^{1}, Laurent Courbin

^{2}, Jérome Crest

^{1}and Howard A. Stone

^{1,a)}

### Abstract

We analyze the inertia dominated flow of thin liquid films on microtextured substrates, which here are assemblies of micron-size posts arranged on regular lattices. We focus on situations for which the thin-film thickness and the roughness characteristic length scale are of the same order of magnitude, i.e., a few hundred microns. We assume that the liquidflows isotropically through the roughness at a flow rate that depends on the geometrical features of the porous layer; above the texture, the flow is characterized by a larger Reynolds number and modeled using a boundary layer approach. The influence of the microtexture on the thin-filmflow above the microposts is captured by a reduction of the flow rate due to the leakage flow through the texture and a slip boundary condition, which depends on the flow direction as well as on the lattice properties. In this way, the velocity field in the free surface flow adopts the symmetry of the microtexture underneath. The results of this model are in good agreement with experimental observations obtained for thin-filmflows formed upon jet impact on microtextures. The characteristics of the polygonal hydraulic jumps that we obtain depend on both the jet parameters and the topographical features of the surface roughness. We use the measurements and the numerical predictions to estimate the flow rate through the shallow porous layer and the effective slip length for this inertia dominated flow regime. We also discuss the limitations of the model.

The authors thank A. Ajdari, M. Bazant, T. Bohr, J. Bush, and C. Clanet for insightful conversations. We also thank the Harvard MRSEC (Grant No. DMR-0820484) and Schlumberger-Doll Research for support of this research.

I. INTRODUCTION

II. PROBLEM FORMULATION

A. Slip over microtextured surfaces

1. Definition of the effective slip length

2. Flow through roughness: Leakage flow and effective slip length

B. Inertia dominated thin-filmflows on smooth substrates

C. Boundary layer development on a composite interface

D. Thickness of the boundary layer

E. Position of the hydraulic jump

III. DISCUSSION OF EXPERIMENTAL RESULTS

A. Main features of the jumps and modeling assumptions

1. Macroscopic consequences of the micron-scale topology

2. Experimental assessment of the modeling assumptions

B. Discussion of the experimental results

1. Evolution of the jump as function of the total flow rate

2. Determination of two free parameters

C. Limits of the model

1. Deformations of the jumps

2. Shape of the jump for fluids of large viscosity

IV. CONCLUSION

### Key Topics

- Thin film flows
- 45.0
- Boundary value problems
- 27.0
- Viscosity
- 24.0
- Microscale flows
- 17.0
- Surface tension
- 16.0

## Figures

Schematic of thin-film flow over rough substrates. (a) When the thickness of the film is large compared to the wavelength of the topographical features , the influence of flow through the roughness is captured by the slip velocity in the plane defined by the dotted line. (b) When is smaller than , the thickness of the film and the position of the free surface become nonuniform. (c) When is of the same order as , the roughness is modeled using a slip boundary condition, which modifies the development of the viscous boundary layer, as discussed in this paper.

Schematic of thin-film flow over rough substrates. (a) When the thickness of the film is large compared to the wavelength of the topographical features , the influence of flow through the roughness is captured by the slip velocity in the plane defined by the dotted line. (b) When is smaller than , the thickness of the film and the position of the free surface become nonuniform. (c) When is of the same order as , the roughness is modeled using a slip boundary condition, which modifies the development of the viscous boundary layer, as discussed in this paper.

(a) Schematic representation of a unit cell of the square lattice in the plane defined by the top of the posts. On the posts, there is no slip as the liquid flows on a solid substrate. Between the posts, the liquid flows on top of a liquid layer, so the fluid velocity is nonzero (modelled by an effective slip boundary condition). (b) Schematic representation of the thin-film flow over a square lattice of cylindrical posts of radius and height , typically . and are, respectively, the thickness and the total flow rate of the liquid layer. is referred to as the leakage flow rate through the texture.

