Volume 24, Issue 3, March 2012

In this paper we study decaying turbulence in fixed and rotating boxes in two dimensions using the particle method smoothed particle hydrodynamics (SPH). The boundaries are specified by boundary force particles, and the turbulence is initiated by a set of Gaussian vortices. In the case of fixed boxes we recover the results of Clercx and his colleagues obtained using both a high accuracy spectral method and experiments. Our results for fixed boxes are also in close agreement with those of Monaghan [Eur. J. Mech. – B/Fluids30, 360370 (2011)] and Robinson and Monaghan [Int. J. Numer. Methods Fluids (in press)] obtained using SPH. A feature of decaying turbulence in noslip, square, fixed boundaries is that the angular momentum of the fluid varies with time because of the reaction on the fluid of the viscous stresses on the boundary. We find that when the box is allowed to rotate freely, so that the total angular momentum of box and fluid is constant, the change in the angular momentum of the fluid is a factor of ∼500 smaller than in the case for the fixed box, and the final vorticity distribution is different. We also simulate the behaviour of the turbulence when the box is forced to rotate with small and large Rossby number, and the turbulence is initiated by Gaussian vortices as before. If the rotation of the box is maintained after the turbulence is initiated we find that in the rotating frame the decay of kinetic energy, enstrophy, and the vortex structure is insensitive to the angular velocity of the box. On the other hand, if the box is allowed to rotate freely after the turbulence is initiated, the evolved vortex structure is completely different.
 LETTERS


A new way to describe the transition characteristics of a rotatingdisk boundarylayer flow
View Description Hide DescriptionA new method of graphically representing the transition stages of a rotatingdisk flow is presented. The probability density function contour map of the fluctuating azimuthal disturbance velocity is used to show the characteristics of the boundarylayer flow over the rotating disk as a function of Reynolds numbers. Compared with the variation of the disturbance amplitude (rms) or spectral distribution, this map more clearly shows the changing flow characteristics through the laminar, transitional, and turbulent regions. This method may also be useful to characterize the different stages in the transition process not only for the rotatingdisk flow but also for other flows.

 ARTICLES

 Biofluid Mechanics

Vesicle tumbling inhibited by inertia
View Description Hide DescriptionVesicles under flow constitute a model system for the study of red blood cells (RBCs) dynamics and blood rheology. In the blood circulatory system the Reynolds number (at the scale of the RBC) is not always small enough for the Stokes limit to be valid. We develop a numerical method in two dimensions based on the level set approach and solve the fluid/membrane coupling by using an adaptive finite element technique. We find that a Reynolds number of order one can destroy completely the vesicle tumbling motion obtained in the Stokes regime. We analyze in details this phenomenon and discuss some of the far reaching consequences. We suggest experimental tests on vesicles.

Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels
View Description Hide DescriptionThreedimensional simulations of blood flow usually produce such large quantities of data that they are unlikely to be of clinical use unless methods are available to simplify our understanding of the flowdynamics. We present a new method to investigate the mechanisms by which vascular curvature and torsion affect blood flow, and we apply it to the steadystate flow in single bends, helices, double bends, and a rabbit thoracic aorta based on image data. By calculating forces and accelerations in an orthogonal coordinate system following the centreline of each vessel, we obtain the inertial forces (centrifugal, Coriolis, and torsional) explicitly, which directly depend on vascular curvature and torsion. We then analyse the individual roles of the inertial, pressure gradient, and viscous forces on the patterns of primary and secondary velocities, vortical structures, and wall stresses in each cross section. We also consider crosssectional averages of the inplane components of these forces, which can be thought of as reducing the dynamics of secondary flows onto the vessel centreline. At Reynolds numbers between 50 and 500, secondary motions in the directions of the local normals and binormals behave as two underdamped oscillators. These oscillate around the fully developed state and are coupled by torsional forces that break the symmetry of the flow. Secondary flows are driven by the centrifugal and torsional forces, and these are counterbalanced by the inplane pressure gradients generated by the wall reaction. The viscous force primarily opposes the pressure gradient, rather than the inertial forces. In the axial direction, and depending on the secondary motion, the curvaturedependent Coriolis force can either enhance or oppose the bulk of the axial flow, and this shapes the velocity profile. For bends with little or no torsion, the Coriolis force tends to restore flow axisymmetry. The maximum circumferential and axial wall shear stresses along the centreline correlate well with the averaged inplane pressure gradient and the radial displacement of the peak axial velocity, respectively. We conclude with a discussion of the physiological implications of these results.
 Micro and Nanofluid Mechanics

