Volume 25, Issue 8, August 2013

We report a highresolution numerical study of twodimensional (2D) miscible RayleighTaylor (RT) incompressible turbulence with the Boussinesq approximation. An ensemble of 100 independent realizations were performed at small Atwood number and unit Prandtl number with a spatial resolution of 2048 × 8193 grid points. Our main focus is on the temporal evolution and the scaling behavior of global quantities and of smallscale turbulence properties. Our results show that the buoyancy force balances the inertial force at all scales below the integral length scale and thus validate the basic forcebalance assumption of the BolgianoObukhov scenario in 2D RT turbulence. It is further found that the Kolmogorov dissipation scale η(t) ∼ t 1/8, the kineticenergy dissipation rate ɛ u (t) ∼ t −1/2, and the thermal dissipation rate ɛθ(t) ∼ t −1. All of these scaling properties are in excellent agreement with the theoretical predictions of the Chertkov model [“Phenomenology of RayleighTaylor turbulence,” Phys. Rev. Lett.91, 115001 (Year: 2003)]10.1103/PhysRevLett.91.115001. We further discuss the emergence of intermittency and anomalous scaling for high order moments of velocity and temperature differences. The scaling exponents of the pthorder temperature structure functions are shown to saturate to for the highest orders, p ∼ 10. The value of and the order at which saturation occurs are compatible with those of turbulent RayleighBénard (RB) convection[A. Celani, T. Matsumoto, A. Mazzino, and M. Vergassola, “Scaling and universality in turbulent convection,” Phys. Rev. Lett.88, 054503 (Year: 2002)]10.1103/PhysRevLett.88.054503, supporting the scenario of universality of buoyancydriven turbulence with respect to the different boundary conditions characterizing the RT and RB systems.
 LETTERS


Retention and entrainment effects: Experiments and theory for porous spheres settling in sharply stratified fluids
View Description Hide DescriptionWe present an experimental study of single porous spheres settling in a near twolayer ambient density fluid. Data are compared with a firstprinciple model based on diffusive processes. The model correctly predicts accelerations of the sphere but does not capture the retention time at the density transition quantitatively. Entrainment of lighter fluid through a shell encapsulating the sphere is included in this model empirically. With this parametrization, which exhibits a power law dependence on Reynolds numbers, retention times are accurately captured. Extrapolating from our experimental data, model predictions are presented.

 ARTICLES

 Biofluid Mechanics

Flow structure on a rotating wing: Effect of steady incident flow
View Description Hide DescriptionThe flow structure along a rotating wing in steady incident flow is compared to the structure on a rotating wing in quiescent fluid, in order to clarify the effect of advance ratio J (ratio of freestream velocity to tip velocity of wing). Stereoscopic particle image velocimetry leads to patterns of vorticity, velocity, and Qcriterion (constant values of the second invariant of the velocity gradient tensor), as well as streamlines, which allow identification of critical points of the flow. The effective angle of attack is held constant over the range of J, and the wing rotates from rest to a large angle that corresponds to attainment of the asymptotic state of the flow structure. Prior to the onset of motion, the wing is at high angle of attack and the steady incident flow yields a fully stalled state along the wing. After the onset of rotation, the stalled region quickly gives rise to a stable leading edge vortex. Throughout the rotation maneuver, the development of the flow structure in the leading edge region is relatively insensitive to the value of J. In the trailingedge region, however, the structure of the shed vorticity layer is strongly dependent on the value of J. Further insight into the effects of J is provided by threedimensional patterns of spanwiseoriented vorticity, spanwise velocity, and Qcriterion.

