Session W13: General Fluid Dynamics: Viscous Flows (10:00am - 10:45am CST)

A New Dual, Complex Wave/Particle Theoretical View of Viscous Flow Turbulence

The mechanics of turbulence in fluid flows is considered to be the most difficult theoretical problem of Physics. Present, most successful theories rely on a heavy statistical approach. It is experimentally well known, that, according to the rate of flow, there are two distinct regimes: laminar, with very ordered stream lines and turbulent, with a chaotic behavior of the fluid motions. The transition between these two regimes is characterized by a dimensionless Reynolds Number. In a tube, for example, the transition happens at Re=2100. In a falling film at Re=10. In the 2020 APS, Denver and Washington Meetings, I did present my own complex algebra Physics leading to a non relativist theory, expressing the dual de Broglie wave/particle as an entity with a complex mass: a real mass for the particle and an imaginary mass for the associated wave. Here I apply this new concept to fluid dynamics, thus extending Navier-Stockes equation toward Schrodinger’s equation. For falling films my approach yields a good prediction of the critical Reynolds Number and shows a metastability of laminar flow in tube due to confinement of the constrained fluid.\\ \\These results may open a path toward suprafluidity.   

Chaotic elastic filament in periodically driven Stokes flow

We numerically study the dynamics of a free elastic filament in a periodically driven (period T) Stokes flow in the absence of inertia and Brownian motion. We use a bead-spring model with particle-particle interaction. We find that the dynamics depends on the elasto-viscous number, $\mu = \frac{8\pi\eta SL^4}{B}$ (where $\eta$ is viscosity, $S$ is shear rate of the fluid, $L$ is length, and $B$ is bending rigidity of the filament)-- the dimensionless ratio of viscous and elastic stress. For small $\mu$, the filament remains straight. As $\mu$ increases, we observe buckling, the appearance of two-period, and complex and even chaotic behavior.\\ To understand the dynamics of the non-autonomous system for small and intermediate $\mu$, we consider the map obtained by integrating the dynamical equations over exactly one period. We calculate the fixed points and some of the periodic orbits of this map and their stability. We further characterize the chaotic state at large $\mu$ in terms of the space-time correlation function of the curvature as a function of arc-length. We also calculate the phase-portrait of the time series of velocity at a fixed Eulerian point. For large $\mu$, the phase portrait converges to an attractor with a fractal (non-integer) dimension.   

Two-layer fluid flows on inclined surfaces

We present a theoretical and experimental study of the dynamics of two-layer viscous fluid flows on inclined surfaces, motivated by natural and industrial phenomena involving the interactions between two fluid layers. A general model describing the evolution of two fluids on an inclined substrate is developed and explored to reveal a rich variety of flow regimes for different modes of release. For the canonical example in which two fluids are introduced at a constant flux, the flow forms two regions: an upstream region containing both fluids, and a downstream region comprised purely of the lighter fluid, with a sharp intervening jump in thicknesses between the two. By constructing similarity solutions, we establish a full regime diagram of the possible configurations over all asymptotic limits of the viscosity, flux and density ratios. For the release of two fixed volumes of fluid, the layers separate completely into two disjoint but connected regions, contrasting in essential structure from the constant-flux case. Even a small volume of the heavier fluid is able to significantly accelerate the propagation of the lighter fluid in front of it. Excellent agreement is found between our theoretical predictions and the results of a series of laboratory experiments.   

Not Participating

Motivated by the emergence of new drug delivery techniques employing backpacks attached to macrophages (WBCs) and erythrocytes (RBCs), we study theoretically and numerically the motion of conjoined objects in Stokes flow. As a first approximation, the cells and backpacks are modeled as non-deformable conjoined spheres. The motion of this simplified system is investigated, and the resulting velocity and pressure fields are evaluated for a range of physiologically relavent conditions. Of particular interest, is the net disjoining force that acts to separate the conjoined spheres. In the physiological case, the backpacks separate from the cells when the disjoining force exceeds the adhesive force between the two. We report the important disjoining force as a function of physically relavent variables such as the sphere size ratios, sphere orientations and the center to center distance between the spheres. We believe the results from this study will improve our understanding of the conditions that lead to cell-backpack seperation, and will aid the further development of backpack mediated drug delivery.   