Inertial focusing dynamics in spiral microchannels
View Description Hide DescriptionThis report details a comprehensive study of inertial focusing dynamics and particle behavior in low aspect ratio (h/w ∼ 1/1 to 1/8) spiral microchannels. A continuum of particle streak behavior is shown with longitudinal, crosssectional, and velocity resolution, yielding a large analyzed parameter space. The dataset is then summarized and compared to prior results from both straight microchannels and other low aspect ratio spiral microchannel designs. Breakdown of focusing into a primary and secondary fluorescent streak is observed in the lowest aspect ratio channels at high average downstream velocities. Streak movement away from the theoretically predicted near inner wall equilibrium position towards the center of the channel at high average downstream velocities is also detailed as a precursor to breakdown. State diagrams detail the overall performance of each device including values of the required channel lengths and the range of velocities over which quality focusing can be achieved.

Parabolic temperature profile and secondorder temperature jump of a slightly rarefied gas in an unsteady twosurface problem
View Description Hide DescriptionThe behavior of a slightly rarefied monatomic gas between two parallel plates whose temperature grows slowly and linearly in time is investigated on the basis of the kinetic theory of gases. This problem is shown to be equivalent to a boundaryvalue problem of the steady linearized Boltzmann equation describing a rarefied gas subject to constant volumetric heating. The latter has been recently studied by Radtke, Hadjiconstantinou, Takata, and Aoki (RHTA) as a means of extracting the secondorder temperature jump coefficient. This correspondence between the two problems gives a natural interpretation to the volumetric heating source and explains why the secondorder temperature jump observed by RHTA is not covered by the general theory of slip flow for steady problems. A systematic asymptotic analysis of the timedependent problem for small Knudsen numbers is carried out and the complete fluiddynamic description, as well as the related halfspace problems that determine the structure of the Knudsen layer and the coefficients of temperature jump, are obtained. Finally, a numerical solution is presented for both the BhatnagarGrossKrook model and hardsphere molecules. The jump coefficient is also calculated by the use of a symmetry relation; excellent agreement is found with the result of the numerical computation. The asymptotic solution and associated secondorder jump coefficient obtained in the present paper agree well with the results by RHTA that are obtained by a low variance stochastic method.

Gas flow and heat transfer in nanotube and nanowire arrays
View Description Hide DescriptionGas flow through arrays of nanotube or nanowirestructures is modeled by combining the onedimensional equations for conservation of mass, momentum, and energy with the linearized freemolecular drag and heat transfer for a cylinder. The results show that the pressure gradient, temperature, and local velocity of the gas are governed by coupled ordinary differential equations. Three cases are considered: an isothermal system, a constant wall temperature, and a constant wall heat flux. While the coupled momentum, heat transfer, and continuity equations are nonlinear, the relatively low velocities encountered in these systems cause the nonlinear portions of pressure drops and thermal phenomena to be relatively small.

High accuracy numerical solutions of the Boltzmann BhatnagarGrossKrook equation for steady and oscillatory Couette flows
View Description Hide DescriptionModeling gas flows generated by micro and nanodevices often requires the use of kinetic theory. To facilitate implementation, various approximate formulations have been proposed based on the BhatnagarGrossKrook (BGK) kinetic model, including most recently, the lattice Boltzmann (LB) method. While there exists a comprehensive numerical data set for the hard sphere linearized Boltzmann equation for steady Couette flow, no such set of data is available for the BoltzmannBGK equation. The purpose of this article is to present a high accuracy data set for the linearized BoltzmannBGK equation over the full range of Knudsen numbers and normalized oscillation frequencies – this encompasses both steady and unsteady Couette flows. This data set is expected to be of particular value in the benchmarking and validation of computational methods such as the LB method and other approaches based on the BoltzmannBGK equation.
 Interfacial Flows

Nonlinear Marangoni waves in a twolayer film in the presence of gravity
View Description Hide DescriptionLongwave Marangoni convection in twolayer films under the action of gravity is considered. The analysis is carried out in the lubrication approximation. A linear stability analysis reveals the existence of monotonic and oscillatory instability modes, depending on the way of heating and the value of the Biot number. Numerical simulations are performed in the case of an oscillatory instability, which takes place by heating from above. Periodic boundary conditions are applied on the boundaries of the computational region. A sequence of nonlinear wavy regimes, which develop by the increase of the Galileo number, is studied. That sequence includes threedimensional and twodimensional structures. The multistability of wavy patterns with different spatial periods has been revealed.