Nematode locomotion in unconfined and confined fluids
View Description Hide DescriptionThe millimeterlong soildwelling nematode Caenorhabditis elegans propels itself by producing undulations that propagate along its body and turns by assuming highly curved shapes. According to our recent study [V. Padmanabhan et al. , PLoS ONE7, e40121 (Year: 2012)10.1371/journal.pone.0040121] all these postures can be accurately described by a piecewiseharmoniccurvature model. We combine this curvaturebased description with highly accurate hydrodynamic bead models to evaluate the normalized velocity and turning angles for a worm swimming in an unconfined fluid and in a parallelwall cell. We find that the worm moves twice as fast and navigates more effectively under a strong confinement, due to the large transversetolongitudinal resistancecoefficient ratio resulting from the wallmediated farfield hydrodynamic coupling between body segments. We also note that the optimal swimming gait is similar to the gait observed for nematodes swimming in highviscosity fluids. Our bead models allow us to determine the effects of confinement and finite thickness of the body of the nematode on its locomotion. These effects are not accounted for by the classical resistiveforce and slenderbody theories.

Physics of rheologically enhanced propulsion: Different strokes in generalized Stokes
View Description Hide DescriptionShearthinning is an important rheological property of many biological fluids, such as mucus, whereby the apparent viscosity of the fluid decreases with shear. Certain microscopic swimmers have been shown to progress more rapidly through shearthinning fluids, but is this behavior generic to all microscopic swimmers, and what are the physics through which shearthinning rheology affects a swimmer's propulsion? We examine swimmers employing prescribed stroke kinematics in twodimensional, inertialess Carreau fluid: shearthinning “generalized Stokes” flow. Swimmers are modeled, using the method of femlets, by a set of immersed, regularized forces. The equations governing the fluid dynamics are then discretized over a bodyfitted mesh and solved with the finite element method. We analyze the locomotion of three distinct classes of microswimmer: (1) conceptual swimmers comprising sliding spheres employing both one and twodimensional strokes, (2) slipvelocity envelope models of ciliates commonly referred to as “squirmers,” and (3) monoflagellate pushers, such as sperm. We find that morphologically identical swimmers with different strokes may swim either faster or slower in shearthinning fluids than in Newtonian fluids. We explain this kinematic sensitivity by considering differences in the viscosity of the fluid surrounding propulsive and payload elements of the swimmer, and using this insight suggest two reciprocal sliding sphere swimmers which violate Purcell's Scallop theorem in shearthinning fluids. We also show that an increased flow decay rate arising from shearthinning rheology is associated with a reduction in the swimming speed of slipvelocity squirmers. For spermlike swimmers, a gradient of thick to thin fluid along the flagellum alters the force it exerts upon the fluid, flattening trajectories and increasing instantaneous swimming speed.
 Micro and Nanofluid Mechanics

A volumeoffluid formulation for the study of coflowing fluids governed by the HeleShaw equations
View Description Hide DescriptionWe present a computational framework to address the flow of two immiscible viscous liquids which coflow into a shallow rectangular container at one side, and flow out into a holding container at the opposite side. Assumptions based on the shallow depth of the domain are used to reduce the governing equations to one of HeleShaw type. The distinctive feature of the numerical method is the accurate modeling of the capillary effects. A continuum approach coupled with a volumeoffluid formulation for computing the interface motion and for modeling the interfacial tension in HeleShaw flows is formulated and implemented. The interface is reconstructed with a heightfunction algorithm. The combination of these algorithms is a novel development for the investigation of HeleShaw flows. The order of accuracy and convergence properties of the method are discussed with benchmark simulations. A microfluidic flow of a ribbon of fluid which coflows with a second liquid is simulated. We show that for small capillary numbers of O(0.01), there is an abrupt change in interface curvature and focusing occurs close to the exit.

Exotic states of bouncing and walking droplets
View Description Hide DescriptionWe present the results of an integrated experimental and theoretical investigation of droplets bouncing on a vibrating fluid bath. A comprehensive series of experiments provides the most detailed characterisation to date of the system's dependence on fluid properties, droplet size, and vibrational forcing. A number of new bouncing and walking states are reported, including complex periodic and aperiodic motions. Particular attention is given to the first characterisation of the different gaits arising within the walking regime. In addition to complex periodic walkers and limping droplets, we highlight a previously unreported mixed state, in which the droplet switches periodically between two distinct walking modes. Our experiments are complemented by a theoretical study based on our previous developments [J. Molacek and J. W. M. Bush, J. Fluid Mech.727, 582–611 (Year: 2013);10.1017/jfm.2013.279J. Molacek and J. W. M. Bush, J. Fluid Mech.727, 612–647 (Year: 2013)]10.1017/jfm.2013.280, which provide a basis for rationalising all observed bouncing and walking states.

