Volume 25, Issue 10, October 2013
Index of content:

The topological and dynamical features of small scales are studied in the context of decaying magnetohydrodynamic turbulent flows using direct numerical simulations. Joint probability density functions (PDFs) of the invariants of gradient quantities related to the velocity and the magnetic fields demonstrate that structures and dynamics at the time of maximum dissipation depend on the large scale initial conditions at the examined Reynolds numbers. This is evident in particular from the fact that each flow has a different shape for the joint PDF of the invariants of the velocity gradient in contrast to the universal teardrop shape of hydrodynamic turbulence. The general picture that emerges from the analysis of the invariants is that regions of high vorticity are correlated with regions of high strain rate S also in contrast to hydrodynamic turbulent flows. Magnetic strain dominated regions are also well correlated with region of high current density j . Viscous dissipation ( ) as well as Ohmic dissipation ( ) resides in regions where strain and rotation are locally almost in balance. The structures related to the velocity gradient possess different characteristics than those associated with the magnetic field gradient with the latter being locally more quasitwo dimensional.
 ARTICLES


Biofluid Mechanics

Swimming near deformable membranes at low Reynolds number
View Description Hide DescriptionMicroorganisms are rarely found in nature swimming freely in an unbounded fluid. Instead, they typically encounter other organisms, hard walls, or deformable boundaries, such as free interfaces or membranes. Hydrodynamic interactions between the swimmer and nearby objects lead to many interesting phenomena, such as changes in swimming speed, tendencies to accumulate or turn, and coordinated flagellar beating. Inspired by this class of problems, we investigate locomotion of microorganisms near deformable boundaries. We calculate the speed of an infinitely long swimmer close to a flexible surface separating two fluids; we also calculate the deformation and swimming speed of the flexible surface. When the viscosities on either side of the flexible interface differ, we find that fluid is pumped along or against the swimming direction, depending on which viscosity is greater.

Micro and Nanofluid Mechanics

Mixing by chaotic advection in a magnetohydrodynamic driven flow
View Description Hide DescriptionA new device containing three circular electrodes and where very small quantities of a weakly electrically conductive liquid are propelled and mixed by chaotic advection is designed and constructed. The liquid, a copper sulfate solution, is propelled by the Lorentz body force, i.e., a magnetic field perpendicular to an electrical current. When the potentials of the electrodes are constant and the Lorentz force is small enough so that at the free surface the vertical velocity is practically zero, the flow field exhibits there a saddle point when the three circular electrodes are not in a concentric position. By modulating the electrical potential between the electrodes, the position of the saddle point changes. This slowly varying system is far from integrable and exhibits largescale chaos, the nonintegrability is due to the slow continuous modulation of the position of the saddle stagnation point and the two streamlines stagnating on it. Dye advection experiments are compared successfully to a numerical solution of the 3D equations of motion under these assumptions. We have also defined a potential mixing zone to predict the location of the chaotic region and calculated Poincaré sections. These two tools give results which are in excellent agreement, they are used, with others, to adjust the mixing protocol parameters and the geometry in order to improve mixing.

Interfacial Flows

Experimental study on the evolution of traveling waves over an undulated incline
View Description Hide DescriptionWe present experimental results on the evolution of traveling waves over a strongly undulated incline. In order to investigate the difference between waves in the linearly stable and unstable region, we set the Reynolds number near the neutral curve. That way, we were able to cross the neutral curve by increasing the frequency of excitation, without changing the velocity field of the basic flow. The amplitude of excitation was also varied, to analyze the evolution of both linear and nonlinear waves. We report on a rich variety of phenomena, including: (a) energy transfer from the excitation frequency to its higher harmonics, (b) the growth rate of the traveling waves, (c) the stability of traveling waves depending on its amplitude, and (d) the amplitude of saturation depending on the excitation frequency. We compare our results to those so far available in the literature. To our knowledge, this is the first experimental work on the development of traveling waves over strongly undulated substrate geometries.

Stokes flow in a drop evaporating from a liquid subphase
View Description Hide DescriptionThe evaporation of a drop from a liquid subphase is investigated. The two liquids are immiscible, and the contact angles between them are given by the Neumann construction. The evaporation of the drop gives rise to flows in both liquids, which are coupled by the continuity of velocity and shearstress conditions. We derive selfsimilar solutions to the velocity fields in both liquids close to the threephase contact line, where the drop geometry can be approximated by a wedge. We focus on the case where Marangoni stresses are negligible, for which the flow field consists of three contributions: flow driven by the evaporative flux from the drop surface, flow induced by the receding motion of the contact line, and an eigenmode flow that is due to the stirring of the fluid in the corner by the largescale flow in the drop. The eigenmode flow is asymptotically subdominant for all contact angles. The moving contactline flow dominates when the angle between the liquid drop and the horizontal surface of the liquid subphase is smaller than 90°, while the evaporativeflux driven flow dominates for larger angles. A parametric study is performed to show how the velocity fields in the two liquids depend on the contact angles between the liquids and their viscosity ratio.

