Flow induced patterning at the air/water interface

Transcription

1 Flow induced patterning at the air/water interface R. Miraghaie Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY x J. M. Lopez Department of Mathematics and Statistics, Arizona State University, Tempe AZ A. H. Hirsa Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY (Dated: March 11, 2003) Patterns on the air/water interface of a swirling cylinder flow are produced via hydrodynamic symmetry-breaking instability of the bulk flow. The patterns are rotating waves breaking the axisymmetry of the system and are longitudinal at the free surface (i.e., not surface deforming). Qualitative observations and quantitative measurements of velocity and vorticity are provided. Threedimensional Navier-Stokes computations identify the symmetry-breaking mode responsible for the waves. These waves are then used to pattern Langmuir monolayers at concentrations sufficiently below saturation. Well-ordered patterns in nature are ubiquitous and have been a source of fascination for ages. Their manifestation has allowed for a deeper understanding of many natural phenomena in non-equilibrium systems, and the ability of complex systems to form patterns has been extensively exploited in science and technology. With interfacial phenomena, pattern formation has been focused on structured films, including self-assembled monolayers and two-dimensional crystallization of proteins. These processes have for the most part relied on the microscopic structure of the monolayer as well as of the substrate, and have been dominated by diffusive time-scales. Structured monolayers can form on solid surfaces (gas/solid or liquid/solid interfaces) 1,2 as well as liquid surfaces (gas/liquid or liquid/liquid interfaces), 3,4 a major difference being that liquid surfaces allow fluid motion at the interface. This potential advantage has yet to be fully exploited. Only fluid flow in the linear regime (i.e., in the inertia-less Stokes flow limit) has so far been investigated for its influence on monolayers. For example, the effect of shear on monolayer structure has recently been studied by several groups. 5 7 Nonlinearity of fluid flow with inertia can also lead to pattern formation. Here, the susceptibility of a flow system to hydrodynamic instability when driven far from equilibrium has been exploited to produce a steadily precessing macroscopic pattern by controlling the flow parameters. One advantage of flow-induced patterning is that the length scale at which patterning takes place may be influenced by the flow geometry as well as dynamic Presently at UCLA, MAE Department, Los Angeles, CA Electronic mail:miraghaie@fusion.ucla.edu Electronic mail:lopez@math.asu.edu Electronic mail:hirsaa@rpi.edu parameters such as Reynolds number. Transverse surface waves (i.e. capillary/gravity waves) can also produce patterns at the interface, Faraday waves are a paradigm problem. 8 However, these surface deforming waves are not the types of pattern forming waves addressed here. The present interest is in patterns formed at flat gas/liquid interfaces, i.e., patterns in longitudinal motion and in the absence of traverse waves. Our interest in pattern formation on flat interfaces is due to the fact that they are more tractable to Navier-Stokes computations as well as the relative ease with which optical measurements can be made at flat interfaces. For example, probing of a flat gas/liquid interface via nonlinear optical methods for spectroscopic purposes (e.g. second-harmonic generation 9 ) is less complicated than on a highly deformed free surface. Further, flow at a flat interface allows for observation of interfacial films via microscopy as the focal distance remains constant. Resolving the free surface boundary layer to determine the velocity and shear stress is also straightforward in the case of interfaces that are essentially flat. 10 In this Letter, patterns on flat free surfaces are produced by inertial instabilities of the bulk flow. Flow with inertia was selected over other means of pattern generation. Longitudinal surface patterns are well documented in thermal Marangoni flow, where the paradigm problem is Bénard convection, 11,12 driven by interfacial temperature gradients. The present interest however is in inertia driven patterns as these can be produced isothermally, and since one of our goals is to pattern Langmuir monolayers, strong surface flows are needed to overcome their visco-elastic resistance to motion. A simple flow geometry consisting of a stationary open cylinder driven by the constant rotation of the floor was selected to investigate interfacial structuring by flow with inertia at an essentially flat free surface. A precisionbore glass cylinder, with an inner radius R = 2.5 cm,

