Why bumpy is better: The role of the dissipation distribution in slip flow over a bubble mattress

It has been observed that the amount of effective slip for transverse flow over a bubble mattress is maximum for bubbles that protrude somewhat in the channel flow. In this paper we provide an explanation for this characteristic feature by analyzing the spatial distribution of viscous dissipation for bubbles of varying protrusion angles. Bubbles protruding in the channel act as obstacles and reduce the effective channel height, thereby increasing the viscous dissipation in the bulk flow. At small scales, however, our numerical analysis reveals that increasing the bubble protrusion angle reduces the dissipation near the contact points of the no-slip channel wall and the no-shear bubble surface. We obtain an analytical expression to quantify this effect based on classical corner flow solutions. The two antagonistic effects, decreased dissipation near the bubble corners and increased dissipation on larger scale, explain why the effective slip length is maximum for a bubble mattress that is slightly bumpy.

Figure 1

Geometry of a viscous flow over a bubble mattress. A no-shear bubble with a protrusion angle $\vartheta$ is sticking out into the channel flow at the lower boundary. The boundary conditions in the horizontal direction are periodic. In the left figure, the color map represents the velocity magnitude $\left|\tilde{\mathbf{u}}\right|$, while in the right figure the color map shows the local viscous dissipation rate $\tilde{\mathrm{\Phi }}$. The white lines are streamlines of the velocity field $\tilde{\mathbf{u}}$. The relevant dimensions are the channel height $H$, the width of periodicity $L$, and the bubble size $L_g$. Throughout the paper, all length scales are nondimensionalized by the channel height $H$.

Figure 2

(a) Dimensionless slip length $\tilde{b}$ and total dissipation ${\tilde{\mathrm{\Phi }}}_A$ plotted as a function of the bubble protrusion angle $\vartheta$ for $a=2$ and $\varepsilon =1/2$. The effective slip length and the total dissipation are related to each other by Eq. (13). When $\vartheta =9^{\circ },\tilde{b}$ is maximum and ${\tilde{\mathrm{\Phi }}}_A$ is minimum. (b) Dependence of the optimal protrusion angle ${\vartheta }_o$ on the confinement factor $a=H/L$ for $\varepsilon =1/2$.

Figure 3

The color map in (a) shows the distribution of the dissipation rate $\tilde{\mathrm{\Phi }}$ for four different protrusion angles for $0\le \tilde{y}\le 0.5$: from left to right, $\vartheta =1^{\circ },9^{\circ },{33}^{\circ },{65}^{\circ }$. The streamlines of the velocity field $\tilde{\mathbf{u}}$ are shown in white. The black lines are contour lines of the dissipation rate $\tilde{\mathrm{\Phi }}$. In (b) and (c) the viscous dissipation ${\tilde{\mathrm{\Phi }}}_{A,\mathrm{str}}$ in a streamtube of width $\mathrm{\Delta }\tilde{y}=0.01$ is plotted against the midpoint of the streamtube ${\tilde{y}}_0$ at $\tilde{x}=0$. In (b) $a=2$ and $\varepsilon =1/2$ and (c) is a magnification of (b).

Figure 4

To prove that the viscous dissipation in the vicinity of the bubble corners decreases with increasing protrusion angle $\vartheta$, the flow near the corner is analyzed using the Moffatt corner flow solution $\psi =Af\left(\varphi \right)r^{\lambda }$ [18]. (a) Exponent $\lambda$ plotted as a function of $\vartheta$, obtained from fitting the Moffatt solution to the analytical solution of Davis and Lauga [11] (circles) and by solving Eq. (15) (solid line). (b) Plot of $Af\left(\varphi \right)$ for various $\vartheta$. Both the amplitude $A$ and the function $f\left(\varphi \right)$ are approximately constant. (c) Maximum dissipation rate ${\tilde{\mathrm{\Phi }}}_{\max }\left(\vartheta \right)$, determined at $\tilde{r}={10}^{-5}$ and normalized by ${\tilde{\mathrm{\Phi }}}_{\max }\left(0\right)$. Based on a theoretical analysis, it is predicted that near the corner $\tilde{\mathrm{\Phi }}\left(\vartheta \right)/\tilde{\mathrm{\Phi }}\left(0\right)\sim {\tilde{r}}^{2\lambda -3}$ [Eq. (17)]. (d) Dissipation rate $\tilde{\mathrm{\Phi }}$ for a protrusion angle of $\vartheta ={12}^{\circ }$, with some streamlines of $\tilde{\mathbf{u}}$ plotted in white. The bubble surface is located at $(\tilde{r},{168}^{\circ })$.

Figure 5

For four different protrusion angles $\vartheta$, the difference in dissipation rate $\mathrm{\Delta }\tilde{\mathrm{\Phi }}=\tilde{\mathrm{\Phi }}-{\tilde{\mathrm{\Phi }}}_{\mathrm{HP}}$ is plotted. Each contour level corresponds to $\mathrm{\Delta }\tilde{\mathrm{\Phi }}=25$. Here $\mathrm{\Delta }\tilde{\mathrm{\Phi }}<0$ for the blue contour levels and $\mathrm{\Delta }\tilde{\mathrm{\Phi }}>0$ for the green and red contour levels. When $\vartheta$ is larger than the critical protrusion angle of, in this case, ${61}^{\circ }$, in the channel mainly $\mathrm{\Delta }\tilde{\mathrm{\Phi }}>0$, indicating increased friction with respect to Hagen-Poseuille flow through a nonslippery channel.