Papers
Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition
- STEFAN G. SIEGEL, JÜRGEN SEIDEL, CASEY FAGLEY, D. M. LUCHTENBURG, KELLY COHEN, THOMAS MCLAUGHLIN
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- 08 August 2008, pp. 1-42
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For the systematic development of feedback flow controllers, a numerical model that captures the dynamic behaviour of the flow field to be controlled is required. This poses a particular challenge for flow fields where the dynamic behaviour is nonlinear, and the governing equations cannot easily be solved in closed form. This has led to many versions of low-dimensional modelling techniques, which we extend in this work to represent better the impact of actuation on the flow. For the benchmark problem of a circular cylinder wake in the laminar regime, we introduce a novel extension to the proper orthogonal decomposition (POD) procedure that facilitates mode construction from transient data sets. We demonstrate the performance of this new decomposition by applying it to a data set from the development of the limit cycle oscillation of a circular cylinder wake simulation as well as an ensemble of transient forced simulation results. The modes obtained from this decomposition, which we refer to as the double POD (DPOD) method, correctly track the changes of the spatial modes both during the evolution of the limit cycle and when forcing is applied by transverse translation of the cylinder. The mode amplitudes, which are obtained by projecting the original data sets onto the truncated DPOD modes, can be used to construct a dynamic mathematical model of the wake that accurately predicts the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using nonlinear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. We demonstrate this ability not just for unforced and open-loop forced data, but also for a feedback-controlled simulation that leads to a 90% reduction in lift fluctuations. This indicates the possibility of constructing accurate dynamic low-dimensional models for feedback control by using unforced and transient forced data only.
The effects of flow stratification by non-cohesive sediment on transport in high-energy wave-driven flows
- DANIEL C. CONLEY, SILVIA FALCHETTI, IRIS P. LOHMANN, MAURIZIO BROCCHINI
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- 08 August 2008, pp. 43-67
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The two-way effects of the time-varying suppression of turbulence by gradients in suspended sediment concentration have been investigated using a modified form of the Generalized Ocean Turbulence Model (GOTM). Field measurements of fluid velocities and sediment concentrations collected under high-energy conditions (mobility number ≈ 900) have been simulated both including and neglecting the feedback between sediment and turbulence. The results show that, when present, this feedback increases the wave-coherent component of transport relative to the mean component of transport, which can even change the direction of transport. Comparisons between measured and simulated time series of near-bed sediment concentrations show great coherence (0.95 correlation) and it is argued that the differences in net transport rates may be partially explained by the use of a uniform grain size in the simulations. It is seen that the effects of sediment stratification scale with orbital velocity divided by sediment setting velocity, um/ws, for all grain sizes.
Kinematics and statistics of dense, slow granular flow through vertical channels
- ANANDA K. S., SUDHESHNA MOKA, PRABHU R. NOTT
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- 08 August 2008, pp. 69-97
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We have investigated the flow of dry granular materials through vertical channels in the regime of dense slow flow using video imaging of the particles adjacent to a transparent wall. Using an image processing technique based on particle tracking velocimetry, the video movies were analysed to obtain the velocities of individual particles. Experiments were conducted in two- and three-dimensional channels. In the latter, glass beads and mustard seeds were used as model granular materials, and their translational velocities were measured. In the former, aluminium disks with a dark diametral stripe were used and their translational velocities and spin were measured. Experiments in the three-dimensional channels were conducted for a range of the channel width W, and for smooth and rough sidewalls. As in earlier studies, we find that shearing takes place predominantly in thin layers adjacent to the walls, while the rest of the material appears to move as a plug. However, there are large velocity fluctuations even in the plug, where the macroscopic deformation rate is negligibly small. The thickness of the shear layer, scaled by the particle diameter dp, increases weakly with W/dp. The experimental data for the velocity field are in good agreement with the Cosserat plasticity model proposed recently. We also measured the mean spin of the particles in the two-dimensional channel, and its deviation from half the vorticity. There is a clear, measurable deviation, which too is in qualitative agreement with the Cosserat plasticity model. The statistics of particle velocity and spin fluctuations in the two-dimensional channel were analysed by determining their probability distribution function, and their spatial and temporal correlation. They were all found to be broadly similar to previous observations for three-dimensional channels, but some differences are evident. The spatial correlation of the velocity fluctuations are much stronger in the two-dimensional channel, implying a pronounced solid-like motion superimposed over an uncorrelated fluid-like motion. The strong spatial correlation over large distances has led us to propose a mechanism for the production of velocity fluctuations in the absence of a macroscopic deformation rate.
