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

Entry by Leon Furchtgott, APP 225 Fall 2010.

Thin-Film Fluid Flows over Microdecorated Surfaces: Observation of Polygonal Hydraulic Jumps (2010). PRL, 102, 194503.

## Summary

The paper is interested in the behavior of liquid films flowing over rough substrates whose asperities are the same sizes as the film thickness. The flow of thin films on substrates is of interest in many different fields, from industrial processes to biology and vulcanology. There has been some research on how individual isolated features of substrates affect flow, but little on the collective influence of patterns of micron-size asperities. In addition, previous studies have examined films thicker or thinner than the characteristic length scales of the substrate, while this study looks at thin liquid films with thickness on the same order of magnitude as the roughness of the substrate. The authors investigate flows through experimental observations of flows on regular arrays of micron-size pores with different sizes and arrangements of pores. The authors then model the flow using a boundary-layer approach and fit experimental results to the model.

## Experimental Setup

Water is pumped into a nozzle of radius 1.2 mm at a flow rate Q between 0.5 and 2.5 L/min (Fig. 1a). The jet impacts the center of a patterned disc of PDMS of radius 2.5 cm, which is embedded in a smooth clear acrylic plate (Fig. 1b). The patterned surface consists of square or hexagonal arrays of cylindrical posts with height H, radius R and lattice distance D of the order of 100 $\mu m$. The water spreads radially over the rough substrate, then over the smooth surrounding area, before spilling into the collection reservoir. The water sheet experiences a hydraulic jump at a particular radius. The thickness $h(r)$ of the thin film propagating before the jump is of the order of a few hundreds of microns, whereas the depth of the water layer outside the jump is d = 4.3 mm. Measurements show that d does not depend on the flow rate and the substrate topography.

Fig. 1. (a). Schematic of the experimental setup. (b). Micropattern in the heart of the impact plate. The pattern depends on 3 factors: R and H, the radius and height of the posts, and D, the lattice distance. (c). Photograph of a polygonal jump taken from above. The jump is formed by a water jet (Q = 1 L/min) impacting a square lattice ($D = 200 \mu m, R = 50 \mu m, H = 50 \mu m$).

Intriguingly, patterning of asperities on the substrate can break the symmetry of the typically circular hydraulic jump. In Fig. 1c, the jump adopts an eight-corner star shape. The remainder of the paper examines the role of the lattice geometry and the fluid properties. The paper concentrates on effects on the thin film and assumes that the flow in the outer layer are constant.

## Results

The authors begin by varying the symmetry of the lattice (Fig. 2). They look at jumps of the same mean radius but formed over different micropatterned surfaces, with hexagonal (b) and square (c) shapes. These have an effect on the jump symmetries.

Fig. 2. Influence of pattern structure. Impact of water jets on (a) a smooth surface, (b) a square lattice, and (c) a square lattice. Lattice parameters: $D = 200 \mu m, R = 50 \mu m, H = 50 \mu m$.

This shape selection is robust in that it is independent of the viscosity and surface tension of the water that is being flowed. This tells us that the symmetry breaking is not caused by viscocapillarity or inertiocapillary instability.

The authors vary systematically the parameters of a square lattice, both the lattice distance D and the height of the posts H (see Fig. 3). For each set (D, H), they look at the average jump radius as a function of the flow rate Q:

- For all (D, H), the mean jump radius increases nonlinearly with the flow rate, and the corners of the polygonal structures become sharper with the flow rate.

- For a given flow rate Q, the radius decreases for smaller D and larger H, and the polygon corners become sharper.

Fig. 3. A. Optical micrographs and renderings of nonrigid structures at N = 9 and (B) N = 10 (C) Structures of 3N – 5 = 25 bond packings at N = 10. The anharmonic vibrational modes of the nonrigid structures are shown by red arrows. Experimentally measured probabilities are listed at top. Annotations in micrographs indicate clusters corresponding to subsets of HCP or FCC lattices. Scale bars, 1 µm.

Fig. 4. Calculated minima of the free-energy landscape for 6 $\leq$ N $\leq$ 10. The x axis is in units of the rotational partition function, where I is the moment of inertia (calculated for a particle mass equal to 1) and $\sigma$ is the rotational symmetry number. Each black symbol represents the free energy of an individual cluster. The number of spokes in each symbol indicates the symmetry number (dot = 1, line segment = 2, and so on). Orange symbols are nonrigid structures, which first appear at N = 9, and violet symbols have extra bonds, first appearing at N = 10. Vertical gray lines indicate the contribution to the free energy due to rotational and vibrational entropy. The reference states are chosen to be the highest free-energy states at each value of N.

## Relation to Soft Matter

This paper fits well with our discussions of