2 2 (a) (b) (c) Clean free surface (a) z = 0.5D/R (b) z = 0 FIG. 1: Photos of the rotating wave (RW 3) at the air/water interface (Re = 2000 and D/R = 0.25), visualized using laserinduced fluorescence. The rotating wave, RW 3, steadily precesses with the floor (clockwise, as seen by the camera); parts (a), (b) and (c) are photos taken approximately 0.37 s apart, showing about 1/3 of the precession period. and depth D = 0.63 cm was utilized. The system was filled with double-distilled water and the floor was rotated at Ω = 3.08 rad s 1, resulting in a Reynolds number Re = ΩR 2 /ν = 2000 (ν = cm 2 s 1 is the kinematic viscosity of water at 22 C), signifying inertiadominated flow. A flat free surface was achieved by filling the system exactly to the top rim of the cylinder. A horizontal light sheet (< 0.05 cm in thickness) was used for illumination of the plane of interest for visualization via laser-induced fluorescence (LIF) and quantitative measurements of velocity and vorticity via digital particle image velocimetry (DPIV). For DPIV measurements, water was seeded with 22 micron polystyrene particles (Duke Scientific, 7520A). For D/R = 0.25 and Re > 1450, the axisymmetric flow observed at lower Re is unstable, resulting in a steadily precessing three-dimensional flow structure whose shape is invariant in a co-rotating frame of reference, i.e., a rotating wave (RW ). Since the Froude number Ω 2 R 2 /gd = (g is the gravitational acceleration) is small, the air/water interface remains essentially flat (i.e., no surface deforming waves). The relative flatness of the interface was confirmed independently by direct determination of the surface slope via refraction measurements using a He-Ne laser. The maximum surface slope was , which occurred at a radius of about 0.6R, and the root-mean square (rms) of the slope across the entire surface was The maximum surface deformation was about 0.3 mm, which is less than 5% of the depth. More importantly, the shape of the surface remained essentially constant with time. The rms of the surface slope as a function of time, averaged over the entire surface, was about This is only larger than the rms of the surface slope due to background vibrations, i.e. in the absence of flow. A rotating wave with azimuthal wavenumber 3, RW 3, is shown in Fig. 1, visualized by using a fluorescent dye (Aldrich 25,246-8) and illuminating with a thin laser light sheet placed at the air/water interface (z = 0). The wave precesses prograde with the floor (clockwise, as seen in the figure), but at a fraction of its speed (0.62Ω), i.e. FIG. 2: Contours of z-vorticity and horizontal velocity vectors measured via DPIV at Re = 2000 and D/R = 0.25; measurements with a clean free surface at (a) at mid-depth (z = 0.5D/R) and (b) at the free surface (z = 0). the precession period is about 3.3 s. The photos in parts (a), (b) and (c) of the figure are taken approximately 0.37 s apart. It takes about 1 minute (approximately the viscous diffusion time, D 2 /ν) after the floor is set into motion for the axisymmetric flow to develop and then a further 5 10 minutes for the rotating wave to establish itself for the present geometry (D/R = 0.25) and Re = Subsequently, the structure of the rotating wave remains unchanged indefinitely. What we have demonstrated here, for the first time, are longitudinal waves of non-trivial patterns on a flat air/water surface that are driven by hydrodynamic instabilities of the substrate. In order to understand better the flow field associated with the rotating wave and its manifestation at the air/water interface, DPIV measurements were performed in horizontal planes, including the free surface. Measurements of the velocity taken at mid-depth (z = 0.5D/R) and at the free surface (z = 0) are shown in Figs. 2(a) and (b), respectively, for a clean free surface. Each plot shows the velocity in the plane (vectors) as well as the z-vorticity component (contours). The mode-3 nature of the rotating wave is evident in the velocity vectors, and the contours of z-vorticity show the details more clearly. Although the structure of the rotating wave varies with depth, its precession period (3.3 ± 0.1 s) is identical at all depths and prograde with the floor rotation. Rotating waves in the interior of enclosed cylinder flows have previously been observed experimentally and numerically In those cases, the hydrodynamic instability breaking SO(2) symmetry, and leading to the rotating wave state, is associated with the boundary layer on the rotating disk separating and being turned into the interior to form an azimuthal shear layer with a jetlike meridional structure. For sufficiently large Re, this jet-shear layer is unstable to azimuthal disturbances. In those enclosed flow cases, the three-dimensional perturbations are localized in the interior flow and the flow near all boundaries is axisymmetric. In an open cylinder flow, 17 rotating waves similar to those in the enclosed