On the mechanism of wetting failure during fluid displacement along a moving substrate
View Description Hide DescriptionThis work investigates the onset of wetting failure for displacement of Newtonian fluids in parallel channels. A hydrodynamic model is developed for planar geometries where an advancing fluid displaces a receding fluid along a moving substrate. The model is evaluated with three distinct approaches: (i) the lowspeed asymptotic theory of Cox [J. Fluid Mech.168, 169–194 (1986)], (ii) a onedimensional (1D) lubrication approach, and (iii) a twodimensional (2D) flow model solved with the Galerkin finite element method (FEM). Approaches (ii) and (iii) predict the onset of wetting failure at a critical capillary number Ca ^{ crit }, which coincides with a turning point in the steadystate solution family for a given set of system parameters. The 1D model fails to accurately describe interface shapes near the threephase contact line when air is the receding fluid, producing large errors in estimates of Ca ^{ crit } for these systems. Analysis of the 2D flow solution reveals that strong pressure gradients are needed to pump the receding fluid away from the contact line. A mechanism is proposed in which wetting failure results when capillary forces can no longer support the pressure gradients necessary to steadily displace the receding fluid. The effects of viscosity ratio, substrate wettability, and fluid inertia are then investigated through comparisons of Ca ^{ crit } values and characteristics of the interface shape. Surprisingly, the lowspeed asymptotic theory (i) matches trends computed from (iii) throughout the entire investigated parameter space. Furthermore, predictions of Ca ^{ crit } from the 2D flow model compare favorably to values measured in experimental airentrainment studies, supporting the proposed wettingfailure mechanism.

Numerical simulations of bubble formation from submerged needles under nonuniform direct current electric field
View Description Hide DescriptionIn several chemical and space industries, small bubbles are desired for efficient interaction between the liquid and gas phases. In the present study, we show that nonuniform electric field with appropriate electrode configurations can reduce the volume of the bubbles forming at submerged needles by up to three orders of magnitude. We show that localized high electric stresses at the base of the bubbles result in slipping of the contact line on the inner surface of the needle and subsequent bubble formation occurs with contact line inside the needle. We also show that for bubble formation in the presence of highly nonuniform electric field, due to high detachment frequency, the bubbles go through multiple coalescences and thus increase the apparent volume of the detached bubbles.

On selfsimilar thermal rupture of thin liquid sheets
View Description Hide DescriptionWe consider the dynamics of a symmetrically heated thin incompressible viscous fluid sheet. We take surface tension to be temperature dependent and consequently the streamwise momentum equation includes the effects of thermocapillarity, inertia, viscous stresses, and capillarity. Energy transport to the surrounding environment is also included. We use a longwave analysis to derive a single nondimensional system which, with appropriate choices of Reynolds number, recovers two previously studied cases. In both cases, we find conditions under which sufficiently largeamplitude initial temperature profiles induce film rupture in finite time, notably without the inclusion of disjoining pressures from van der Waals effects. When the Reynolds number is large, the similarity solution is governed by a balance of inertia and capillarity near the rupture location, analogous to the isothermal case. When the Reynolds number is small, the thermocapillary transients induce the same similarity solution over intermediate times that is found for the drainage of lamellae in foams. For O(1) Reynolds numbers, the dynamics are governed initially by the large Reynolds number evolution, and then a transition over several orders of magnitude in the sheet thickness needs to take place before the small Reynolds number similarity solution is observed.

Effect of ambient air on liquid jet impingement on a moving substrate
View Description Hide DescriptionAn experimental investigation into the effect of surrounding air pressure on liquid jet impingement on a moving substrate was performed. The study was carried out with Newtonian liquids impacting dry substrates. A variety of jet speeds, substrate speeds, and liquid viscosities were studied. It was observed that, as is the case for Newtonian droplet impact, the surrounding air pressure plays a crucial role in the splashing behaviour of jet impingement. There exists a threshold pressure below which splash does not occur. It is proposed that for certain impingement conditions lamella detachment from the substrate occurs due to aerodynamic forces acting on the leading edge of the lamella, which destabilizes the balance between surface tension and fluid pressure forces.

