Spheroidal approximation for finite-amplitude highly viscous axisymmetric drop/bubble free shape relaxation.

A common simplification used in different physical contexts by both experimentalists and theoreticians when dealing with essentially non-spherical drops is treating them as ellipsoids or, in the axisymmetric case, spheroids. In the present theoretical study, we are concerned with such a spheroidal approximation for free viscous shape relaxation of strongly deformed axisymmetric drops towards a sphere. A general case of a drop in an immiscible fluid medium is considered, which includes the particular cases of high and low inside-to-outside viscosity ratios (e.g., liquid drops in air and bubbles in liquid, respectively). The analysis involves solving for the accompanying Stokes (creeping) flow inside and outside a spheroid of an evolving aspect ratio. Here this is accomplished by an analytical solution in the form of infinite series whose coefficients are evaluated numerically. The study aims at the aspect ratios up to about 3 at most in both the oblate and prolate domains. The inconsistency of the spheroidal approximation and the associated non-spheroidal tendencies are quantified from within the approach. The spheroidal approach turns out to work remarkably well for the relaxation of drops of relatively very low viscosity (e.g., bubbles). It is somewhat less accurate for drops in air. A semi-heuristic result encountered in the literature, according to which the difference of the squares of the two axes keeps following the near-spherical linear evolution law even for appreciable deformations, is put into context and verified against the present results.

Acoustofluidics 16: acoustics streaming near liquid-gas interfaces: drops and bubbles.

In this sixteenth part of the series on "Acoustofluidics-exploiting ultrasonic standing waves forces and acoustic streaming in microfluidic systems for cell and particle manipulation," we continue our discussion on the analytical aspects of the streaming phenomenon. In particular, the use of the singular perturbation technique for this class of problems is delineated with a set of examples where fluid-fluid interaction takes place. In this category, we focus on drops and bubbles, and deal specifically with the effect of interfacial mobility on the streaming flow.

Acoustofluidics 15: streaming with sound waves interacting with solid particles.

In Part 15 of the tutorial series "Acoustofluidics-exploiting ultrasonic standing waves forces and acoustic streaming in microfluidic systems for cell and particle manipulation," we examine the interaction of acoustic fields with solid particles. The main focus here is the interaction of standing waves with spherical particles leading to streaming, together with some discussion on one non-spherical case. We begin with the classical problem of a particle at the velocity antinode of a standing wave, and then treat the problem of a sphere at the velocity node, followed by the intermediate situation of a particle between nodes. Finally, we discuss the effect of deviation from sphericity which brings about interesting fluid mechanics. The entire Focus article is devoted to the analysis of the nonlinear fluid mechanics by singular perturbation methods, and the study of the streaming phenomenon that ensues from the nonlinear interaction. With the intention of being instructive material, this tutorial cannot by any means be considered 'complete and comprehensive' owing to the complexity of the class of problems being covered herein.

Acoustofluidics 13: Analysis of acoustic streaming by perturbation methods.

In this Part 13 of the tutorial series "Acoustofluidics--exploiting ultrasonic standing waves forces and acoustic streaming in microfluidic systems for cell and particle manipulation," the streaming phenomenon is presented from an analytical standpoint, and perturbation methods are developed for analyzing such flows. Acoustic streaming is the phenomenon that takes place when a steady flow field is generated by the absorption of an oscillatory field. This can happen either by attenuation (quartz wind) or by interaction with a boundary. The latter type of streaming can also be generated by an oscillating solid in an otherwise still fluid medium or vibrating enclosure of a fluid body. While we address the first kind of streaming, our focus is largely on the second kind from a practical standpoint for application to microfluidic systems. In this Focus article, we limit the analysis to one- and two-dimensional problems in order to understand the analytical techniques with examples that most-easily illustrate the streaming phenomenon.

Fluid dynamical analysis of a particle with large vapor transport in poiseuille flow.