Scaling NavierStokes equation in nanotubes
View Description Hide DescriptionOn one hand, classical Monte Carlo and molecular dynamics simulations have been very useful in the study of liquids in nanotubes, enabling a wide variety of properties to be calculated in intuitive agreement with experiments. On the other hand, recent studies indicate that the theory of continuum breaks down only at the nanometer level; consequently flows through nanotubes still can be investigated with NavierStokes equations if we take suitable boundary conditions into account. The aim of this paper is to study the statics and dynamics of liquids in nanotubes by using methods of nonlinear continuum mechanics. We assume that the nanotube is filled with only a liquid phase; by using a second gradient theory the static profile of the liquid density in the tube is analytically obtained and compared with the profile issued from molecular dynamics simulation. Inside the tube there are two domains: a thin layer near the solid wall where the liquid density is nonuniform and a central core where the liquid density is uniform. In the dynamic case a closed form analytic solution seems to be no more possible, but by a scaling argument it is shown that, in the tube, two distinct domains connected at their frontiers still exist. The thin inhomogeneous layer near the solid wall can be interpreted in relation with the Navier length when the liquid slips on the boundary as it is expected by experiments and molecular dynamics calculations.

Electricfieldinduced response of a droplet embedded in a polyelectrolyte gel
View Description Hide DescriptionThe electricfield induced response of a droplet embedded in a quenched polyelectrolyte gel is calculated theoretically. The response comprises the droplet translation and the electricfield induced flow fields within the droplet. The gel is modeled as a soft, and electrically charged porous solid saturated with a salted Newtonian fluid. The droplet is considered an incompressible Newtonian fluid with no free charge. An analytical solution, using the perturbation methodology and linear superposition, is obtained for the leadingorder steady response to a DC electricfield. The fluid within the droplet is driven due to hydrodynamic coupling with the electroosmotic flow. The fluid velocity within the droplet is linearly proportional to the electroosmotic flow. Moreover, the microrheological response function of a droplet within a polyelectrolyte gel is also provided, highlighting the importance of boundary conditions at the dropletgel interface on microrheological measurements.

Bubble formation during the collision of a sessile drop with a meniscus
View Description Hide DescriptionThe impact of a sessile droplet with a moving meniscus, as encountered in processes such as dipcoating, generically leads to the entrapment of small air bubbles. Here we experimentally study this process of bubble formation by looking through the liquid using highspeed imaging. Our central finding is that the size of the entrapped bubble crucially depends on the location where coalescence between the drop and the moving meniscus is initiated: (i) at a finite height above the substrate, or (ii) exactly at the contact line. In the first case, we typically find bubble sizes of the order of a few μm, independent of the size and speed of the impacting drop. By contrast, the bubbles that are formed when coalescence starts at the contact line become increasingly large, as the size or the velocity of the impacting drop is increased. We show how these observations can be explained from a balance between the lubrication pressure in the air layer and the capillary pressure of the drop.

Numerical study of thermocoalescence of microdroplets in a microfluidic chamber
View Description Hide DescriptionThe present paper reports the numerical investigation of thermocoalescence of droplets in a microchannel network consisting of a droplet formation section connecting to a temperatureinduced merging chamber. The numerical model is formulated as an incompressible immiscible twophase flow problem with oil and water as the continuous and dispersed phase, respectively. The governing equations are solved using finite volume method on a staggered mesh. The interface is captured by a narrowband particle levelset method. The paper examines the droplet formation process and droplet size at 4 different ratios of oil and water flow rate. The motion of the droplets from the formation section into and through the heatinduced merging chamber is analyzed. The numerical method is able to provide a visual presentation of the droplet movement in a heated environment under the influence of thermocapillarity. The relationship between the critical merging temperature and the fluid flow rate is also analyzed and discussed.
 Interfacial Flows