(a) Schematic representation of a unit cell of the square lattice in the plane defined by the top of the posts. On the posts, there is no slip as the liquid flows on a solid substrate. Between the posts, the liquid flows on top of a liquid layer, so the fluid velocity is nonzero (modelled by an effective slip boundary condition). (b) Schematic representation of the thin-film flow over a square lattice of cylindrical posts of radius and height , typically . and are, respectively, the thickness and the total flow rate of the liquid layer. is referred to as the leakage flow rate through the texture.

Definition of for square (a) and hexagonal (b) lattices. Each lattice is defined by its unit cell, and represented here in the plane defined by the top of the posts, i.e., at the composite interface. is the angle between the direction of the flow, indicated by the black arrow and the main axis of the lattice (white arrow). The projected segment lines of total length are represented in gray. The effective slip length is assumed proportional to , the total length of the black segment lines. The angular dependence of is represented for a square (c) and a hexagonal lattice (d). The two dotted lines correspond to values of equal to 200 and . Lattice parameters are as follows: lattice spacing , post radius , and post height .

Definition of for square (a) and hexagonal (b) lattices. Each lattice is defined by its unit cell, and represented here in the plane defined by the top of the posts, i.e., at the composite interface. is the angle between the direction of the flow, indicated by the black arrow and the main axis of the lattice (white arrow). The projected segment lines of total length are represented in gray. The effective slip length is assumed proportional to , the total length of the black segment lines. The angular dependence of is represented for a square (c) and a hexagonal lattice (d). The two dotted lines correspond to values of equal to 200 and . Lattice parameters are as follows: lattice spacing , post radius , and post height .

Schematic representation of the two-dimensional thin-film flow. The discontinuous line delimits the boundary layer of thickness .

Schematic representation of the two-dimensional thin-film flow. The discontinuous line delimits the boundary layer of thickness .

(a) Experimental setup and (b) image of a circular hydraulic jump, indicated by an arrow on the photograph, formed upon impact of a water jet on a smooth substrate (the total flow rate and the jet radius ). The jet impacts a smooth substrate, which presents a circle of diameter 5 cm centered on the impact point of the jet. The circle delimits the region that is removable to permit the study of the surface roughness effects [indicated in dark gray on the horizontal plate shown in (a)].

(a) Experimental setup and (b) image of a circular hydraulic jump, indicated by an arrow on the photograph, formed upon impact of a water jet on a smooth substrate (the total flow rate and the jet radius ). The jet impacts a smooth substrate, which presents a circle of diameter 5 cm centered on the impact point of the jet. The circle delimits the region that is removable to permit the study of the surface roughness effects [indicated in dark gray on the horizontal plate shown in (a)].

Evolution of flow properties for a slip coefficient varying between 0 (no slip) and 5 [slip length equal to ]. (a) Self-similar solution at . The velocity at the composite interface increases with the slip coefficient. (b) Difference between and : the value of is smaller for larger values of the slip coefficient.

Evolution of flow properties for a slip coefficient varying between 0 (no slip) and 5 [slip length equal to ]. (a) Self-similar solution at . The velocity at the composite interface increases with the slip coefficient. (b) Difference between and : the value of is smaller for larger values of the slip coefficient.

Evolution of flow properties, and , reported in for a slip coefficient varying between 0 (no slip) and 5 [slip length equal to ]. (a) . (b) Thickness of the boundary layer for a radial position . We use parameters whose values are typical of the experimental study described in Sec. III: , , and . We chose for . The curves correspond to composite substrates characterized by a leakage flow parameter varying between 0 (bottom curve) and 0.5 (top curve) in increments of 0.1. The thickness of the boundary layer above a smooth substrate is equal to (Ref. 25).

Evolution of flow properties, and , reported in for a slip coefficient varying between 0 (no slip) and 5 [slip length equal to ]. (a) . (b) Thickness of the boundary layer for a radial position . We use parameters whose values are typical of the experimental study described in Sec. III: , , and . We chose for . The curves correspond to composite substrates characterized by a leakage flow parameter varying between 0 (bottom curve) and 0.5 (top curve) in increments of 0.1. The thickness of the boundary layer above a smooth substrate is equal to (Ref. 25).