Thermocapillary flows and interface deformations produced by localized laser heating in confined environment
View Description Hide DescriptionThe deformation of a fluidfluidinterface due to the thermocapillary stress induced by a continuous Gaussian laser wave is investigated analytically. We show that the direction of deformation of the liquidinterface strongly depends on the viscosities and the thicknesses of the involved liquid layers. We first investigate the case of an interface separating two different liquid layers while a second part is dedicated to a thin film squeezed by two external layers of same liquid. These results are predictive for applications fields where localized thermocapillary stresses are used to produce flows or to deform interfaces in presence of confinement, such as optofluidics.

Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate
View Description Hide DescriptionThe present study is an experimental investigation regarding the evaporation of ethanoldrops deposited onto a heated substrate in a partial wetting situation. The originality of this work is based on the simultaneous observation of the kinetics of evaporation, heat and mass transfers, the tripleline dynamic, and thermal motions inside the drop. The triple line recedes during the dropevaporation and a spontaneous development of thermalconvective instabilities driven by the evaporation are observed. These instabilities are interpreted as hydrothermal waves induced by surface tension gradient along the free surface. An infrared technique is used to investigate the temporal and spatial dynamics of the hydrothermal waves. Results reveal a nonlinear evolution of the number of waves as well as several instability regimes. A complete description of the dropevaporation with the evidence of several phases is provided. The influence of geometrical and thermal parameters has been analyzed and raised scaling laws on hydrodynamic and energy transport. The dropevaporation appears to be characterized by a constant drop Nusselt number of a value 1.7 during all the process which highlights both the importance of conduction and convection in the energy transport in an evaporating drop.

Instability of a transverse liquid rivulet on an inclined plane
View Description Hide DescriptionThis work concentrates on the stability of a viscous liquid rivulet positioned across an inclined plane under partial wetting conditions. The study is performed within the framework of lubrication approximation by employing a slip model. Both normal and parallel components of gravity are considered. We find the stability regions for given area of the cross section of the rivulet, A, plane inclination angle, α, and static contact angle, θ_{0}, characterizing the wettability of the substrate. For α’s smaller than some critical angle, α*, a staticsolution exists. This solution is characterized by rear/front contact angles given by θ_{0} ± δ. The linear stability analysis of this solution is performed using an efficient pseudospectral Chebyshev method. We analyze the effects of A, θ_{0}, and α on the predictions of the model, such as the dominant wavelength, the maximum growth rate, and the behavior of the most unstable perturbation mode. To verify them, we also carry out experiments with silicone oils spreading on a coated glass substrate for several different fluid volumes and inclination angles. We find very good agreement between the wavelength of maximum growth rate given by the theory and the average distance between the drops after rivulet breakup. An analysis of finite size effects shows that the inclusion of normal gravity effects leads to a better agreement between theoretical and experimental results.

Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field
View Description Hide DescriptionA detailed nonlinear analysis of the deformation and breakup of a perfect dielectric (PD) drop, suspended in another perfect dielectric fluid, in the presence of a quadrupoleelectric field is presented using analytical (asymptotic) and numerical (boundary integral) methods. The quadrupole field is the simplest kind of an axisymmetric nonuniform electric field. A drop, when placed at the center of such a field, does not translate, thus allowing systematic investigation of the effect of nonuniformity of the electric field. The deformation of a drop under a quadrupole field for PDPD systems exhibits several novel features as compared to that of a drop under a uniform electric field. The first order analysis predicts oblate deformation for a PDPD system when the dielectric constant of the suspending medium is larger than that of the drop (Q = ε_{ i }/ε_{ e } < 1). This is in sharp contrast to uniform electric fields where oblate shapes are observed only in leaky dielectric systems. Prolate shapes are observed for Q > 1, and the deformation is larger than that for uniform fields for similar electric capillary numbers. The steady state shapes are defined by higher harmonics as compared to the uniform field. At large capillary numbers, prolate deformations (Q > 1) show breakup whereas oblate deformations (Q < 1) do not. Positive and negative dielectrophoresis is observed when the drop is placed off center, and its translation and simultaneous deformation under quadrupole fields is also investigated. The electrohydrostatics is unaffected by the viscosity ratio. However, the breakup of the drop and the dielectrophoretic motion and deformation strongly depend upon the viscosity ratio.