Odd viscosity in three dimensional flows

A fluid that breaks microscopic time-reversal symmetry, for example, by being composed of spinning units, can acquire a so-called “odd-viscosity”, an antisymmetric contribution to the viscosity tensor. In isotropic two-dimensional fluids, it has been shown that the odd viscosity does not modify the flow in the incompressible limit. In this work, we consider the extension of odd viscosity to three dimensional flows. We find a host of additional odd viscosity coefficients that give rise to dramatic changes in the flow even in incompressible fluids. By revisiting classic hydrodynamic problems in the Stokes limit, we elucidate the effects of these additional coefficients, and discuss the modifications in the presence of odd stress. Our work provides guiding principles for experimental and numerical investigations of complex fluids in suitable biological and active matter systems.   

Deformation of a flexible hair in a Poiseuille flow

Numerous biological systems rely on hair-like structures to sense their fluidic environment. Fluid-structure interactions can result in the elastic bending of flexible hairs. Recent theoretical work has shown that clamped elastic beams can act as direct strain sensors or as strain amplifiers by deforming their substrate. In this study, we investigate the deformation of a clamped elastic rod in a viscous flow. In our experiments, the elastomer rod and base are mounted on a wall of a rectangular channel, in which the flow field has a Poiseuille profile. We vary the geometry and Young’s modulus of the rod and the flow rate in the channel. We measure the deformation of the rod and show the influence of the confinement on deflection. We analyze our results using numerical simulations and scaling arguments for a slender body in a viscous flow.   

Inertial contributions to friction measurements in thrust bearings

Surface textures decrease friction in lubricated sliding contact. Traditionally, the friction reduction for a given textured surface is determined by using the Reynolds equation, which neglects fluid inertia. However, as the separation and relative motion between the surfaces increase, inertia can affect the measured tangential and normal forces for flow over a textured surface, and thus cause the coefficient of friction to differ from the purely viscous, Stokes flow prediction. Here, the increase in torque and normal force between a moving plate and stationary textured surface, which simulates a textured thrust bearing, are calculated as a function of the Reynolds number in the thin film limit. The predictions for a non-textured thrust bearing are compared to fully 3-D numerical simulations of the incompressible Navier-Stokes equation, and the predictions for textured thrust bearings are compared to experimental data given in the literature. Good agreement is seen between the predictions and the data, validating the predicted scaling laws. This work also suggests that inertia can be used as a secondary effect to reduce friction in lubricated sliding, and textures that take advantage of the inertial effects will have lower friction than those that only use purely viscous effects.   

Viscoacoustic Squeeze Film Force on a Rigid Disk Undergoing Small-amplitude Axial Oscillations

The repulsive force felt by a rigid disk vibrating along its axis of symmetry close to a parallel surface is investigated through asymptotic reduction of the Navier-Stokes equations. It is of great interest to predict the load capacity of such squeeze film bearings due to their ubiquity in high-speed rotary equipment and contactless assembly-line transport of microelectronics. Previous attempts to relate the levitation height and force largely rely on simplifications afforded by neglecting either viscous or inertial effects. The present analysis applies the slender-flow approximation in the limit of small amplitude oscillations to derive explicit closed-form expressions for the time-averaged radial pressure distribution. The resulting levitation force is shown to depend on two dimensionless parameters -- namely -- the Stokes number, quantifying the relative importance of viscous stresses and local acceleration, and a compressibility parameter, comparing the timescales of acoustic wave propagation and disk oscillation. In addition to synthesizing the historically explored lubrication and acoustic limits under a unified framework, the analytical results presented serve to potentially reduce computational costs involved in feedback control of hydrodynamic bearings.   

Mutual Capture Of Two Charged Particles Settling Under Gravity In A Viscous Fluid

Interacting particles in a fluid are said to capture each other if their relative orbits are bounded. Such particle capture in a fluid is a topic of interest in both theory and application. One important feature of the global dynamics is the size of the set of all initial conditions in which two particles capture each other, especially as parameterized by physical ratios such as particle radii and densities. We explore this problem for a pair of charged point particles settling under gravity in a Stokes flow. Having previously demonstrated that vertical stable stationary states are possible, we now also find a family of inclined stable stationary states. We demonstrate how a vertical non-stable stationary state is essential for establishing the size of the capturing set. We show that for a region in the parameter space of the ratios of the particle radii \& densities, and of electrostatic to gravitational forces, the capturing set of particle relative positions is large compared to their radii.