The fluid dynamics of a particle with vapor transport in the presence of gaseous Poiseuille flow is examined in detail for low translational Reynolds number. Poiseuille flow is used to represent an undisturbed steady flow in a cylindrical tube. The particle has a dominant radial field of condensation, evaporation, sublimation, or other form of gasification, with a corresponding radial Reynolds number of order unity. An analysis is carried out by using the perturbation method in which the purely radial flow is used as the leading order, and Poiseuille flow together with particle translation is a perturbation of higher order. Whereas the leading order motion may be described by potential flow, the higher order involves nonlinear interaction of viscous and inertial forces. With the perturbation process bringing about linearization of this interaction, an Oseen-like solution is obtained. However, with the dominant radial flow being strongly diminishing in the far field, a regular perturbation (instead of singular) is sufficient for the perturbed flow description. Presently, the axisymmetric case of a particle along the centerline of the cylinder is considered. The asymmetric case of the off-center particle is also under examination. The results show that the drag components decrease as the radial Reynolds number increases. The influence of the paraboloidal component of Poiseuille flow is a slight increase in the pressure drag as the ratio of the particle size and the tube radius (a/R(0)) increases for moderate values of the radial Reynolds number (Re(R) < 5). For higher values of Re(R), the pressure drag decreases with increasing (a/R(0)). The viscous drag, on the other hand, consistently decreases as the ratio (a/R(0)) increases.

Shape relaxation of liquid drops in a microgravity environment.

We investigated shape relaxation of liquid drops in a microgravity environment that was created by letting the drops fall freely. The drops were initially levitated in air by an acoustic/electrostatic hybrid levitator. The levitated drops were deformed due to the force balance among the levitating force, surface tension, and gravity. During the free fall, the deformed drops underwent shape relaxation driven by the surface tension to restore a spherical shape. The progress of the shape relaxation was characterized by measuring the aspect ratio as a function of time, and was compared to a simple linear relaxation model (in which only the fundamental mode was considered) for perfectly conductive drops. The results show that the model quite adequately describes the shape relaxation of uncharged/charged drops released from an acoustically levitated state. However, the model is less successful in describing the relaxation of drops that were levitated electrostatically before the free fall. This may be due to finite electrical conductivities of liquids, which somehow affects the initial stage of the shape relaxation process.

Thermal diffusivity coefficient of glycerin determined on an acoustically levitated drop.

We present a technique that can be used to determine the thermal diffusivity coefficient of undercooled liquids that exist at temperatures below their freezing points. The technique involves levitation of a small amount of liquid in the shape of a flattened drop using an acoustic levitator and heating it with a CO2 laser. The heated drop is then allowed to cool naturally by heat loss from the surface. Due to acoustic streaming, heat loss is highly non-uniform and appears to mainly occur at the drop circumference (equatorial region). This fact allows us to relate the heat loss rate with a heat transfer model to determine the thermal diffusion coefficient. We demonstrate the feasibility of the technique using glycerin drops as a model liquid.

Internal circulation in a drop in an acoustic field.

An investigation of the internal flow field for a drop at the antinode of a standing wave has been carried out. The main difference from the solid sphere case is the inclusion of the shear stress and velocity continuity conditions at the liquid-gas interface. To the leading order of calculation, the internal flow field was found to be quite weak. Also, this order being fully time dependent has a zero mean flow. At the next higher order, steady internal flows are predicted and, as in the case of a solid sphere, there is a recirculating layer consisting of closed streamlines near the surface. In the case of a liquid drop, however, the behavior of this recirculating Stokes layer is quite interesting. It is predicted that the layer ceases to have recirculation when [formula: see text], where [symbol: see text] is the liquid viscosity, mu is the exterior gas-phase viscosity, and M is the dimensionless frequency parameter for the gas phase, defined by M = i omega a2 rho/mu, with a being the drop radius. Thorough experimental confirmation of this interesting new development needs to be conducted. Although it seems to agree with many experiments with levitated drops where no recirculating layer has been clearly observed, a new set of experiments for specifically testing this interesting development need to be carried out.