Bulletin of the American Physical Society
2006 59th Annual Meeting of the APS Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2006; Tampa Bay, Florida
Session FP: Viscous Flows II |
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Chair: Jim Duncan, University of Maryland Room: Tampa Marriott Waterside Hotel and Marina Meeting Room 12 |
Monday, November 20, 2006 8:00AM - 8:13AM |
FP.00001: The Viscous Catenary: Experiments and Simple Theory Catalin D. Mitescu, John Koulakis, Fran\c{c}oise Brochard, Pierre-Gilles deGennes, Etienne Guyon Detailed experimental observations have been carried out on the evolution of a viscous catenary, a filament of (100,000 centistokes kinematic viscosity) silicone oil stretched between two points, falling under the influence of a uniform gravitational field. Precise data describing this evolution have been obtained by digitizing high-resolution, multiple-exposure stroboscopic pictures. The shape of the filament is observed to depend on the quantity of fluid it contains: the thicker filaments assuming the catenary-like shape, while thinner ones evolve in a more surprising and complicated manner. In the first case, we have found that the evolution is well described by a simple theoretical model involving a variational calculation of the energy balance between the rate of viscous dissipation and the decrease in gravitational energy of the filament. For the thinner filaments, on which theoretical work is in progress, surface tension effects, complicating the analysis, appear to be responsible for the more complex structure. [Preview Abstract] |
Monday, November 20, 2006 8:13AM - 8:26AM |
FP.00002: Rigorous Error Estimates for Reynolds' Lubrication Approximation Jon Wilkening Reynolds' lubrication equation is used extensively in engineering calculations to study flows between moving machine parts, e.g. in journal bearings or computer disk drives. It is also used extensively in micro- and bio-fluid mechanics to model creeping flows through narrow channels and in thin films. To date, the only rigorous justification of this equation (due to Bayada and Chambat in 1986 and to Nazarov in 1987) states that the solution of the Navier-Stokes equations converges to the solution of Reynolds' equation in the limit as the aspect ratio $\varepsilon$ approaches zero. In this talk, I will show how the constants in these error bounds depend on the geometry. More specifically, I will show how to compute expansion solutions of the Stokes equations in a 2-d periodic geometry to arbitrary order and exhibit error estimates with constants which are either (1) given in the problem statement or easily computable from $h(x)$, or (2) difficult to compute but universal (independent of $h(x)$). Studying the constants in the latter category, we find that the effective radius of convergence actually increases through 10th order, but then begins to decrease as the inverse of the order, indicating that the expansion solution is probably an asymptotic series rather than a convergent series. [Preview Abstract] |
Monday, November 20, 2006 8:26AM - 8:39AM |
FP.00003: Flow between a cavity and a flexible wall: Lubrication model and finite-element calculations Satish Kumar, Xiuyan Yin Flows near deformable solid boundaries occur in a diverse range of settings including coating and printing processes, biological systems, and suspensions. In order to examine the effect of surface topography on the elastohydrodynamic interactions that arise in these flows, the flow between a rigid cavity and a flexible wall is studied using a lubrication model and finite-element calculations. In the lubrication model, Reynolds equation for the fluid is coupled to a model for a uniformly tensioned wall, resulting in a coupled system of nonlinear ordinary differential equations. When the wall tension is small relative to viscous forces, the wall easily deforms and assumes a shape similar to that of the cavity. The pressure profiles are also dramatically altered and in some cases show only a valley without a mountain. Cavity shape is found to have a significant influence on both the pressure profiles and the wall deformation. Predictions from the lubrication model agree remarkably well with those from finite-element solution of Stokes' equations coupled with the wall model. The finite-element calculations also suggest that replacing the bottom of a cavity with a flexible wall and applying a time-periodic pressure to it may be a potentially useful way to improve mixing and heat/mass transport in the cavity. [Preview Abstract] |
Monday, November 20, 2006 8:39AM - 8:52AM |
FP.00004: Thin film flows over Structured Substrates Alex Oron, Shomeek Mukhopadhyay, Robert Behringer The dynamics of liquids over patterned substrates has been of tremendous interest because of applications in microfluidics and in creating specially engineered `non -- stick' surfaces. We study both experimentally and numerically, the limiting case of completely wetting silicone oil (PDMS) on a silicone wafer on which is deposited a `step like' precursor layer of the oil. Step like precursors of PDMS are deposited ranging in viscosity from 50 centistokes to 10000 centistokes. Silicone oil is driven up an inclined plane against gravity by imposing a temperature gradient. As the contact line emerges from the reservoir it shows a `hyberbolic tamgent' shape near the step and the flat film near the front is non trivially modified in the presence of the step. We describe the formation and evolution of the perturbed contact line in the `lubrication theory' approximation. In this particular case the flow field is analyzed in three separate regions and then `matched' across the step. The presence of the `hyberbolic tangent' shape is markedly different from other corner flow problems in thin film flows. [Preview Abstract] |