The circular internal hydraulic jump
- S. A. THORPE, I. KAVCIC
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- 08 August 2008, pp. 99-129
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Circular hydraulic jumps are familiar in single layers. Here we report the discovery of similar jumps in two-layer flows. A thin jet of fluid impinging vertically onto a rigid horizontal plane surface submerged in a deep layer of less-dense miscible fluid spreads radially, and a near-circular internal jump forms within a few centimetres from the point of impact with the plane surface. A jump is similarly formed as a jet of relatively less-dense fluid rises to the surface of a deep layer of fluid, but it appears less stable or permanent in form. Several experiments are made to examine the case of a downward jet onto a horizontal plate, the base of a square or circular container. The inlet Reynolds numbers, Re, of the jet range from 112 to 1790. Initially jumps have an undular, laminar form with typically 2–4 stationary waves on the interface between the dense and less-dense layers but, as the depth of the dense layer beyond the jump increases, the transitions become more abrupt and turbulent, resulting in mixing between the two layers. During the transition to a turbulent regime, single and sometimes moving multiple cusps are observed around the periphery of jumps. A semi-empirical model is devised that relates the parameters of the laboratory experiment, i.e. flow rate, inlet nozzle radius, kinematic viscosity and reduced gravity, to the layer depth beyond the jump and the radius at which an undular jump occurs. The experiments imply that surface tension is not an essential ingredient in the formation of circular hydraulic jumps and demonstrate that stationary jumps can exist in stratified shear flows which can be represented as two discrete layers. No stationary circular undular jumps are found, however, in the case of a downward jet of dense fluid when the overlying, less-dense, fluid is stratified, but a stationary turbulent transition is observed. This has implications for the existence of stationary jumps in continuously stratified geophysical flows: results based on two-layer models may be misleading. It is shown that the Froude number at which a transition of finite width occurs in a radially diverging flow may be less than unity.
Constrained flow around a magnetic obstacle
- EVGENY V. VOTYAKOV, EGBERT ZIENICKE, YURI B. KOLESNIKOV
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- 08 August 2008, pp. 131-156
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Many practical applications exploit an external local magnetic field – magnetic obstacle – as an essential part of their operation. It has been demonstrated that the flow of an electrically conducting fluid influenced by an external field can show several kinds of recirculation. The present paper reports a three-dimensional numerical study, some results of which are compared with an experiment on such a flow in a rectangular duct. First, we derive equations to compute analytically the external magnetic field and verify these equations by comparing with experimentally measured field intensity. Then, we study flow characteristics for different magnetic field configurations. The flow inside the magnetic gap is dependent mainly on the interaction parameter N, which represents the ratio of the Lorentz force to the inertial force. Depending on the constrainment factor κ = My/Ly, where My and Ly are the half-widths of the external magnet and duct, the flow can show different stationary recirculation patterns: two magnetic vortices at small κ, a six-vortex ensemble at moderate κ, and no vortices at large κ. Recirculation appears when N is higher than a critical value Nc,m. The driving force for the recirculation is the reverse electromotive force that arises to balance the reverse electrostatic field. The reversal of the electrostatic field is caused by the concurrence of internal and external vorticity respectively related to the internal and external slopes in the M-shaped velocity profile. The critical value of Nc,m grows quickly as κ increases. For the case of well-developed recirculation, the numerical reverse velocity agrees well with that obtained in experiments. Two different magnetic systems can induce the same electric field and stagnation region provided these systems have the same power of recirculation, given by the N/Nc,m ratio. The three-dimensional helical characteristics of the vortices are elaborated, and an analogy is shown to exist between helical motion inside the recirculation studied and secondary motion in Ekman pumping. Finally, it is shown that a two-dimensional model fails to properly produce stable two- and six-vortex structures as found in the three-dimensional system. Interestingly, these recirculation patterns appear only as time-dependent and unstable transitional states before a Kármán vortex street forms, when one suddenly applies a retarding local magnetic field to a constant flow.