Photographs of polygonal hydraulic jumps observed on microtextured substrates: (a) hexagonal lattice and (b) square lattice. The parameters of the water jets are and . The lattice parameters are , , and . The insets are photographs of the lattices of posts.

Photographs of polygonal hydraulic jumps observed on microtextured substrates: (a) hexagonal lattice and (b) square lattice. The parameters of the water jets are and . The lattice parameters are , , and . The insets are photographs of the lattices of posts.

Photographs of polygonal hydraulic jumps observed on a square lattice (, , and ). The radius of the liquid jet is . Different viscosities are obtained by adding glycerol to water: (a) 0 wt % glycerol, , and . (b) 10 wt % glycerol, , and . (c) 20 wt % glycerol, , and . (d) 40 wt % glycerol, , and .

Photographs of polygonal hydraulic jumps observed on a square lattice (, , and ). The radius of the liquid jet is . Different viscosities are obtained by adding glycerol to water: (a) 0 wt % glycerol, , and . (b) 10 wt % glycerol, , and . (c) 20 wt % glycerol, , and . (d) 40 wt % glycerol, , and .

Influence of the surface tension on the size of the jumps: photographs of polygonal hydraulic jumps over a square lattice (, , and ) for two different surface tensions. Liquid jets have the same radius . The addition of SDS at a concentration equal to the CMC leads to a reduction in surface tension from (a) for pure water to (b) .

Influence of the surface tension on the size of the jumps: photographs of polygonal hydraulic jumps over a square lattice (, , and ) for two different surface tensions. Liquid jets have the same radius . The addition of SDS at a concentration equal to the CMC leads to a reduction in surface tension from (a) for pure water to (b) .

Influence of the location of the posts on the shape of the jump. Photographs of polygonal hydraulic jumps over square lattice of posts (, , and ). The parameters of the water jets are and . In (a) and (b), the surface area covered by microposts is indicated by the dashed lines. The patterned disc is rotated around the water jet, which remains fixed. (c) Microtextured stripe of width 2 mm and length 5 cm. (d) Complementary coverage to that of (a) and (b): the microtexture covers the whole disc of diameter 5 cm except for the disc sector used in (a) and (b).

Influence of the location of the posts on the shape of the jump. Photographs of polygonal hydraulic jumps over square lattice of posts (, , and ). The parameters of the water jets are and . In (a) and (b), the surface area covered by microposts is indicated by the dashed lines. The patterned disc is rotated around the water jet, which remains fixed. (c) Microtextured stripe of width 2 mm and length 5 cm. (d) Complementary coverage to that of (a) and (b): the microtexture covers the whole disc of diameter 5 cm except for the disc sector used in (a) and (b).

Influence of the surface area covered by posts on jumps. Photographs of polygonal hydraulic jumps over substrates with different coverages by the microtexture. The surface area covered by the posts is delimited by a dashed line. The parameters of the square lattice are , , and , and the parameters of the water jets are equal to and . The microtextured surface is centered on the impact point and covers (a) a disc of diameter 5 cm, (b) a disc of diameter 1.25 cm, and (c) a square of side length 2.5 cm.

Influence of the surface area covered by posts on jumps. Photographs of polygonal hydraulic jumps over substrates with different coverages by the microtexture. The surface area covered by the posts is delimited by a dashed line. The parameters of the square lattice are , , and , and the parameters of the water jets are equal to and . The microtextured surface is centered on the impact point and covers (a) a disc of diameter 5 cm, (b) a disc of diameter 1.25 cm, and (c) a square of side length 2.5 cm.

Jump radius as a function of the applied flow rate: photographs of polygonal hydraulic jumps over a square lattice (, , and ). The parameters of the water jets are for a total flow rate of (a) , (b) , and (c) .

