Session GR: Viscous Flows I

Low Reynolds Flow Visualization Revisited: Free-Surface and Wall Effects

Many of the seminal experimental flow visualizations at low Reynolds number can be attributed to the pioneering works of S. Taneda. These classic investigations are still considered today benchmark visualizations and are widely used as textbook examples (Van Dyke, \textit{An Album of Fluid Motion}, 1982). With the advent of modern quantitative flow visualization techniques, we are in a position to revisit in more detail some of the original questions posed by Taneda, including boundary effects on viscous flows surrounding objects (\textit{J Phys Soc Jpn}, 1964). In the present talk, we conduct experimental flow visualizations around three-dimensional objects at low Reynolds number (\textit{Re}=$O$(10$^{-3}$-10$^{-1}))$. Quantitative visualizations are implemented in a tow tank using velocimetry measurements (PIV); models including cubes and spheres are submerged in a highly viscous Newtonian fluid (silicon oil, 5000x viscosity of water). Here, we discuss wall effects on velocity profiles in the near- and far-field surrounding such objects. Moreover, we interrogate the influence of the free surface of the tank on the resulting viscous flow fields. The present experimental setup offers a versatile framework to investigate a wide range of fundamental fluid mechanical problems relating flows at low Reynolds number.   

Transition from Hele-Shaw Flow to 2-D Creeping Flow

In the Hele-Shaw experimental technique, liquid flows at very low Reynolds number through the narrow gap $b$ between parallel plates. When a body is inserted between the plates, and dye is introduced upstream, the streaklines appear nearly identical to streamlines of steady 2-D potential flow over a body of the same shape. For example, Hele-Shaw flow does not separate at sharp corners, just like potential flow. However, if the plates are very far apart (large $b$), the resulting creeping flow at the same low Reynolds number is observed to separate at sharp corners, unlike potential flow. Here, we investigate how the flow changes from Hele-Shaw flow (small $b$) to 2-D creeping flow (large $b$). Low Reynolds number CFD simulations of a fence of height $s$ along a wall in a channel reveal that the transition from Hele-Shaw flow to 2-D creeping flow is not sudden, but rather quite gradual as channel gap width is increased. Separation bubbles appear at small $b$/$s$, and grow in size as $b$/$s$ increases. The reattachment length reaches 1{\%} of the 2-D value at $b$/$s \approx 0.21$, but it does not reach 99{\%} of the 2-D value until $b$/$s \approx 150$. Furthermore, for all values of $b$/$s$ for which separation and reattachment are observed, even for large $b$/$s$ ($>$ 100), the reattachment length of the separation bubble is non-uniform across the span; it starts high, dips to a minimum, and then slowly rises, reaching 99{\%} of the center plane value beyond about 15$s$ to 20$s$ from the wall.   

Three-dimensional corner flows in microchannels

We study, by means of three-dimensional numerical simulations and analytical investigations, low Reynolds number fluid flows in rectangular micro-channels that present sharp angles, bends or curved boundaries. These flows are characterized by the generation of secondary streamwise vorticity, adjacent to the boundary, whose intensity is related to the rate of change of the curvature of the boundary (Balsa, 1998). We also study how this not well-known, yet relevant phenomenon affects the transport of scalar quantities at the boundary.   

Exchange flows in the low Reynolds number flow limit

We analyze the viscous gravity current that occurs when two fluids with different densities flow into each other in a two-dimensional channel. Assuming that the mixing between them and the surface tension at their interface are negligible, we study the flow within the lubrication approximation. For the general case of two fluids with different viscosities and in the presence of an imposed flow rate, the evolution of the current can be described by a single nonlinear PDE. When the mean flow rate is zero (closed channel) the model admits self-similar solutions for the thickness of the gravity current and solutions are obtained for different viscosity ratios. We also present numerical solutions for the gravity current in the cases of a non-zero imposed flow rate (open channel).   

Balancing a ball on a moving vertical wall covered in viscous fluid

We present the results of experimental investigations into balancing heavy balls and cylinders on a vertical moving wall using a thin layer of viscous fluid. It is found that balance can be achieved over a very narrow range of speeds and the critical speed for fixed point behavior scales with the surface area of the cylinders and spheres. Surprising data collapse is achieved using the density of the particles.   

