Session EP: Viscous Flows I

The Effect of Confining Boundaries on Viscous Gravity Currents.

We examine the influence of different bottom boundaries on viscous gravity currents by considering two cases: horizontal flow along pipes with (i) semi-circular and (ii) V-shaped cross-sections. The main motivations of the study are to understand the underlying structure of the solutions and also to indicate how to analyze lava flows down confined channels. Similarity solutions are obtained for both cases treating the problem as a thin layer of unidirectional viscous flow. Laboratory experiments are set up so that a viscous liquid (glycerine) is initially suspended by a lock gate and released at one end of the pipe into open air. The rate of spread and shape of the resulting currents will be discussed quantitatively. We will conclude by comparing the analytic solutions with the experimental data and showing that they are in accurate agreement.   

Viscous gravity currents in the presence of inhomogeneous ambient flows

Many practical situations involve gravity currents moving in external flows. We present a study of gravity currents moving in spatially inhomogeneous flows. We begin by considering the effect of a linear straining flow on the spreading of steady viscous gravity currents originating from localised sources. The spreading of the current is altered by the presence of the external flow; with the cross stream flow acting to reduce the extent of spreading for accelerating flows, and enhance it in the case of decelerating flows. Similarity solutions are developed to describe the spreading behaviour. By applying a conformal mapping technique, these results are generalised to steady viscous gravity currents moving in the presence of a layerwise irrotational ambient flow. These results are illustrated with examples of spreading in the presence of linear straining flows, radial source/sink flows and flows past rigid bodies. The spreading behaviour is described using closed form expressions and near and far field similarity solutions are developed. A laboratory study confirms the salient features of the models used.   

Axisymmetric viscous gravity currents flowing over a deep porous medium

When a viscous fluid flows over a porous substrate, it not only spreads but also seeps into the underlying medium. Such flows have relevance to the design of shingle beds for use as safety features around storage facilities of dense fluids and to flow through fissures in porous rocks. Whereas previous investigations have been confined to two-dimensional flows of fixed volume, we have investigated currents fed by a constant fluid flux flowing axisymmetrically over a deep porous bed. Our experimental system consisted of glycerin spreading over monodisperse glass spheres of known permeability and the data were analyzed using scaling analyses. We have also solved a mathematical model using the well-known equations for a viscous gravity current spreading due to the slope of its free surface augmented by a simple draining law. Its predictions agree well with our experimental results and quantify, in particular, the maximum distance to which the current spreads as a function of the material and input properties.   

Squeezing and de-wetting of a drop between plane-parallel walls: a model problem for understanding capillary adhesion

The radial squeezing and de-wetting of a thin film of viscous Newtonian fluid filling the gap between parallel plane walls is examined both experimentally and theoretically for gap spacings much smaller than the capillary length ($<$ 1 $mm$). The interaction between squeezing or de-wetting and surface tension is parameterized by a single dimensionless variable, $F$, which is the ratio of the constant force supplied by the top plate to surface tension acting on the spreading drops circumference. In the de-wetting problem the analytical solution reveals the formation of a singularity, leading to capillary adhesion, as the gap spacing approaches a critical value that depends on $F$ and the contact angle. Experiments are performed to test the analytical predictions for both squeezing and de-wetting by using 3 fluids with different viscosity and surface tension for gap spacings $O$(10-1000 $\mu m$). There is excellent agreement between theory for the change in gap spacing and the experimental results for either squeezing or de-wetting.   

Asymptotic Analysis of Colloid Diffusivity Altered by Polymer Depletion

Self-diffusion of Brownian particles plays an important role controlling transport in many biological and colloidal systems. In a homogeneous fluid, single particle' long-time self-diffusivity is characterized by the Stokes-Einstein relation. In polymer solutions, such a prediction is not accurate because particle dynamics is strongly affected by the background polymer chains suspended in the medium. For solutions containing non-adsorbing chains, a depletion zone immediately adjacent to the surface can reduce the viscous drag a particle experiences. This is known as non-Stokes-Einsteinian behavior. Based on the mean-field approximation, once the molecular weight, the degree of polymerization, intrinsic viscosity, and the correlation length of the polymer chains are found, the depletion zone can be characterized by a polymer density function. By bridging the microscopic thermodynamic theory and continuum fluid flow analysis, we present theoretical prediction of the resistance force a moving sphere experiences within non-adsorbing polymer solutions, for both translational and rotational modes. We develop the Green function based second-order perturbation approximation by solving the modified Stokes equation with nonuniform viscosity. The analytical result can be easily applied to quantify how polymer depletion alters the long-time diffusivity of Brownian particles.   

The motion of an ellipsoid in a rotating Stokes flow

Experimental observations of the motion of ellipsoids in a rotating horizontal drum, fully filled with very viscous fluid, have been recorded. For a given drum speed an ellipsoid lies adjacent to the rising drum wall at a fixed position, with its long axis $a$ parallel to the direction of wall motion. For ellipsoids with maximum radius of curvature less than that of the drum wall, the positions at which they lie appears to be independent of $a$. Ellipsoids with maximum radius of curvature greater than that of the drum wall tilt with respect to the direction of drum motion and then undergo lateral translations from end to end of the drum.   

Force and torque on a rotating sphere close to and within a fluid-filled rotating sphere

The logarithmic term in each of the force and torque formulas can be readily found from lubrication theory (O'Neill \& Majumdar,ZAMP,1970). The challenge is to find exactly the O(1) terms, a feat achieved by O'Neill \& Stewartson (JFM,1967) for a fixed sphere close to a translating plane by means of deceptively intricate asymptotic manipulations. These are now strategically mimicked to determine the exact O(1) terms for the more complicated rotating sphere within a rotating sphere case, generating a pair of functions and second order linear D.E's to be solved numerically. The resulting force and torque coefficients facilitate the determination of the position and rotational speed of a sphere that is heavier than the fluid. Recent experiments by Mullin and coworkers have demonstrated, in slow viscous flow, the possible existence, when a sphere rotates close to a translating plane or rotating sphere, of the cavitation bubble suggested by Taylor (JFM 1963) and posited by Goldman, Cox \& Brenner (ChEngSci,1967) as an explanation of the observed disparity between theory and experiment. The latter's asymptotic estimates for small gaps were obtained by extrapolation from the exact series solutions for translation and rotation at arbitrary separation and are more commonly cited. Since the bubble occurs in the `inner field', when it exists, the lubrication theory suffices to determine its shape, the normal force due to the induced asymmetry and adjustments to the sphere's position and possibly reversed rotational speed.   

Viscous coordination in systems of rotating flagella

Bacterial flagella are helical filaments, 5-10 microns long and 20 nm in diameter that rotate at approximately 100 Hz and are responsible for cell motility and, in engineered systems, enhanced mixing and pumping. It has been previously hypothesized that adjacent flagella can coordinate, presumably due to viscous interactions between nearby filaments. To explore the physics of this phenomenon, we present results from a model experiment performed in a low-Re tank in which asymmetric paddles (representing the asymmetry of helical flagella) are rotated using servo motors. The first series of experiments explore the dynamics of a single ``flagella'' close to a planar wall, and the torque-speed behavior is characterized as a function of the flagella-wall separation and other geometric and dynamic parameters. The second series of experiments explores the dynamics of two flagella rotating in close proximity. Scaling arguments for the observed behavior are presented based on related theoretical results.