Jump radius as a function of the applied flow rate: photographs of polygonal hydraulic jumps over a square lattice (, , and ). The parameters of the water jets are for a total flow rate of (a) , (b) , and (c) .

Photographs of hydraulic jumps over (a) smooth substrate and (b) square lattice of posts (, , and ). The dotted line indicates the mean radius of the jump on the smooth substrates. Some corners of the polygonal jump are located outside the circle delimited by the dotted line. The water jet parameters are and .

Photographs of hydraulic jumps over (a) smooth substrate and (b) square lattice of posts (, , and ). The dotted line indicates the mean radius of the jump on the smooth substrates. Some corners of the polygonal jump are located outside the circle delimited by the dotted line. The water jet parameters are and .

Influence of the shape of the posts on the jumps: photographs of polygonal hydraulic jumps over square lattices (, , and ). The (a) circular, (b) square, and (c) star-shaped microposts cover a disc of diameter of 5 cm centered at the impact point. Parameters of the water jets: and a total flow rate of .

Influence of the shape of the posts on the jumps: photographs of polygonal hydraulic jumps over square lattices (, , and ). The (a) circular, (b) square, and (c) star-shaped microposts cover a disc of diameter of 5 cm centered at the impact point. Parameters of the water jets: and a total flow rate of .

Influence of topographical parameters on the jump formed from jets of water. (a) Average radius and (b) maximum deformation vs flow rate for four square lattices with post radius and different lattice spacings and post heights: (△) and , and , (▽) and , (○) and , (◻) and , and smooth substrate.

Influence of topographical parameters on the jump formed from jets of water. (a) Average radius and (b) maximum deformation vs flow rate for four square lattices with post radius and different lattice spacings and post heights: (△) and , and , (▽) and , (○) and , (◻) and , and smooth substrate.

Coefficients of the model determined by fitting the experimental value of and with . Leakage flow and slip length coefficients as functions of (a) the aspect ratio of the posts and (b) the roughness porosity . Symbols are identical to those in Fig. 16.

Coefficients of the model determined by fitting the experimental value of and with . Leakage flow and slip length coefficients as functions of (a) the aspect ratio of the posts and (b) the roughness porosity . Symbols are identical to those in Fig. 16.

Results of the modeling for square lattices of posts. (a) Evolution of the shape of the jump for increasing flow rates from left to right. Lattice parameters: post radius , lattice spacing , and post height . (b) Shapes predicted for jumps over square lattices with different lattice parameters. Symbols as in Fig. 16. In these simulations, we use the values of and determined from experimental data (Fig. 17).

Results of the modeling for square lattices of posts. (a) Evolution of the shape of the jump for increasing flow rates from left to right. Lattice parameters: post radius , lattice spacing , and post height . (b) Shapes predicted for jumps over square lattices with different lattice parameters. Symbols as in Fig. 16. In these simulations, we use the values of and determined from experimental data (Fig. 17).

Results of the modeling. (a) Comparison between the results of the model (solid line) and the experimental data. The results obtained in Ref. 26 are also represented (●). (b) Maximum deformation of the shape predicted by the model. Symbols as in Fig. 16.

Results of the modeling. (a) Comparison between the results of the model (solid line) and the experimental data. The results obtained in Ref. 26 are also represented (●). (b) Maximum deformation of the shape predicted by the model. Symbols as in Fig. 16.

Influence of the surface tension predicted by the model. Prediction of the shape of the jump over a square lattice with parameters , and . The parameters of the jet are and . In (a) , and in (b) .

Influence of the surface tension predicted by the model. Prediction of the shape of the jump over a square lattice with parameters , and . The parameters of the jet are and . In (a) , and in (b) .

Influence of the kinematic viscosity on the results of the modeling. Prediction of the shape of the jump over a square lattice with parameters , , and . The jet is described by and and (a) , (b) , and (c) .

Influence of the kinematic viscosity on the results of the modeling. Prediction of the shape of the jump over a square lattice with parameters , , and . The jet is described by and and (a) , (b) , and (c) .

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