The acceleration of solid particles subjected to cavitation nucleation
- BRAM M. BORKENT, MANISH ARORA, CLAUS-DIETER OHL, NICO DE JONG, MICHEL VERSLUIS, DETLEF LOHSE, KNUD AAGE MØRCH, EVERT KLASEBOER, BOO CHEONG KHOO
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- 08 August 2008, pp. 157-182
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The cavity–particle dynamics at cavitation inception on the surface of spherical particles suspended in water and exposed to a strong tensile stress wave is experimentally studied with high-speed photography. Particles, which serve as nucleation sites for cavitation bubbles, are set into a fast translatory motion during the explosive growth of the cavity. They reach velocities of ~40 ms−1 and even higher. When the volume growth of the cavity slows down, the particle detaches from the cavity through a process of neck-breaking, and the particle is shot away. The experimental observations are simulated with (i) a spherical cavity model and (ii) with an axisymmetric boundary element method (BEM). The input for both models is a pressure pulse, which is obtained from the observed radial cavity dynamics during an individual experiment. The model then allows us to calculate the resulting particle trajectory. The cavity shapes obtained from the BEM calculations compare well with the photographs until neck formation occurs. In several cases we observed inception at two or more locations on a single particle. Moreover, after collapse of the primary cavity, a second inception was often observed. Finally, an example is presented to demonstrate the potential application of the cavity–particle system as a particle cannon, e.g. in the context of drug delivery into tissue.
An analytical model for bore-driven run-up
- DAVID PRITCHARD, PAUL A. GUARD, TOM E. BALDOCK
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- 08 August 2008, pp. 183-193
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We use a hodograph transformation and a boundary integral method to derive a new analytical solution to the shallow-water equations describing bore-generated run-up on a plane beach. This analytical solution differs from the classical Shen–Meyer runup solution in giving significantly deeper and less asymmetric swash flows, and also by predicting the inception of a secondary bore in both the backwash and the uprush in long surf. We suggest that this solution provides a significantly improved model for flows including swash events and the run-up following breaking tsunamis.
Vortex flow and cavitation in diesel injector nozzles
- A. ANDRIOTIS, M. GAVAISES, C. ARCOUMANIS
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- 08 August 2008, pp. 195-215
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Flow visualization as well as three-dimensional cavitating flow simulations have been employed for characterizing the formation of cavitation inside transparent replicas of fuel injector valves used in low-speed two-stroke diesel engines. The designs tested have incorporated five-hole nozzles with cylindrical as well as tapered holes operating at different fixed needle lift positions. High-speed images have revealed the formation of an unsteady vapour structure upstream of the injection holes inside the nozzle volume, which is referred to as ‘string-cavitation’. Computation of the flow distribution and combination with three-dimensional reconstruction of the location of the strings inside the nozzle volume has revealed that strings are found at the core of recirculation zones; they originate either from pre-existing cavitation sites forming at sharp corners inside the nozzle where the pressure falls below the vapour pressure of the flowing liquid, or even from suction of outside air downstream of the hole exit. Processing of the acquired images has allowed estimation of the mean location and probability of appearance of the cavitating strings in the three-dimensional space as a function of needle lift, cavitation and Reynolds number. The frequency of appearance of the strings has been correlated with the Strouhal number of the vortices developing inside the sac volume; the latter has been found to be a function of needle lift and hole shape. The presence of strings has significantly affected the flow conditions at the nozzle exit, influencing the injected spray. The cavitation structures formed inside the injection holes are significantly altered by the presence of cavitation strings and are jointly responsible for up to 10% variation in the instantaneous fuel injection quantity. Extrapolation using model predictions for real-size injectors operating at realistic injection pressures indicates that cavitation strings are expected to appear within the time scales of typical injection events, implying significant hole-to-hole and cycle-to-cycle variations during the corresponding spray development.