Visualization of Internal Flows with Pressure Oscillation and Surface Modification

A Stirling engine's displacer piston causes motion in its working fluid that exposes the fluid to pressure oscillations that directly impact flow behavior. Stirling engines are highly efficient external combustion engines that are often used in renewable energy applications and have been identified for use on near space platforms as auxiliary power units. The goal of this study was to identify the basic structures of the transient flow field within the expansion cylinder of a Stirling engine without the added complications introduced by convective heat transfer. A two-dimensional representation of the flow within the expansion cylinder of a Stirling engine was produced using an optically-accessible piston-enclosure configuration. The transient flow field within the enclosure was visualized using a rheoscopic fluid. The Reynolds number, based on the frequency of the piston oscillation and the stroke length, was varied from 1.74 to 9.05. Several transient flow structures are identified and the impact that an array of triangular fins has on these flow structures will be discussed.   

Axisymmetric Ice Shelf Dynamics

West Antarctica is composed principally of marine ice sheets, in which the mainland (grounded) ice sheet extends over the coastline as a floating ice shelf. Fed by snowfall far upstream, these sheets transport ice from the grounded component, over the grounding line, where the ice shelf lifts off, and into the ice shelf, which ultimately calves and adds water to the ocean. An idealized two-dimensional ice shelf has no dynamical influence on the grounded ice sheet or the position of the grounding line. However, horizontal stresses within a three-dimensional ice shelf, caused for example by ice rises or the lateral walls of a bay, can help fix the grounding line and prevent it from receding. This study investigates theoretically and experimentally the dynamics of an idealized three-dimensional ice shelf which flows radially from a point source, to elucidate the controlling influence of circumferential stresses within the shelf.   

A fully 3D experimental and theoretical study of flow patterns and Lagrangian trajectories generated by spinning bent rods in viscous fluids

The fluid motion induced by spinning cilia is fundamental to many living organisms. Under some circumstances it is appropriate to approximate cilia as rigid bent rods. We study the effects of shape and orientation of these idealized cilia upon flow structures in a Stokes fluid. By utilizing slender body theory and image method, an asymptotic solution is constructed for a slender body attached to a no-slip flat plane and rotating about its base sweeping out a cone. Using 3D stereoscopic projection in a table-top experiment we explore the complex flow structures and present quantified comparisons with the theoretical predictions. Intriguing short, intermediate and long time phenomena of particle trajectories are documented, and the intricacies of their theoretical modeling reported.   

A higher-order Hele-Shaw approximation for micro-channel flows

The classic hydrodynamic Hele-Shaw problem is revisited in the context of evaluating the viscous resistance to low-Mach compressible gas flows through shallow non-uniform microfluidic configurations (whose depths are small in comparison with all other characteristic dimensions). Earlier calculations have demonstrated that failure to satisfy the no-slip condition at the channel lateral walls severely restricts the applicability of the resulting approximation. To overcome this we have extended the calculation to incorporate an inner solution in the vicinity of the side walls (which, in turn, allows for the characterization of the effects of non-rectangular channel cross sections) and its matching to an outer correction. Comparison with finite-element simulations demonstrates a remarkably improved accuracy relative to the leading-order Hele-Shaw approximation. This suggests the present scheme as a useful alternative for the rapid performance estimate of microfluidic devices.   

Acoustic Droplet Vaporization through PDMS

Acoustic droplet vaporization (ADV) involves the generation of bubbles from albumin-encapsulated perfluorocarbon (PFC) droplets that have been insonated with high intensity ultrasound (US). Gas embolotherapy, utilizing ADV, may facilitate occlusion of blood flow in the vasculature as bubbles undergo volume expansion of up to 125 times. Cancer therapy could benefit from such occlusions through starvation of the tumor. In order to visualize the detailed mechanics of vaporization and expansion process of the PFC droplets, idealized microvessels were constructed using polydimethylsiloxane (PDMS) channels. Microchannels (20 micron diameter) were fabricated using PDMS with polymer-crosslinker mixing ratios ranging from 5:1 to 20:1. Droplets were introduced into the channels and exposed to US for vaporization. Mixing ratios were observed to impact the impedance matching at the water-PDMS interface, which affected the threshold for ADV. The threshold was lowest for mixing ratios of 5:1 and 20:1, and greatest for 9:1. Final bubble volumes were compared with a computational model by Ye {\&} Bull and were found to be consistent. This work is supported by NIH grant R01EB006476.