Stability of a moving radial liquid sheet: Timedependent equations
View Description Hide DescriptionWe study the stability of a radial liquid sheet produced by headon impingement of two equal laminar liquid jets. Linear stability equations are derived from the inviscid flow equations for a radially expanding sheet that govern the timedependent evolution of the two liquid interfaces. The analysis accounts for the varying liquid sheet thickness while the inertial effects due to the surrounding gas phase are ignored. The analysis results in stability equations for the sinuous and the varicose modes of sheet deformation that are decoupled at the lowest order of approximation. When the sheet is excited at a fixed frequency, a small sinuous displacement introduced at the point of impingement grows as it is convected downstream suggesting that the sheet is unstable at all Weber numbers (We ≡ ρ l U ^{2} h/σ) in the absence of the gas phase. Here, ρ l is the density of the liquid, U is the speed of the liquid jet, h is the local sheet thickness, and σ is the surface tension. The sinuous disturbance diverges at We = 2 which sets the size of the sheet, in agreement with the results of Taylor [“The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets,” Proc. R. Soc. London, Ser. A253, 313 (1959)]. Asymptotic analysis of the sinuous mode for all frequencies shows that the disturbance amplitude diverges inversely with the distance from the edge of the sheet. The varicose waves, on the other hand, are neutrally stable at all frequencies and are convected at the speed of the liquid jet.

Inertial effects at moderate Reynolds number in thinfilm rimming flows driven by surface shear
View Description Hide DescriptionIn this paper, we study twodimensional thinfilm flow inside a stationary circular cylinder driven by an imposed surface shear stress. Modelling is motivated by a need to understand the cooling and film dynamics provided by oil films in an aeroengine bearing chamber characterised by conditions of very high surface shear and additional film mass flux from oil droplets entering the film through the surface. In typical highspeed operation, film inertial effects can provide a significant leadingorder mechanism neglected in existing lubrication theory models. Inertia at leadingorder is included within a depthaveraged formulation where wall friction is evaluated similar to hydraulic models. This allows key nonlinear inertial effects to be included while retaining the ability to analyse the problem in a mathematically tractable formulation and compare with other approaches. In constructing this model, a set of simplified mass and momentum equations are integrated through the depth of the film yielding a spatially onedimensional depthaveraged formulation of the problem. An a priori assumed form of velocity profile is needed to complete the system. In a local Stokes flow analysis, a quadratic profile is the exact solution for the velocity field though it must be modified when inertial effects become important. Extension of the velocity profile to a cubic profile is selected enabling specification of a wall friction model to include the roughness of the cylinder wall. A modelling advantage of including the inertia term, relevant to the applications considered, is that a smooth progression in solution can be obtained between cases of low Reynolds number corresponding to lubrication theory, and high Reynolds number corresponding to uniform rimmingflow. Importantly, we also investigate the effect of inertia on some typical solutions from other studies and present a greater insight to existing and new film solutions which arise from including inertia effects.

Physical and numerical investigation of cavitating flows around a pitching hydrofoil
View Description Hide DescriptionThe objective of this paper is to investigate cavitating flows around a pitching hydrofoil via combined physical and numerical studies. The aims are to (1) improve the understanding of the interplay between unsteady cavitating flow, hydrofoil motion, and hydrodynamic performance, (2) quantify the influence of pitching rate on subcavitating and cavitating responses, and (3) quantify the influence of cavitation on the hydrodynamic load coefficients and surrounding flow structures. Results are presented for a NACA66 hydrofoil undergoing controlled, slow and fast pitching motions from α = 0° to α = 15° and back to α = 0° for both subcavitating and cavitating conditions at a moderate Reynolds number of Re = 750 000. The experimental studies were conducted in a cavitation tunnel at the French Naval Academy, France. The numerical simulations are performed by solving the incompressible, multiphase Unsteady ReynoldsAveraged NavierStokes Equations via the commercial code CFX using a transport equationbased cavitation model; a modified kω SST turbulence model is used to account for the effect of local compressibility on the turbulent eddy viscosity. The results showed that increases in the pitching rate suppressed laminar to turbulent transition, delayed stall, and significantly modified poststall behavior. Cavitation inception at the leading edge modified the pressure distribution, which in turn significantly changed the interaction between leading edge and trailing edge vortices, and hence the magnitude as well as the frequency of the load fluctuations. For a fixed cavitation number, increases in pitching rate lead to increase in cavitation volume, which in turn changed the cavity shedding frequencies and significantly modified the hydrodynamic loads. Inversely, the leading edge cavitation observed for the low pitching velocity case tends to stabilize the stall because of the decrease of the pressure gradient due to the formation of the cavity. The results showed strong correlation between the cavity and vorticity structures, which suggest that the inception, growth, collapse and shedding of sheet/cloud cavities are important mechanisms for vorticity production and modification.