Thermally induced van der Waals rupture of thin viscous fluid sheets
View Description Hide DescriptionWe consider the dynamics of a thin symmetric fluid sheet subject to an initial temperature profile, where inertia, viscous stresses, disjoining pressures, capillarity, and thermocapillarity are important. We apply a longwave analysis in the limit where deviations from the mean sheet velocity are small, but thermocapillary stresses and heat transfer from the sheet to the environment are significant and find a coupled system of partial differential equations that describe the sheet thickness, the mean sheet velocity, and the mean sheet temperature. From a linear stability analysis, we find that a stable thermal mode couples the velocity to the interfacial dynamics. This coupling can be utilized to delay the onset of rupture or to promote an earlier rupture event. In particular, rupture can be induced thermally even in cases when the heat transfer to the surrounding environment is significant, provided that the initial phase shift between the initial velocity and temperature disturbances is close to ϕ = π/2. These effects suggest a strategy that uses phase modulation in the initial temperature perturbation related to the initial velocity perturbation that assigns priority of the rupture events at particular sites over several spatial periods.

Thermocapillaryassisted pulling of contactfree liquid films
View Description Hide DescriptionWe study the formation of a free liquid film that is pulled out of a bath at constant speed and stabilized by the action of thermocapillary stresses prescribed at the free surfaces. The basic concept was introduced recently by Scheid et al. [“Thermocapillaryassisted pulling of thin films: Application to molten metals,” Appl. Phys. Lett.97, 171906 (2010)]10.1063/1.3505523. The theory suggests that very thin ribbons of molten material can be drawn out of a melt by adequately tuning the temperature gradient along the dynamic meniscus that connects the static meniscus at the melting bath to the region of the drawn flat film. In the present paper, we extend our original analysis by investigating the roles of inertia and gravity on the film thickness, and show how the results depend on heat transfer/conduction properties. Furthermore, we analyze the onedimensional transverse stability of the free film with respect to the longwave thermocapillary instability.

Disorderinduced hysteresis and nonlocality of contact line motion in chemically heterogeneous microchannels
View Description Hide DescriptionWe examine the motion of a liquidairmeniscus advancing into a microchannel with chemically heterogeneous walls. We consider the case where a constant flow rate is imposed, so that the mean velocity of the interface is kept constant, and study the effects of the disorder properties on the apparent contact angle for each microchannel surface. We focus here on a large diffusivity regime, where any possible advection effect is not taken into account. To this end, we make use of a phasefield model that enables contact line motion by diffusive interfacial fluxes and takes into account the wetting properties of the walls. We show that in a regime of sufficiently low velocities, the contact angle suffers a hysteresis behavior which is enhanced by the disorder strength. We also show that the contact line dynamics at each surface of the microchannel may become largely coupled with each other when different wetting properties are applied at each wall, reflecting that the dynamics of the interface is dominated by nonlocal effects.

Linear oscillations of constrained drops, bubbles, and plane liquid surfaces
View Description Hide DescriptionThe smallamplitude oscillations of constrained drops, bubbles, and plane liquid surfaces are studied theoretically. The constraints have the form of closed lines of zero thickness which prevent the motion of the liquid in the direction normal to the undisturbed free surface. It is shown that, by accounting explicitly for the singular nature of the curvature of the interface and the force exerted by the constraint, methods of analysis very close to the standard ones applicable to the unconstrained case can be followed. Weak viscous effects are accounted for by means of the dissipation function. Graphical and numerical results for the oscillations of constrained drops and bubbles are presented. Examples of two and threedimensional gravitycapillary waves are treated by the same method. A brief consideration of the RayleighTaylor unstable configuration shows that the nature of the instability is not affected, although its growth rate is decreased.