Beating of a circular cylinder mounted as an inverted pendulum
- A. VOORHEES, P. DONG, P. ATSAVAPRANEE, H. BENAROYA, T. WEI
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- 08 August 2008, pp. 217-247
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This paper contains temporally and spatially resolved flow visualization and DPIV measurements characterizing the frequency–amplitude response and three-dimensional vortex structure of a circular cylinder mounted like an inverted pendulum. Two circular cylinders were examined in this investigation. Both were 2.54 cm in diameter and ~140 cm long with low mass ratios, m* = 0.65 and 1.90, and mass–damping ratios, m*ζ = 0.038 and 0.103, respectively. Frequency–amplitude response analysis was done with the lighter cylinder while detailed wake structure visualization and measurements were done using the slightly higher-mass-ratio cylinder. Experiments were conducted over the Reynolds number range 1900≤Re≤6800 corresponding to a reduced velocity range of 3.7 ≤ U* ≤ 9.6. Detailed examination of the upper branch of the synchronization regime permitted, for the first time, the identification of short-time deviations in cylinder oscillation and vortex-shedding frequencies that give rise to beating behaviour. That is, while long-time averages of cylinder oscillation and vortex-shedding frequencies are identical, i.e. synchronized, it is shown that there is a slight mismatch between these frequencies over much shorter periods when the cylinder exhibits quasi-periodic beating. Data are also presented which show that vortex strength is also modulated from one cylinder oscillation to the next. Physical arguments are presented to explain both the origins of beating as well as why the quasi-periodicity of the beating envelopes becomes irregular; it is believed that this result may be generalized to a broader class of fluid–structure interactions. In addition, observations of strong vertical flows associated with the Kármán vortices developing 2–3 diameters downstream of the cylinder are presented. It is hypothesized that these three-dimensionalities result from both the inverted pendulum motion as well as free-surface effects.
Viscoelastic effects on the jetting–dripping transition in co-flowing capillary jets
- J. M. MONTANERO, A. M. GAÑÁN-CALVO
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- 08 August 2008, pp. 249-260
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Linear hydrodynamics stability analysis is used to determine the influence of elasticity on the jetting–dripping transition and on the temporal stability of non-axisymmetric modes in co-flowing capillary jets. The critical Weber number for which axisymmetric perturbations undergo a transition from convective to absolute instability is calculated from the spatio-temporal analysis of the dispersion relation for Oldroyd-B liquids, as a function of the density and viscosity ratios, and the Reynolds and Deborah numbers. Elasticity increases the critical Weber number for all cases analysed and, consequently, fosters the transition from jetting to dripping. The temporal analysis of the dispersion relation for the m = 1 lateral mode shows that elasticity does not affect its stability.
On the Lamb vector divergence in Navier–Stokes flows
- CURTIS W. HAMMAN, JOSEPH C. KLEWICKI, ROBERT M. KIRBY
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- 08 August 2008, pp. 261-284
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The mathematical and physical properties of the Lamb vector divergence are explored. Toward this aim, the instantaneous and mean dynamics of the Lamb vector divergence are examined in several analytic and turbulent flow examples relative to its capacity to identify and characterize spatially localized motions having a distinct capacity to effect a time rate of change of momentum. In this context, the transport equation for the Lamb vector divergence is developed and shown to accurately describe the dynamical mechanisms by which adjacent high- and low-momentum fluid parcels interact to effect a time rate of change of momentum and generate forces such as drag. From this, a transport-equation-based framework is developed that captures the self-sustaining spatiotemporal interactions between coherent motions, e.g. ejections and sweeps in turbulent wall flows, as predicted by the binary source–sink distribution of the Lamb vector divergence. New insight into coherent motion development and evolution is found through the analysis of the Lamb vector divergence.