Transition between regimes of a vertical channel bubbly upflow due to bubble deformability
View Description Hide DescriptionThe effect of the deformability of viscous bubbles on the flow rate of bubbly upflow in a vertical channel is examined using direct numerical simulations. A sharp transition between two different flow regimes has been observed. At large bubble deformability, characterized by large Eötvös number (Eo), the flow rate is close to the single phase flow rate, with adjusted pressure gradient, and the bubbles are almost uniformly distributed in the middle of the channel. On the other hand, at low Eo the bubbles are concentrated near channel walls and flow rates are much smaller than the single phase flow. The transition from high flow rate to low flow rate occurs rather abruptly. It is found that the transition occurs when the less deformable bubbles enter the viscous sublayer due to the lateral lift force on the bubbles. This leads to an increase in the viscous dissipation near the wall which leads to a decrease in the flow rate.

Viscous and NonNewtonian Flows

Dynamics of an electrically charged polymer jet
View Description Hide DescriptionElectrohydrodynamic equations describing the behavior of a charged polymer jet are analyzed by analytical methods and scaling approach. A FENEP constitutive equation is employed to describe the viscoelastic properties of a conducting polymer liquid. Effects of the electric field, the flow rate, and the material parameters on the jet dynamics are investigated. Four different regimes are examined. In particular, a regime in which the electric current is linearly proportional to the electric field and independent on the flow rate and a regime in which the electric current is linearly proportional to the flow rate and independent on the electric field are identified. An operating window limiting the region of a stable conejet mode is also considered.

Electroosmotic flow of a viscoplastic material through a slit channel with walls of arbitrary zeta potential
View Description Hide DescriptionElectroosmotic (EO) flow is known to have a nearly uniform velocity profile, but such a plugflow velocity can be considerably diminished if the fluid is a viscoplastic material having a yield stress. This paper aims to investigate the reduction of EO velocity (also known as Smoluchowski slip velocity) due to a yield stress as a function of the material rheological parameters and the zeta potential. Three rheological models are considered: Casson, Herschel–Bulkley, and Bingham fluids. In the absence of pressure forcing and without the Debye–Hückel approximation, the problems of EO flow of these materials in a slit channel with walls uniformly charged with an arbitrary zeta potential are analytically solved. Analytical expressions are deduced for the reduced Smoluchowski velocity under the limiting conditions of very small and very large zeta potentials. It is shown that qualitatively different asymptotic behaviors will be exhibited by materials of different models.

Particulate, Multiphase, and Granular Flows

Stochastic vortex structure method for modeling particle clustering and collisions in homogeneous turbulence
View Description Hide DescriptionCurrent particle dispersion models do not accurately predict the particle clustering that occurs in turbulent flow due to interaction of the particles with turbulent eddies. This clustering arises due to the effects of centrifugal forces which act to throw heavy particles out of the turbulent eddies, causing the particles to collect in highconcentration sheets lying between the eddies. The current paper proposes a stochastic vortex structure (SVS) model for simulation of particle clustering and collisions in turbulent flows. A new measure for particle drift relative to the fluid velocity is proposed that is related to the cross product of the fluid acceleration and velocity fields. Tests were conducted comparing the predictions of the SVS model with direct numerical simulation (DNS) and with three different stochastic Lagrangian methods in statistically stationary homogeneous isotropic turbulence with particles having Stokes numbers based on integral length scaling of order unity, assuming oneway fluidparticle coupling. The tests examined different turbulent flow features that are important for particle dispersion and clustering, as well as for prediction of the particle collision rate and collision distribution. The results indicate that the SVS model performs reasonably well for predicting particle concentration heterogeneity and collision rate, and that differences between the SVS and DNS results can be attributed to the fact that the SVS model neglects the smallscale velocity fluctuations within the turbulent flow.

On the rising motion of a drop in stratified fluids
View Description Hide DescriptionThe rising dynamics of a deformable drop in a linearly stratified fluid is numerically obtained using a finitevolume/fronttracking method. Our results show that the drag coefficient of a spherical drop in a stratified fluid enhances as for drop Froude numbers in the range of 4 < Fr d < 16. The role of the deformability of the drop on the temporal evolution of the motion is investigated along with stratification and inertial effects. We also present the important role of stratification on the transient rising motion of the drop. It is shown that a drop can levitate in the presence of a vertical density gradient. The drop undergoes a fading oscillatory motion around its neutrally buoyant position except for high viscosity ratio drops where the oscillation occurs around a density level lighter than the neutral buoyancy level. In addition, a detailed characterization of the flow signature of a rising drop in a linearly stratified fluid including the buoyancy induced vortices and the resultant buoyant jet is presented.