3 3 flows have been observed, arising from the same jet-shear layer instability. In that open flow case (D/R = 2), the three-dimensional instabilities are absent at the free surface at z = 0, where the experimental measurements showed an axisymmetric flow. Computations of the axisymmetric basic state with a flat stress-free surface at z = 0 show that there are two regions with distinct flow characteristics. 18,19 For small radii, the flow is essentially in solid-body-rotation, with zero meridional flow (axial and radial velocities) and an azimuthal velocity that is proportional to r. Whereas for larger radii, the flow has a strong meridional component that includes the jet-shear layer described earlier for the enclosed flows. For deep systems (D/R 1), the region of solid-body-rotation is small and the dynamics are dominated by the jet-shear layer. For shallow systems however, such as the case presented here, the solid-body-rotation region extends to about mid-radius. The boundary between these two regions extends along the entire depth. Here, we demonstrate that in a shallow system, this boundary is unstable to azimuthal perturbations leading to the rotating waves. Modeling these phenomena in systems with an air/water interface is not trivial. Flows with clean nondeforming gas/liquid interfaces are typically modeled as being flat stress-free interfaces. This is certainly an appropriate model for the steady axisymmetric basic state in our present problem. But, when the flow becomes unstable, the model may not be appropriate due to hidden symmetries. The hidden symmetry in our problem is due to the imposed flat stress-free interface which implies that the velocity has even z-parity. The hidden symmetry is revealed by considering the extended problem of a cylinder of twice the depth with both top and bottom endwalls (at z = ±D/R) exactly corotating, then the system is invariant to reflections about z = 0, i.e. (u, v, w)(r, θ, z, t) = (u, v, w)(r, θ, z, t) is the representation of this Z 2 reflection symmetry. Solutions invariant to this symmetry have even z-parity. Imposing a flat stress-free boundary condition on the air/water interface is equivalent to restricting solutions in the extended problem to a Z 2 -invariant subspace. However, the physical experiment is not restricted to such a subspace. Any imperfection, e.g. meniscus effects so that the surface is not perfectly flat or the presence of residual surfactants so that it is not perfectly stress-free, destroys the symmetry. In the extended system with Z 2 symmetry not imposed, the basic axisymmetric reflection-symmetric state can spontaneously lose stability to modes that have either even or odd z-parity (whereas models with a stressfree boundary can only have even z-parity modes). We have found that for deep cylinders the system loses stability to an even z-parity that also breaks axisymmetry, 17 whereas for the shallow systems considered here, the bifurcating state is to an odd z-parity mode with azimuthal wavenumber 3. In the extended system model, the odd parity means that the z-velocity at the mid-plane is non-zero, and so this plane is no longer a symmetry plane. How this mode saturates nonlinearly in the exz-velocity (a) z = 0.5D/R (b) z = 0 z-vorticity (c) z = 0.5D/R (d) z = 0 FIG. 3: Computed contours of z-velocity and z-vorticity at z = 0 and z = 0.5D/R for Re = 2000 and D/R = Vitamin K 1 monolayer (a) 0.8 mg m 2 (b) 1.6 mg m 2 FIG. 4: Contours of z-vorticity and horizontal velocity vectors measured via DPIV at the air/water interface (z = 0) with a spread monolayer of vitamin K 1 with initial concentration of (a) 0.8 mg m 2 and (b) 1.6 mg m 2 ; Re = 2000 and D/R = tended model is then quite different to how it evolves in the physical experiment (the extended model does not have an air/water interface with the surface tension force due to curvature and tangential gradients). Nevertheless, the extended model is able to correctly determine the mode responsible for the onset of instability (at onset, its magnitude is infinitesimal). Figure 3 shows computed contours of the z-velocity and z-vorticity at both z = 0.5D/R and z = 0, which should be compared with the DPIV measurements in Fig. 2. The 3D Navier-Stokes solutions were computed using a spectral code. 20 Rotating wave patterns can form with Langmuir monolayers present on the interface, and hence Langmuir monolayers can be shaped by non-axisymmetric sur-

4 4 face flow patterns. The shaping of Langmuir monolayers by axisymmetric flow patterns has recently been reported. 21 In this Letter, we present experiments that were conducted with an insoluble surfactant monolayer (amphiphile), vitamin K 1, spread on the air/water interface prior to starting the flow. DPIV measurements with a relatively low concentration of the surfactant monolayer, 0.8 mg m 2 (liquid-expanded phase; see Ref. 22 for the equation-of-state), presented in Fig. 4(a), shows that the monolayer does not disrupt the surface structuring (compare with Fig. 2b). However, Fig. 4(b) shows that with a concentration of 1.6 mg m 2 (surface-pressuresaturation level, i.e. the concentration above which surface tension is no further reduced), the surfactant inhibited symmetry breaking and the transition to a rotating wave, i.e., it stabilized the axisymmetric state. It is well established that surfactant monolayers nonlinearly couple to free surface flows (even at high Re). This occurs via tangential stress balances at the interface between the shear stress in the bulk and monolayer concentration gradient induced Marangoni stress. 23,24 This coupling alters both the bulk and interfacial flows, and in turn the monolayer is altered by the flow. In summary, a hydrodynamic instability in an autonomous system, in the form of a longitudinal rotating wave arising from the breaking of axisymmetry, has for the first time been observed at the surface of a liquid. The rotating wave was found to be capable of (macroscopically) shaping an insoluble surfactant monolayer, provided its concentration is sufficiently below saturation. Acknowledgments We wish to thank Dr. M. J. Vogel for many useful discussions and assistance with the DPIV measurements and data analysis, and Mr. J. Leung for measuring the surface slopes. This work was supported by NSF through grants CTS , CTS and CTS P. Schwartz, F. Schreiber, P. Eisenberger, and G. 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