Long-range interaction and elastic collisions of isolated vortices
- TIMOUR RADKO
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- 08 August 2008, pp. 285-310
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This study explores the interaction of two nearly axisymmetric two-dimensional vortices using a combination of numerical simulations and analytical arguments. We consider isolated or ‘shielded’ eddies, characterized by zero net vorticity. The ability of such vortices to propagate and interact is associated with the small dipolar component that is introduced initially. Numerical contour dynamics experiments indicate that the interaction of shielded eddies takes one of two forms, depending on their initial separation and on the relative orientation of their dipolar components. Eddies can influence each other by remotely modifying the dipolar moments of partner vortices, an effect manifested in a gentle deflection of their trajectories from a straight course. Strong interactions occur when eddies collide and rebound. The remote interaction is explained by weakly nonlinear theory in which the basic state consists of identical circularly symmetric eddies and the perturbation is assumed to be small. It is argued that the elastic rebounds observed during direct collisions are induced by the exchange of fluid between colliding vortices.
Destabilization of mixed Rossby gravity waves and the formation of equatorial zonal jets
- BACH LIEN HUA, MARC D'ORGEVILLE, MARK D. FRUMAN, CLAIRE MENESGUEN, RICHARD SCHOPP, PATRICE KLEIN, HIDEHARU SASAKI
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- 08 August 2008, pp. 311-341
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The stability of mixed Rossby gravity (MRG) waves has been investigated numerically using three-dimensionally consistent high-resolution simulations of the continuously stratified primitive equations. For short enough zonal wavelength, the westward phase propagating MRG wave is strongly destabilized by barotropic shear instability leading to the formation of zonal jets. The large-scale instability of the zonally short wave generates zonal jets because it consists primarily of sheared meridional motions, as shown recently for the short barotropic Rossby wave problem.
Simulations were done in a variety of domain geometries: a periodic re-entrant channel, a basin with a short MRG wave forced in its western part and a very long channel initialized with a zonally localized MRG wave. The characteristics of the zonal jets vary with the geometry. In the periodic re-entrant channel, barotropic zonal jets dominate the total flow response at the equator and its immediate vicinity. In the other cases, the destabilization leads to zonal jets with quite different characteristics, especially in the eastward group propagating part of the signal. The most striking result concerns the formation of zonal jets at the equator, alternating in sign in the vertical, with vertical scale short compared to the scale of the forcing or initial conditions.
A stability analysis of a simplified perturbation vorticity equation is formulated to explain the spatial scale selection and growth rate of the zonal jets as functions of the characteristics of the basic state MRG wave. For both types of zonal jets, the model predicts that their meridional scales are comparable to the zonal scale of the MRG wave basic state, while their growth rates scale as μ ∝ Fr |k|, where Fr is the Froude number of the meridional velocity component of the basic state and k its non-dimensional zonal wavenumber. The vertical scale of the baroclinic zonal jets corresponds to the dominant harmonic ppeak of the basic state in the fastest growing mode, given by ppeak≈0.55k2. Thus, the shorter the zonal wavelength of the basic state MRG wave, the narrower the meridional scale of the zonal jets, both barotropic and baroclinic, with the vertical scale of the baroclinic jets being tied to their meridional scale through the equatorial radius of deformation, which decreases as the square root of the vertical wavenumber. The predictions of the spatial scales are in both qualitative and quantitative agreement with the numerical simulations, where shorter vertical scale baroclinic zonal jets are favoured by shorter-wavelength longer-period MRG wave basic states, with the vertical mode number increasing as the square of the MRG wave period.
An Appendix deals with the case of zonally long and intermediate wavelength MRG waves, where a weak instability regime causes a moderate adjustment involving resonant triad interactions without leading to jet formation. For eastward phase propagating waves, adjustment does not lead to significant angular momentum redistribution.
Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment
- CHRISTOF SODTKE, VLADIMIR S. AJAEV, PETER STEPHAN
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- 08 August 2008, pp. 343-362
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We consider the evaporation of volatile liquid droplets deposited on a heated substrate in a pure saturated vapour environment. A mathematical model is developed that incorporates the effects of surface tension, evaporation, thermocapillarity, gravity, disjoining pressure, as well as unsteady heat conduction in the solid substrate. The apparent contact line is treated mathematically as a transition region between the macroscopic droplet shape and the adsorbed film of liquid on the heated substrate. Theoretical parametric studies are conducted to clarify the effects of thermocapillarity and wetting properties on the droplet dynamics. An experimental study is conducted in a closed container with de-ionized water droplets on a stainless steel foil heated by an electric current. The interface shapes are recorded together with the temperature profiles under the droplets, measured using thermochromic liquid crystals. Experiment and theory are in very good agreement as long as the conditions of applicability of our lubrication-type mathematical model are satisfied.