Monday, November 20, 2006 8:52AM - 9:05AM |
FP.00005: Mixing in Thin Flows over a Curved Substrate Jean-Luc Thiffeault, Khalid Kamhawi We consider steady gravity-driven flow of a thin layer of viscous fluid over a curved substrate. The substrate has topographical variations (`bumps') on a large scale compared to the layer thickness. Using lubrication theory, we find the velocity field in generalized curvilinear coordinates. We correct the velocity field so as to satisfy kinematic constraints, which is essential to avoid particles escaping the fluid when computing their trajectories. We then investigate the particle transport properties of flows over substrates with translational symmetry, where chaotic motion is precluded. The existence of trapped and untrapped trajectories leads to complicated transport properties even for this simple case. For more general substrate shapes, the trajectories chaotically jump between trapped and untrapped motions [1].\\[12pt] \noindent [1] J.-L. Thiffeault and K. Kamhawi, http://nlin/0607075 [Preview Abstract] |
Monday, November 20, 2006 9:05AM - 9:18AM |
FP.00006: Slow rupture of viscous film: theory and experiment Igor Kliakhandler, Sofya Chepushtanova A rupture of viscous levitating horizontal film between parallel needles is considered. The system reaches remarkable steady-state propagation mode. Profile of the rupture is similar to U-shape. Visually, the system resembles many classical problems such as rising long bubble in the tube, or Hele-Shaw tongue. The system has a clear separation of scales: the rim on the rupture front is substantially thicker than the film itself, but much smaller than distance between the needles. This allows to develop a simple theory of the rupture propagation. The theory agrees well with the experiments. [Preview Abstract] |
Monday, November 20, 2006 9:18AM - 9:31AM |
FP.00007: Liquid transport \& jetting via light-scattering Robert Schroll, R\'egis Wunenburger, Wendy Zhang, Jean-Pierre Delville Previous works have shown that light scattering by inhomogeneities in the index of refraction of a fluid can drive a large-scale flow. Here we investigate how the interaction of this large-scale flow with radiation pressure across an interface in a phase-separated liquid near a second-order phase transition can create a thin, stable jet. Estimates of the volume flux of liquid transported by light-scattering inside the jet agree with experimental measurements. [Preview Abstract] |
Monday, November 20, 2006 9:31AM - 9:44AM |
FP.00008: Vortex dynamics and nonlinear characterization of the bifurcation in the flow in an oscillatory cavity. Guillermo Ovando, Guadalupe Huelsz, Eduardo Ramos, Hector Ju\'arez The flow in a rectangular cavity driven by the oscillatory motion of the vertical walls has been numerically studied for three different Reynolds numbers (Re) based on the cavity width: 50, 500 and 1000 and three different displacement amplitudes of the vertical oscillatory walls (Y=amplitud/width) of 0.2, 0.4 and 0.8. The vortices cores were identified using the Jeong-Hussain criterium and two vortex formation mechanisms are found: M1) the oscillatory shear motion of the moving walls, combined with the dephasing of the fluid motion inward the cavity and the presence of the fixed walls and M2) the abrupt change in the flow direction at the corners of the cavity. We found that vortices occupy smaller areas as the Reynolds number increases. All cases studied shown cyclic and axial symmetries, except for the largest case (Re=1000, Y=0.8), where the axial symmetry is lost. We interpret this phenomenon as a bifurcation and characterize it by analyzing the variation of the kinetic energy along the vertical central axis (x=0), averaged over one cycle (l$^{2})$ as a function of the Reynolds number (for Y=0.8). The behaviour of l$^{2}$ as a function of Re indicates that the loss of axial symmetry is via a supercritical pitchfork bifurcation. [Preview Abstract] |
Monday, November 20, 2006 9:44AM - 9:57AM |
FP.00009: ABSTRACT WITHDRAWN |
Monday, November 20, 2006 9:57AM - 10:10AM |
FP.00010: Creeping Flows Past a Partially Engulfed $2D$ Multiphase Droplet D. Palaniappan The reflection principle is used to construct closed form exact solutions for the two-dimensional perturbed flow fields in the presence of a $2D$ vapor-liquid compound droplet in the limit of low-Reynolds number. The geometry of the multiphase droplet is composed of two overlapping infinitely long cylinders $C_a$ and $C_b$ of radii $a$ and $b$, respectively, intersecting at a vertex angle $\frac{\pi}{2}$. The composite inclusion has the shape resembling a $2D$ {\it lens} type of object with a vapor cylinder $C_a$ partly protruded into a fluid cylinder $C_b$ with a viscosity different from that of the host fluid. The mathematical problem with this twin-circular assembly in the Stokes flow environment is formulated in terms of Stokes stream function with mixed boundary conditions at the boundary of the hybrid droplet. General expressions for the perturbed stream functions in the two phases are obtained in a straightforward fashion using Kelvin's inversion together with shift and reflection properties of biharmonic functions. The general results are then exploited to derive singularity solutions for the hybrid droplet embedded in several unbounded non-uniform flow fields. The exact solutions are utilised to plot the streamline topologies and they show interesting flow patterns. While the flow fields exterior to the droplet exhibit back flow regions and compartmental divisions, the interior flow fields show existence of attached eddies and stagnation points (hyperbolic points). Surprising, but interesting flow features are observed in the case of two-dimensional flows. [Preview Abstract] |
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