Volume of mixing effect on fluid counterdiffusion
View Description Hide DescriptionThe countercurrent diffusiondriven mixing process of two miscible fluids is studied in the absence of gravity, assuming that the mixture is nonregular, that is its volume is smaller than the sum of the initial volumes of the two components. Two competing effects are present in the mixing region: on one hand, the mass flow rate of each species increases, due to the larger density of the fluid; on the other hand, though, the volumetric flux is retarded by the inward convection due to volume disappearance, which opposes the outward velocity field due to diffusion. This intuition is confirmed by the analytical result of a 1D nonideal mixing process, showing that, in the presence of the convection induced by a volume decrease: (a) the process is selfsimilar; (b) the mass flux of each species at the interface increases by approximately 0.8ε, where ε is the maximum relative volume decrease; and (c) the volume flux of each species decreases by approximately a 0.2ε amount. This result is further confirmed by a perturbation analysis for small ε.

Note on the hydrodynamic description of thin nematic films: Strong anchoring model
View Description Hide DescriptionWe discuss the longwave hydrodynamic model for a thin film of nematic liquid crystal in the limit of strong anchoring at the free surface and at the substrate. We rigorously clarify how the elastic energy enters the evolution equation for the film thickness in order to provide a solid basis for further investigation: several conflicting models exist in the literature that predict qualitatively different behaviour. We consolidate the various approaches and show that the longwave model derived through an asymptotic expansion of the full nematohydrodynamic equations with consistent boundary conditions agrees with the model one obtains by employing a thermodynamically motivated gradient dynamics formulation based on an underlying free energy functional. As a result, we find that in the case of strong anchoring the elastic distortion energy is always stabilising. To support the discussion in the main part of the paper, an appendix gives the full derivation of the evolution equation for the film thickness via asymptotic expansion.

Distinguishing features of shallow angle plunging jets
View Description Hide DescriptionNumerical simulations employing an algebraic volumeoffluid methodology are used to study the air entrainment characteristics of a water jet plunging into a quiescent water pool at angles ranging from θ = 10° to θ = 90° measured from the horizontal. Our previous study of shallow angled jets[S. S. Deshpande, M. F. Trujillo, X. Wu, and G. L. Chahine, “Computational and experimental characterization of a liquid jet plunging into a quiescent pool at shallow inclination,” Int. J. Heat Fluid Flow34, 1–14 (Year: 2012)]10.1016/j.ijheatfluidflow.2012.01.011 revealed the existence of a clearly discernible frequency of ingestion of large air cavities. This is in contrast with chaotic entrainment of small air pockets reported in the literature in case of steeper or vertically plunging jets. In the present work, the differences are addressed by first quantifying the cavity size and entrained air volumes for different impingement angles. The results support the expected trend – reduction in cavity size (D 43) as θ is increased. Time histories of cavity volumes in the vicinity of the impingement region confirm the visual observations pertaining to a nearperiodic ingestion of large air volumes for shallow jets (10°, 12°), and also show that such cavities are not formed for steep or vertical jets. Each large cavity (defined as D c /D j ≳ 3) exists in close association with a stagnation point flow. A local mass and momentum balance shows that the high stagnation pressure causes a radial redirection of the jet, resulting in a flow that resembles the initial impact of a jet on the pool. In fact, for these large cavities, their speed matches closely U impact /2, which coincides with initial cavity propagation for sufficiently high Froude numbers. Furthermore, it is shown that the approximate periodicity of air entrainment scales linearly with Froude number. This finding is confirmed by a number of simulations at θ = 12°. Qualitatively, for steeper jets, such large stagnation pressure region does not exist, and the deflection of the entire incoming jet is nonexistent. In fact, for θ = 25°, 45°, 90°, the jet penetrates the pool nearly undisturbed and consequently large cavities are not formed.

