^{*}=(\textit{KU}^{n})^{1/(n-1)}$ according to (4.4), this implies that favourable wetting can only be achieved when the speed $U$ is below a certain critical value, $U_{c}$ :

Figure 3.

Schematic illustrations showing how the contact line structure for shear-thickening fluids ( $n\gt 1$ ) is influenced by the relative size of the dissipation length $h^{*}$ compared with the microscopic cutoff $h_{m}$ . Results are shown for (a) $h^{*}\ll h_{m}$ and (b) $h_{m}\ll h^{*}$ .

(4.28) \begin{align}U_{c}=\big(K^{-1}h_{m}^{n-1}\big)^{1/n}.\end{align}

In this case, the apparent dynamic angle is

(4.29) \begin{align}\theta _{d} \approx C_{2}\left(\frac{k}{\gamma }\right)^{1/3}U^{n/3}h_{m}^{(1-n)/3},\end{align}

which is (3.13b ), where C ₂ = (c(2 + 1/n) ⁿ )^1/3. The apparent dynamic contact relationship (4.29) is new and stands in contrast to that proposed by Wang et al. (Reference Wang, Lee, Peng and Lai2007a ), where the same functional form derived for shear-thinning fluids was applied to shear-thickening fluids.

On the other hand, if $U\gt U_{c}$ , this means $h^{*}\gt h_{m}$ , which may cause the dynamic contact angle $\theta _{d}$ to no longer be small and even approach $90^{\circ}$ (see figure 3b ). In this case, wetting becomes less favourable. Note from (4.26) that $h^{\prime}(x)$ will be smaller than $h_{m}^{\prime}\equiv h^{\prime}(x_{m})$ when $h\lt h_{m}$ and continue to decrease as $h$ approaches the contact line. If there is a pre-wetted film ahead of the macroscopic contact line, this will allow $h^{\prime}(x)$ to continuously decrease towards zero, with $h^{\prime}=0$ at some $h\lt h_{m}$ or $x\lt x_{m}$ . Since $h^{\prime}$ cannot be further decreased to a negative value below this point, an additional mechanism is required to ensure that this zero-slope location also acts as an inflection point $h^{\prime\prime}=0$ . Below this point, $h^{\prime}$ must necessarily increase as it moves towards the microscopic contact line. Given that $h$ is very small in this region, this inflection point and subsequent film thinning below it can only arise from disjoining pressure effects. For the potentially large $\theta _{d}$ situation when $h^{*}\gt h_{m}$ , the presence of a precursor film due to disjoining pressure would become even more essential to facilitate the forward movement of the contact line. In any case, the thickness of the precursor film, which determines $h_{m}$ , and the extent of its spread will be governed by the detailed profile of the film, which is examined in § 5.

4.5. Summary and comparison with literature

Having completed the derivations of the apparent dynamic contact angle relationships for both shear-thinning and shear-thickening fluids in §§ 4.4.1 and 4.4.2, table 1 summarises these results and the corresponding spreading laws in comparison with those reported by previous work.

Table 1.

Summary of theoretical predictions for dynamic contact angle relationship (called wetting law) and spreading law in power-law fluids.

For shear-thinning fluids (n < 1), how the dynamic contact angle $ \theta_{d}$ varies with the contact line speed U is found to be size dependent, described by $ {\theta_{d}}^{3} \sim (k/ \gamma)U^{n}h^{1-n} $ from (4.15) or $ {\theta_{d}}^{n+2} \sim (k/\gamma)U^{n}x^{1-n}$ from (4.18). The resulting wetting law and spreading law are $ \theta_{d} \propto U^{3n/(2n+7)}$ from (4.19) and $ R \propto t^{n/(3n + 7)}$ in (4.20b ), recovering the results of Starov et al. (Reference Starov, Tyatyushkin, Velarde and Zhdanov2003) using self-similar analysis. This consistency also validates our contact-line-based approach. In difference, Wang et al. (Reference Wang, Lee, Peng and Lai2007a ) obtained ${\theta_{d}}^{n+2} \sim (k/ \gamma)U^{n}{x_{m}}^{1-n}$ with a fixed microscopic cutoff $x_{m}$ that is not required. This yields $ R \propto t^{n/(4n+6)} $ – a result inconsistent with the self-similar solution.

For shear-thickening fluids (n > 1) under complete wetting conditions, our analysis shows $ {\theta_{d}}^{3} \sim (k/ \gamma)U^{n}{h_{m}}^{1-n} $ from (4.29) with a required microscopic h _m . As shown later in § 5, if h _m is provided by the microscopic precursor film induced by disjoining pressure, it can vary with the contact line speed U as $ h_{m} \propto U^{-n/(4 - n)} $ , leading to $\theta _{d} \propto U^{n/(4-n)}$ reported previously by Wang & Zhu (Reference Wang and Zhu2014). For partial wetting where the required microscopic cutoff $x_{m}$ becomes a fixed quantity depending on the substrate’s wettability, the dynamic contact angle relationship is modified to $ {\theta_{d}}^{n+2} \sim {\theta_{d}}^{n} {\theta_{e}}^{2} + (k/\gamma) U^{n}x_{m}^{1-n}$ from (3.15). For Newtonian fluids (n = 1), the theory reduces to the classical Cox–Voinov law $ {\theta_{d}} \sim \textit{Ca} ^{1/3} \propto U^{1/3} $ given by (2.3) and to Tanner’s spreading law $ R \propto t^{1/10} $ .