Field-induced motion of ferrofluid droplets through immiscible viscous media
- S. AFKHAMI, Y. RENARDY, M. RENARDY, J. S. RIFFLE, T. St PIERRE
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- 08 August 2008, pp. 363-380
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The motion of a hydrophobic ferrofluid droplet placed in a viscous medium and driven by an externally applied magnetic field is investigated numerically in an axisymmetric geometry. Initially, the drop is spherical and placed at a distance away from the magnet. The governing equations are the Maxwell equations for a non-conducting flow, momentum equation and incompressibility. A numerical algorithm is derived to model the interface between a magnetized fluid and a non-magnetic fluid via a volume-of-fluid framework. A continuum-surface-force formulation is used to model the interfacial tension force as a body force, and the placement of the liquids is tracked by a volume fraction function. Three cases are studied. First, where inertia is dominant, the magnetic Laplace number is varied while the Laplace number is fixed. Secondly, where inertial effects are negligible, the Laplace number is varied while the magnetic Laplace number is fixed. In the third case, the magnetic Bond number and inertial effects are both small, and the magnetic force is of the order of the viscous drag force. The time taken by the droplet to travel through the medium and the deformations in the drop are investigated and compared with a previous experimental study and accompanying simpler model. The transit times are found to compare more favourably than with the simpler model.
Bending of floating flexible legs
- KUN JOONG PARK, HO-YOUNG KIM
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- 08 August 2008, pp. 381-390
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When long thin flexible solid objects, such as the legs of water striders, disposable spoons and human hairs, are pressed against a liquid surface, they bend due to interfacial and hydrostatic forces. To understand the phenomenon, we study the bending of a sheet touching the liquid surface at an angle while clamped at the other end, to find its deflection and the load that the sheet can support before sinking. The theoretically predicted shapes of the sheet and the meniscus match well with experiments. Our theory shows that flexible sheets can support more load than rigid ones before sinking when the sheets are highly hydrophobic.
Linear instability of annular Poiseuille flow
- C. J. HEATON
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- 08 August 2008, pp. 391-406
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The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.
Surfactant- and elasticity-induced inertialess instabilities in vertically vibrated liquids
- BALRAM SUMAN, SATISH KUMAR
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- 08 August 2008, pp. 407-423
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We investigate instabilities that arise when the free surface of a liquid covered with an insoluble surfactant is vertically vibrated and inertial effects are negligible. In the absence of surfactants, the inertialess Newtonian system is found to be stable, in contrast to the case where inertia is present. Linear stability analysis and Floquet theory are applied to calculate the critical vibration amplitude needed to excite the instability and the corresponding wavenumber. A previously reported long-wavelength instability is found to persist to finite wavelengths, and the connection between the long-wavelength and finite-wavelength theories is explored in detail. The instability mechanism is also probed and requires the Marangoni flows to be sufficiently strong and in the appropriate phase with respect to the gravity modulation. For viscoelastic liquids, we find that instability can arise even in the absence of surfactants and inertia. Mathieu equations describing this are derived and these show that elasticity introduces an effective inertia into the system.
On the initial-value problem in a rotating circular cylinder
- KEKE ZHANG, XINHAO LIAO
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- 08 August 2008, pp. 425-443
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The initial-value problem in rapidly rotating circular cylinders is revisited. Four different but related analyses are carried out: (i) we derive a modified asymptotic expression for the viscous decay factors valid for the inertial modes of a broad range of frequencies that are required for an asymptotic solution of the initial value problem at an arbitrarily small but fixed Ekman number; (ii) we perform a fully numerical analysis to estimate the viscous decay factors, showing satisfactory quantitative agreement between the modified asymptotic expression and the fuller numerics; (iii) we derive a modified time-dependent asymptotic solution of the initial value problem valid for an arbitrarily small but fixed Ekman number and (iv) we perform fully numerical simulations for the initial value problem at a small Ekman number, showing satisfactory quantitative agreement between the modified time-dependent solution and the numerical simulations.