Thin-Film Fluid Flows over Microdecorated Surfaces: Observation of Polygonal Hydraulic Jumps

Liquid films flowing over rough substrates are influenced by asperities whose sizes are comparable to the film thickness. We investigate flows driven by jet impact on a regular array of micron-size posts. We show that the topography modifies the size and the shape of the observed hydraulic jumps: in addition to circular jumps, we obtain a variety of stable polygonal shapes. We rationalize our results by considering a leakage flow through the texture and the free-surface flow above it, coupled by an effective slip boundary condition accounting for the symmetry of the texture. This model is in good agreement with our experiments and allows us to account for the interplay between flow properties and roughness parameters, which has applicability in many other thin-film configurations.

Figure 1

Experimental method. (a) Schematic of the experimental setup. (b) The micropattern embedded in the center of the impact plate. The photograph shows the cylindrical posts of polydimethylsiloxane (PDMS) arranged on a square lattice. The typical radius $R$ and height $H$ of the posts and lattice distance $D$ are approximately $100\mu \mathrm{m}$. (c) Photograph of a polygonal jump taken from above, at a 45° angle. The jump is formed by a water jet ($Q=1\mathrm{L}·{\min }^{-1}$) impacting a square lattice ($D=200\mu \mathrm{m}$, $R=50\mu \mathrm{m}$, and $H=50\mu \mathrm{m}$).

Figure 2

Influence of pattern structure. Water jets impact (a) a smooth substrate ($Q=1.3\mathrm{L}·{\min }^{-1}$), (b) a hexagonal lattice ($Q=1.7\mathrm{L}·{\min }^{-1}$), and (c) a square lattice ($Q=1.9\mathrm{L}·{\min }^{-1}$). Lattice parameters: $D=200\mu \mathrm{m}$, $R=50\mu \mathrm{m}$, and $H=50\mu \mathrm{m}$.

Figure 3

Influence of pattern geometry on the jump. (a) Average radius versus flow rate for four square lattices with post radius $R=50\mu \mathrm{m}$ and different lattice spacing and posts heights: (▵) $D=200\mu \mathrm{m}$ and $H=23\mu \mathrm{m}$, (▹) $D=200\mu \mathrm{m}$ and $H=50\mu \mathrm{m}$, (▿) $D=200\mu \mathrm{m}$ and $H=88\mu \mathrm{m}$, (○) $D=300\mu \mathrm{m}$ and $H=50\mu \mathrm{m}$, (□) $D=400\mu \mathrm{m}$ and $H=50\mu \mathrm{m}$. Inset, the data are compared to predictions (solid line) with $Y_j={\overline{r}}_{j}g(\frac{da}{Q}{)}^2(1+\frac{2}{{\overline{B}}_0})+\frac{a^2}{2{\pi }^2{\overline{r}}_{j}d}$ and $X_j=\frac{\nu ({\overline{r}}_j^3+{\ell }^3)}{Q{a}^2}$ [19], where ${\overline{B}}_0=\frac{\rho g{\overline{r}}_{j}d}{\gamma }$ is the Bond number. (b) Images of the polygonal jumps over different substrates for $Q=2.5\mathrm{L}·{\min }^{-1}$.

Figure 4

Results of the modeling. (a) View from above of a unit cell of a square lattice. Black and white arrows indicate, respectively, the flow direction and the main axis of the lattice. $\theta$ is the angle between these two directions (here, 0°, 45°, and 90°) and ${\ell }_0$, which scales as the open frontal area, is represented by the black lines. (b) Comparison between the results of the model (solid line) and the experimental data. The results obtained in 15 are also represented (•). $\overline{\ell }(b)$, $\overline{g}(b)$, ${\overline{B}}_0$, and ${\overline{r}}_j$ are averaged for $\theta$ between 0 and $2\pi$. Inset: Maximum deformation of the shape as predicted by the model. (c) Shapes predicted square lattices with different lattice parameters. Symbols are as in Fig. 3.

Figure 5

Coefficients of the model. Leakage flow ($\alpha$, ◆) and slip length ($\xi$, ▪) coefficients as functions of: (a) The aspect ratio of the posts $\kappa$, for a fixed value of the roughness porosity $ϵ$ ($H$ varies, $R$ and $D$ are constant); (b) $ϵ$, for a fixed value of $\kappa$ ($D$ varies, $R$ and $H$ are constant).