# 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 <math>\mu m</math>. 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 <math>h(r)</math> 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.

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.

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.

## Discussion/Modeling

The authors attribute the nonlinearity of the data in Fig. 3 as indicative of the reduction of the average flow rate in the thin film above the posts: some of the liquid flows through the microtexture. Indeed, when comparing experimental data to classical predictions, only data corresponding to nearly circular jumps collapse onto the curve of an existing model for smooth substrates (Fig. 3b, inset).

One can think of the flow rate of the jet as divided between a 'leakage' flow rate through the roughness and a thin-film flow above the posts. Above the posts, the flow has an intermediate Reynolds number. On the other hand, the fluid velocity through the posts decreases rapidly and the 'leakage' flow has low Reynolds number. The authors therefore use a boundary-layer approach to consider the two flows separately and thus find an analytic solution. They use a slip boundary condition at the interface (top of the posts), where the radial velocity u must satisfy <math>u = \lambda \partial u / \partial z</math>.

The model depends on two parameters describing the substrate surface: the roughness porosity <math>\epsilon</math>, equal to <math>1 - \pi R^2/D^2</math>, and the aspect ratio of the posts <math>\kappa = H/R</math>. <math>\epsilon</math> and <math>\kappa</math> affect the model in two ways. First, the leakage rate is modeled as being proportional to the total flow rate Q: <math>q_{leak} = \alpha Q </math>, with <math>\alpha</math> a function of <math>\epsilon</math> and <math>\kappa</math>. Second, <math>\lambda</math> is proportional to the magnitude of the slip <math>\xi</math>, which is also a function of <math>\epsilon</math> and <math>\kappa</math>.

Using this approach, the authors obtain an expression of the jump radius as a function of angle. Then, for each lattice structure, one can determine the parameters <math>\alpha</math> and <math>\xi</math> by fitting to experimental data. Given this model, the authors can plot the evolution of the mean radius of the jump with the flow rate of the jet, and they find that the experimental data collapses to the analytic prediction (Fig. 4b). The authors also find that the prediction for the jump deformation conforms to experimental results (Fig. 4b, inset).

## Relation to Soft Matter

In class, we talked about calculating the contact angle for liquids on rough surfaces using the Wenzel model (in spite of all of its problems). Here the authors are addressing a similar problem, that of the shape of the hydraulic jump for surfaces with spatially patterned roughness. The paper shows that regular roughness can have a large effect on the properties of the hydraulic jump, with some fairly surprising results.

A note about the paper: the paper is written quite well, and it is fairly straightforward to understand. But the figures, especially the plots in Figures 3 and 4, are quite difficult to interpret. The different marker shapes are quite distracting, and they seem to have been picked pretty arbitrarily. Fig 3 talks about the flow rate for 4 different lattices, but it shows plots for 6 flows in part (a), and shows 5 patterns in part (b). The marker '+' in Fig 4a is never explained, and it does not reappear anywhere in the paper.