Discrete particle simulations and experiments on the collapse of wet granular columns
View Description Hide DescriptionSmall quantities of liquid in a granular material control the flow dynamics as well as the triggering and jamming phases. In order to study this problem, some experimental collapse tests conducted in a rectangular box were reproduced with a 1:1 scale numerical model using the Discrete Element Method. In simulations the effect of the capillary bridges has been investigated implementing a midrange attractive force between particles based on the minimum energy approach. Also a bondingdebonding mechanism was incorporated in the algorithm and the volume of each sessile drop on the particle surface was considered during its motion. The influence of some variables was investigated with respect to the final slope profiles and the runout lengths: the initial liquid content, the particle size, the solid density, the liquid surface tension, and the liquidsolid contact angle. Also the crucial effect of the confinement walls on the collapse phenomenon was investigated: wet particles adhere to the lateral walls providing a higher flow resistance in comparison to the same material in dry conditions. It was observed that particles with largest pathlengths are localized near the movable wall at a middleheight of the initial column sample. Other particles at the surface moves in a rigid way especially if they were wet and with a low solid density. The “fidelity” of each particle with respect to the nearest neighbours was evaluated allowing to recognize the emergence of clusters of particles and rigid parts, to extract the failure surface and to localize where debonding mechanisms concentrate in the wet case.

Laminar Flows

Single fiber transport in a confined channel: Microfluidic experiments and numerical study
View Description Hide DescriptionWe present an experimental and numerical study on the transport of a single fiber confined in a microfluidic HeleShaw geometry. The fiber has a square crosssection and a typical aspect ratio of ten. We address the question of the fiber velocity as it is freely transported by the flow, and study in particular its dependence on the fiber orientation and confinement in the channel, defined as the ratio of the fiber height with the channel height. Both experiments and simulations are set so that the fiber suspended in the middle of the channel height does not interact with the lateral flow boundaries. At low confinements, the fiber velocity is independent of the fiber orientation with the flow direction and tends to the maximal velocity of the fluid when the confinement tends to zero. The fiber slows down as the confinement increases. We find that as the confinement reaches approximately 0.5, the orientation affects the fiber velocity: a fiber perpendicular to the flow direction moves faster than a parallel one. Consequently, a confined fiber transported in a microchannel at an angle different from 0° or 90° with the flow direction will drift towards a lateral wall, in the opposite direction found in sedimenting fibers. We also characterize the perturbation caused by the presence of the fiber on the flow field, and find that it drops very quickly as the fiber confinement decreases.

Instability and Transition

Symmetry analysis in linear hydrodynamic stability theory: Classical and new modes in linear shear
View Description Hide DescriptionWe present a symmetry classification of the linearised NavierStokes equations for a twodimensional unbounded linear shear flow of an incompressible fluid. The full set of symmetries is employed to systematically derive invariant ansatz functions. The symmetry analysis grasps three approaches. Two of them are existing ones, representing the classical normal modes and the Kelvin modes, while the third is a novel approach and leads to a new closedform solution of traveling modes, showing qualitatively different behaviour in energetics, shape, and kinematics when compared to the classical approaches. The last modes are energy conserving in the inviscid case. They are localized in the crossstream direction and periodic in the streamwise direction. As for the kinematics, they travel at constant velocity in the crossstream direction, while in the streamwise direction they are accelerated by the base flow. In the viscous case, the modes break down due to damping of high wavenumber contributions.

Linear stability of miscible twofluid flow down an incline
View Description Hide DescriptionWe show that miscible twolayer freesurface flows of varying viscosity down an inclined substrate are different in their stability characteristics from both immiscible twolayer flows, and flows with viscosity gradients spanning the entire flow. New instability modes arise when the critical layer of the viscosity transport equation overlaps the viscosity gradient. A lubricating configuration with a less viscous wall layer is identified to be the most stabilizing at moderate miscibility (moderate Peclet numbers). This also is in contrast with the immiscible case, where the lubrication configuration is always destabilizing. The coexistence that we find under certain circumstances, of several growing overlap modes, the usual surface mode, and a TollmienSchlichting mode, presents interesting new possibilities for nonlinear breakdown.