Nonaxisymmetric annular curtain stability
View Description Hide DescriptionA stability analysis of nonaxisymmetric annular curtain is carried out for an axially moving viscous jet subject in surrounding viscous gas media. The effect of inertia, surface tension, gastoliquid density ratio, innertoouter radius ratio, and gastoliquid viscosity ratio on the stability of the jet is studied. In general, the axisymmetric disturbance is found to be the dominant mode. However, for small wavenumber, the nonaxisymmetric mode is the most unstable mode and the one likely observed in reality. Inertia and the viscosity ratio for nonaxisymmetric disturbances show a similar stability influence as observed for axisymmetric disturbances. The maximum growth rate in nonaxisymmetric flow, interestingly, appears at very small wavenumber for all inertia levels. The dominant wavenumber increases (decreases) with inertia for nonaxisymmetric (axisymmetric) flow. Gastoliquid density ratio, curvature effect, and surface tension, however, exhibit an opposite influence on growth rate compared to axisymmetric disturbances. Surface tension tends to stabilize the flow with reductions of the unstable wavenumber range and the maximum growth rate as well as the dominant wavenumber. The dominant wavenumber remains independent of viscosity ratio indicating the viscosity ratio increases the breakup length of the sheet with very little influence on the size of the drops. The range of unstable wavenumbers is affected only by curvature in axisymmetric flow, whereas all the stability parameters control the range of unstable wavenumbers in nonaxisymmetric flow. Inertia and gas density increase the unstable wavenumber range, whereas the radius ratio, surface tension, and the viscosity ratio decrease the unstable wavenumber range. Neutral curves are plotted to separate the stable and unstable domains. Critical radius ratio decreases linearly and nonlinearly with the wavenumber for axisymmetric and nonaxisymmetric disturbances, respectively. At smaller Weber numbers, a wider unstable domain is predicted for nonaxisymmetric modes. For both axisymmetric and nonaxisymmetric modes, the disturbance frequency is found to be the same and equal to the negative of axial wavenumber. Finally, comparison between theory and existing experiment leads to good qualitative agreement. A more accurate comparison is not possible given the difference in flow conditions.

Dynamics of a Janus drop in an external flow
View Description Hide DescriptionThe steady motion of a Janus drop under a uniform external flow is considered. First, we analyze the equilibrium shape of a Januslike drop in a motionless ambient fluid, i.e., the special case of a nearly spherical compound drop with a nearly flat internal interface. This configuration is realizable when the liquids comprising the drop have close interfacial tensions with the ambient fluid, but a small interfacial tension between each other. Then, we consider the flow past a perfect Janus drop composed of two hemispherical domains each occupied by a different fluid. For the sake of simplicity, all the interfaces are assumed nondeformable. The problem is solved both analytically, by means of the Lamb expansion, and numerically. The relation between the flow velocity and the force imposed on the drop, which is a generalization of the classical Hadamard–Rybczynski formula, is found. A torque is also imposed on the drop in the general case. The stable regime of motion of a torquefree drop is found to be axisymmetric, with the less viscous fluid at the upstream face. For this particular configuration, the deformation of the internal interface is also found employing a perturbation technique, whereas the distortion of the drop surface can be safely neglected.

Dynamics and fracture of ligaments from a droplet on a vibrating surface
View Description Hide DescriptionA droplet residing on a vibrating surface and in the pressure antinode of an asymmetric standing wave can spread radially outward and atomize. In this work, proper orthogonal decomposition through high speed imaging is shown to predict the likelihood of atomization for various viscous fluids based on prior information in the droplet spreading phase. Capillary instabilities are seen to affect ligament rupture. Viscous dissipation plays an important role in determining the wavelength of the most unstable mode during the inception phase of the ligaments. However, the highest ligament capillary number achieved was less than 1, and the influence of viscosity in the ligament growth and breakup phases is quite minimal. It is inferred from the data that the growth of a typical ligament is governed by a balance between the inertial force obtained from the inception phase and capillary forces. By including the effect of acoustic pressure field around the droplet, the dynamics of the ligament growth phase is revealed and the ligament growth profiles for different fluids are shown to collapse on a straight line using a new characteristic time scale.

