Inelastic non-Newtonian flow over heterogeneously slippery surfaces

In this study, we investigated inelastic non-Newtonian fluid flow over heterogeneously slippery surfaces. First, we simulated the flow of aqueous xanthan gum solutions over a bubble mattress, which is a superhydrophobic surface consisting of transversely positioned no-slip walls and no-shear gas bubbles. The results reveal that for shear-thinning fluids wall slip can be increased significantly, provided that the system is operated in the shear-thinning regime. For a 0.2 wt% xanthan gum solution with a power-law index of $n=0.4$, the numerical results indicate that wall slip can be enhanced 3.2 times when compared to a Newtonian liquid. This enhancement factor was also predicted from a theoretical analysis, which gave an expression for the maximum slip length that can be attained over flat, heterogeneously slippery surfaces. Although this equation was derived for a no-slip/no-shear unit length that is much larger than the typical size of the system, we found that it can also be used to predict the enhancement in the regime where the slip length is proportional to the size of the no-shear region or the bubble width. The results could be coupled to the hydrodynamic development or entrance length of the system, as maximum wall slip is only reached when the fluid flow can fully adapt to the no-slip and no-shear conditions at the wall.

Figure 1

Viscosity $\mu$ for multiple xanthan gum solutions. The lines are the fitted Carreau expressions to the experimental data (the circles and squares correspond to different samples).

Figure 2

Example of the periodic bubble mattress model, consisting of $N=5$ bubble units with height $H=100\mu \mathrm{m}$, unit length $L=20\mu \mathrm{m}$, and bubble width $L_g=10\mu \mathrm{m}$. The bubble protrusion angle of, in this case, $\vartheta ={45}^{\circ }$ defines the gas-liquid interface curvature. The surface porosity is $\varepsilon =1/2$. The plot in (a) shows the local viscosity of a 0.2 wt% xanthan gum solution for an applied average liquid velocity of $u_{\mathrm{av}}={10}^{-5}$ m/s. In (b) and (c) are the velocity magnitude $\left|\mathbf{u}\right|$ and the shear rate $d\mathbf{u}/d\mathbf{x}$ with $\mathbf{x}=(x,y)$ plotted for a single unit.

Figure 3

A cylindrical geometry is used to investigate the relationship between the effective slip length $\tilde{b}=b/R$ and the relative bubble unit confinement $\tilde{L}=L/R$. In (a), the velocity magnitude $\left|\mathbf{u}\right|$ for $\tilde{L}=5.0,n=1$, and $\varepsilon =1/2$ is shown. The color scale varies from red for $\left|\mathbf{u}\right|=2$ to dark blue for $\left|\mathbf{u}\right|=0$. In (b), a schematic of the system is given. The flow pattern over the no-slip (${\tilde{L}}_1$ with ${\tilde{b}}_1=0$) and no-shear regions (${\tilde{L}}_2$ with ${\tilde{b}}_2=\infty$) can be divided into four different subregions: ${\tilde{l}}_i$ with $i=[1\cdots 4]$. When $\tilde{L}\to \infty$, the effective pressure drop in the domain is dominated by the no-slip region ${\tilde{l}}_2$.

Figure 4

Dimensionless effective slip length $2b/L_g$ for a 0.2 $\mathrm{wt}\%$ xanthan gum (XG) solution as a function of the bubble protrusion angle $\vartheta$ for various average liquid velocities $u_{\mathrm{av}}$ ($H=100\mu \mathrm{m}$, $L_g=10\mu \mathrm{m}$, $\varepsilon =1/2$). A liquid velocity of $u_{\mathrm{av}}={10}^{-5}$ m/s and a viscosity of $\mu =0.89$ mPa s was used to calculate the slip length profile for water.

Figure 5

In (a) the effective slip length $2b/L_g$ is plotted as a function of the average liquid velocity $u_{\text{av}}$ for a protrusion angle of $\vartheta =1^{\circ }$ for various xanthan gum solutions. In (b), the slip enhancement factor $b/b_{{\text{H}}_2\text{O}}$ for $\vartheta =1^{\circ }$ is plotted as a function of the effective pressure gradient $dp/dx$, where $b_{{\text{H}}_2\text{O}}$ has been determined for $u_{\text{av}}={10}^{-5}$ m/s. For both (a) and (b), $H=100\mu \mathrm{m}$, $L_g=10\mu \mathrm{m}$, and $\varepsilon =1/2$. The horizontal dashed lines in (b) are the analytically predicted enhancement factors ${\tilde{b}}_{\max }\left(n\right)/{\tilde{b}}_{\max }\left(1\right)$ as obtained from Eq. (23).

Figure 6

In (a) the effective slip length $\tilde{b}=b/R$ is plotted as a function of the no-slip/no-shear unit length $\tilde{L}=L/R$ for various power-law indices $n$. The same data is replotted in (b), showing now the slip length normalised with the length of the no-shear region $L_g$. The dashed lines in (a) and (b) are the theoretical maxima for $\tilde{b}$ as calculated using Eq. (23). When normalizing $\tilde{b}$ using the maximum effective slip length ${\tilde{b}}_{\max }$, the master curve shown in (c) is obtained. In (d) the slope $d\tilde{b}/d\tilde{L}$ of the slip length profiles shown in (a) is plotted. For all figures, $\varepsilon =1/2$.

Figure 7

Wall-slip enhancement factor ${\tilde{b}}_{\max }\left(n\right)/{\tilde{b}}_{\max }\left(1\right)$, given by Eq. (23), as a function of the power-law index $n$ for three different surface porosities $\varepsilon$.

Figure 8

As long as the flow is hydrodynamically developing, the slip length $\tilde{b}$ grows linearly with the unit length $\tilde{L}$. (a) For $\varepsilon \tilde{L}>2.4$, the flow becomes hydrodynamically developed, here shown for a Newtonian liquid with $n=1$. (b) The liquid velocity at the no-shear wall then also reaches the maximum possible value of ${\left.\tilde{u}\right|}_{\tilde{r}=1}=1$ at, in this case, $\tilde{L}/2$. Here, the velocity profiles are plotted for $\tilde{L}={10}^{-2+0.25k}$ with $k=[0\cdots 12],n=1$, and $\varepsilon =1/2$. The increase in the maximum liquid velocity at the wall with $\tilde{L}$ for different power law indices $n$ (here represented for $n=1$ by the gray dashed curve for $\tilde{L}\le 1$) is correlated with $d\tilde{b}/d\tilde{L}$ as plotted in Fig. 6, showing the same behavior.

Figure 9

In (a), a SEM image is shown of one of the microfluidic devices that were used to study non-Newtonian flow over a bubble mattress. They consist of two main channels, connected to each other by 15 microchannels. Because the wall was hydrophobized, as shown in (b), the side channels were filled with a gas (nitrogen) and the liquid in the upper channel was flowing over an array of no-slip wall segments and gas bubbles. Here, the image was focused at the midplane of the channel, where also the $\mu \text{PIV}$ experiments were performed. In (c), for a chip with $H=50\mu \text{m}$, $L_g=10\mu \text{m}$, and $\varepsilon =2/3$, the effective slip lengths $b$ are plotted as a function of the bubble protrusion angle $\vartheta$ for water and for shear-thinning solutions of 0.1 wt% xanthan gum (XG) dissolved in a mixture of water and either 10 or 20 wt% glycerol (G).