Observations of KelvinHelmholtz instability at the airwater interface in a circular domain
View Description Hide DescriptionWe present an analysis of KelvinHelmholtz instability in a circular domain in the limit of the azimuthal integer wavenumber, n → ∞ which reproduces the classical results for a rectilinear geometry at the rim, provided that the additional condition that the surfacecurrent to surfacewind ratio is (ρ_{1}/ρ_{2})^{1/2} where ρ_{1} and ρ_{2} are respectively the densities of air and water, is satisfied. Experiments were carried out in a circular rig of radius 0.19 m in which a family of unstable waveforms with n ≈ 60 were observed with properties (including the additional condition) in approximate agreement with theory. The additional condition is consistent with the absence of a surface shear stress in the instability process.
 Viscous and NonNewtonian Flows

Reynoldsnumber effect on vortex ring evolution in a viscous fluid
View Description Hide DescriptionIt is known that the cross section of the vortex ring core takes an approximately elliptical shape with increasing Reynolds number. In order to model this feature, the functional form of a vortex ringsolution of the Stokes equations is modified so as to be able to model higher Reynolds numberrings. The model introduces two nondimensional parameters that govern the shape of the vortex core:λ ⩾ 1 and β ⩾ 1. Based on this modification, new expressions for the translation velocity, energy, circulation, and streamfunction are derived for a wide range of section ellipticity that are specific to such vortices. To validate the model, the data adapted from the numerical study of vortex ring at Reynolds numberRe = 1400 performed by Danaila and Helie [Phys. Fluids20, 073602 (2008)], is used. In this case, the appropriate values of λ and β are calculated by equating the normalized energy E _{ d } and circulation Γ_{ d } of the theoreticalvortex to the corresponding values obtained from the numerical data. The model provides a good prediction of the ring velocity evolution at high Reynolds numbers.
 Particulate, Multiphase, and Granular Flows

On the motion of inertial particles by sound waves
View Description Hide DescriptionThis paper describes the numerical simulation of the motion of a heavy spherical particle in an acoustic wave using the equation of motion for a point particle. Our results agree well with the recent experimental data of Gonzàlez, Hoffmann, and Gallego [“Precise measurements of particle entrainment in a standingwave acoustic field between 20 and 3500 Hz,” J. Aerosol Sci. 31, 1461–1468 (2000)]. Our simulations cover a range of particle relaxation number, τ* = ωτ, where τ is the particle relaxation time and ω is the angular acoustic frequency from 0.06 to 10, particle to fluid density ratios, ρ_{ p }/ρ_{ f }, from 2500 to 2, and moderate acoustic velocity amplitudes. The results show that the Stokes force controls particle motion for τ* < 1 and ρ_{ p }/ρ_{ f } > 25. Within this regime it is appropriate to consider the Basset, pressure gradient, and virtual mass forces as “higher order” corrections to the Stokes force. The magnitude of the Basset force exceeds that of the Stokes force for ρ_{ p }/ρ_{ f } ⩾ 25 and τ* ⩾ 4. All the forces in the particle equation of motion should be accounted for when simulating particle motion in an acoustic wave for ρ_{ p }/ρ_{ f } < 25.
 Laminar Flows

Boundary layer development in the flow field between a rotating and a stationary disk
View Description Hide DescriptionThis paper discusses the development of boundary layers in the flow of a Newtonian fluid between two parallel, infinite disks. One of the disks is rotating at a constant angular velocity while the other remains stationary. An analytical series approximation and a numerical solution method are used to describe the velocity profiles of the flow. Both methods rely on the commonly used similarity transformation first proposed by Von Kármán [T. von Kármán, ZAMM1, 233 (1921)]10.1002/zamm.19210010401. For Re _{ h } < 18, the power series analytically describe the complete velocity profile. With the numerical model a Batchelor type of flow was observed for Re _{ h } > 300, with two boundary layers near the disks and a nonviscous core in the middle. A remarkable conclusion of the current work is the coincidence of the power series’ radius of convergence, a somewhat abstract mathematical notion, with the physically tangible concept of the boundary layer thickness. The coincidence shows a small deviation of only 2% to 4%.