Analytical study in the mechanism of flame movement in horizontal tubes. II. Flame acceleration in smooth open tubes
View Description Hide DescriptionThe problem of spontaneous acceleration of premixed flames propagating in open horizontal tubes with smooth walls is revisited. It is proved that in long tubes, this process can be considered quasisteady, and an equation for the flame front position is derived using the onshell description. Numerical solutions of this equation are found which show that as in the case of uniform flame movement, there are two essentially different regimes of flame propagation. In the type I regime, the flame speed and its acceleration are comparatively low, whereas the type II regime is characterized by significant flame acceleration that rapidly increases as the flame travels along the tube. A detailed comparison of the obtained results with the experimental data on flame acceleration in methaneair mixtures is given. In particular, it is confirmed that flames propagating in nearstoichiometric mixtures and mixtures near the limits of inflammability belong to the types II and I, respectively, whereas flames in transient mixtures undergo transitions between the two regimes during their travel.

Numerical investigation for the effect of the liquid film volume on thermocapillary flow direction in a thin circular liquid film
View Description Hide DescriptionNASA Astronaut Dr. Pettit carried out a thermocapillary flow experiment onboard the International Space Station in 2003. In this experiment a thin water film containing milk powder was formed in a stainlesssteel wire ring. Heating a section of the ring by a soldering iron induced in the water film a thermocapillary flow towards the heated section of the ring (outward flow : cold to hot). This flow was in the opposite direction of the usually observed thermocapillary flows (inward flow : hot to cold). To shed light on this interesting phenomenon observed in the space experiment, we have conducted a threedimensional numerical simulation study. Simulation results showed that the film geometry of the water film is a key factor determining flow direction and flow strength. When the liquid film free surfaces are convex, i.e., the water film volume is larger than that when the free surfaces are flat, an outward flow develops in the film as observed in the space experiment. However, when the free surfaces are concave, the simulation predicts an inward flow .

Bubble dynamics in N dimensions
View Description Hide DescriptionCavitation and bubble dynamics are central concepts in engineering, the natural sciences, and the mathematics of fluid mechanics. Due to the nonlinear nature of their dynamics, the governing equations are not fully solvable. Here, the dynamics of a spherical bubble in an Ndimensional fluid are discussed in the hope that examining bubble behavior in N dimensions will add insight to their behavior in three dimensions. Several canonical results in bubble dynamics are rederived, including the Rayleigh collapse time, the RayleighPlesset equation, and the Minnaert frequency. Recent analytical approximations to the Rayleigh collapse are discussed, and the Ndimensional generalization is used to resolve a known discrepancy. Numerical simulations are used to examine the onset of nonlinear behavior. Overall, the dynamics of bubbles are faster at higher dimensions, with nonlinear behavior occurring at lower strain. Several features are found to be unique to three dimensions, including the trend of nonlinear behavior and apparent coincidences in timescales.

Suppression and reversal of drop formation on horizontal cylinders due to surfactant convection
View Description Hide DescriptionWhen a thin liquid film is applied to the surface of a horizontal cylinder, gravity will cause a drainage of liquid from the top and sides of the cylinder towards the cylinder bottom. If surfactant is present on the surface of the film, this will cause a convection of surfactant resulting in a higher concentration of surfactant on the cylinder bottom compared to the top and sides of the cylinder. The result is a surface tension gradient, which is equivalent to a surface shear stress, and acts to oppose the drainage of the coating layer due to gravity. For sufficiently small cylinders, this cannot only slow the drainage but reverse the flow, causing a net flux of liquid upward from the bottom of the cylinder towards the top of the cylinder. If this flux is sufficiently strong, a “collar” of liquid forms around the cylinder. In this paper, we develop a mathematical model, based on the lubrication approximations, of the gravitational, surface tension, and surface tension gradient forces, and their effects on the evolution of a thin liquid film coating a horizontal circular cylinder. Using finite differences and an alternating direction implicit technique, numerical simulations show that even for comparatively weak surfactants, surface tension gradient effects greatly affect the flow history and must be included to accurately model the evolution of the film. They cannot only slow the drainage of liquid towards a pendant drop on the bottom of the cylinder, but reverse the flux, resulting in a thicker coating on the top of the cylinder compared to the surfactantfree case. Results from the simulation are presented over a wide range of the dimensionless parameters which characterize the problem.