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Nearwall turbulence statistics and flow structures over threedimensional roughness in a turbulent channel flow
 JIARONG HONG, JOSEPH KATZ, MICHAEL P. SCHULTZ

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 14 January 2011, pp. 137

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Utilizing an optically indexmatched facility and highresolution particle image velocimetry measurements, this paper examines flow structure and turbulence in a roughwall channel flow for Reτ in the 3520–5360 range. The scales of pyramidal roughness elements satisfy the ‘wellcharacterized’ flow conditions, with h/k ≈ 50 and k+ = 60 ~ 100, where h is half height of the channel and k is the roughness height. The nearwall turbulence measurements are sensitive to spatial resolution, and vary with Reynolds number. Spatial variations in the mean flow, Reynolds stresses, as well as the turbulent kinetic energy (TKE) production and dissipation rates are confined to y < 2k. All the Reynolds stress components have local maxima at slightly higher elevations, but the streamwisenormal component increases rapidly at y < k, peaking at the top of the pyramids. The TKE production and dissipation rates along with turbulence transport also peak near the wall. The spatial energy and shear spectra show an increasing contribution of largescale motions and a diminishing role of small motions with increasing distance from the wall. As the spectra steepen at low wavenumbers, they flatten and develop bumps in wavenumbers corresponding to k − 3k, which fall in the dissipation range. Instantaneous realizations show that roughnessscale eddies are generated near the wall, and lifted up rapidly by largescale structures that populate the outer layer. A linear stochastic estimationbased analysis shows that the latter share common features with hairpin packets. This process floods the outer layer with roughnessscale eddies, in addition to those generated by the energycascading process. Consequently, although the imprints of roughness diminish in the outerlayer Reynolds stresses, consistent with the wall similarity hypothesis, the smallscale turbulence contains a clear roughness signature across the entire channel.
Longitudinal profile of channels cut by springs
 O. DEVAUCHELLE, A. P. PETROFF, A. E. LOBKOVSKY, D. H. ROTHMAN

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 13 December 2010, pp. 3847

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We propose a simple theory for the longitudinal profile of channels incised by groundwater flow. The aquifer surrounding the stream is represented in two dimensions through Darcy's law and the Dupuit approximation. The model is based on the assumption that, everywhere in the stream, the shear stress exerted on the sediment by the flow is close to the minimal intensity required to displace a sand grain. Because of the coupling of the stream discharge with the water table elevation in the neighbourhood of the channel head, the stream elevation decreases as the distance from the stream's tip with an exponent of 2/3. Field measurements of steephead ravines in the Florida Panhandle conform well to this prediction.
Boundarylayer hydrodynamics and bedload sediment transport in oscillating water tunnels
 DAVID GONZALEZRODRIGUEZ, OLE SECHER MADSEN

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 01 November 2010, pp. 4884

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Oscillating water tunnels are experimental facilities commonly used in coastal engineering research. They are intended to reproduce nearbed hydrodynamic and sediment transport phenomena at a realistic scale. In an oscillating water tunnel, a piston generates an oscillatory motion that propagates almost instantaneously to the whole tunnel; consequently, flow is uniform along the tunnel, unlike the propagating wave motion in the sea or in a wave flume. This results in subtle differences between the boundarylayer hydrodynamics of an oscillating water tunnel and of a propagating wave, which may have a significant effect in the resulting sediment transport. In this paper, we present a zerothorder analytical model of the turbulent boundarylayer hydrodynamics in an oscillating water tunnel. By using a timevarying eddy viscosity and by accounting for the constraints arising from the tunnel's geometry, the model predicts the oscillating water tunnel hydrodynamics and yields analytical expressions to compute bed shear stresses for asymmetric and skewed waves, both in the absence or presence of an imposed current. These expressions are applied to successfully quantify bedload sediment transport in oscillating water tunnel experiments.
Pattern formation in drying drops of blood
 D. BRUTIN, B. SOBAC, B. LOQUET, J. SAMPOL

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 16 December 2010, pp. 8595

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The drying of a drop of human blood exhibits coupled physical mechanisms, such as Marangoni flow, evaporation and wettability. The final stage of a whole blood drop evaporation reveals regular patterns with a good reproducibility for a healthy person. Other experiments on anaemic and hyperlipidaemic people were performed, and different patterns were revealed. The flow motion inside the blood drop is observed and analysed with the use of a digital camera: the influence of the red blood cells motion is revealed at the drop periphery as well as its consequences on the final stage of drying. The mechanisms which lead to the final pattern of the dried blood drops are presented and explained on the basis of fluid mechanics in conjunction with the principles of haematology. The blood drop evaporation process is evidenced to be driven only by Marangoni flow. The same axisymmetric pattern formation is observed, and can be forecast for different blood drop diameters. The evaporation mass flux can be predicted with a good agreement, assuming only the knowledge of the colloids mass concentration.
The mechanism of detonation attenuation by a porous medium and its subsequent reinitiation
 MATEI I. RADULESCU, BRIAN McN. MAXWELL

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 14 January 2011, pp. 96134

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The attenuation and reinitiation mechanism of detonations transmitted through a porous section consisting of a twodimensional array of staggered cylinders was investigated experimentally and numerically for acetylene–oxygen mixtures. It was found that the leading order attenuation mechanism is the wave diffraction around the cylinders. The local reamplification permitting the selfpropagation of the wave was due to wave reflections from adjacent obstacles. The critical conditions for transmittance of a detonation wave were found to correspond approximately to a pore size equal to approximately 30–60 detonation induction lengths, or one to two cell sizes. For quenched detonations, the reinitiation mechanism was found to rely on wave reflections from neighbouring pores. Depending on the mixture sensitivity, one or several shock reflections may be necessary to reamplify the attenuated detonation wave back to a selfsustained wave. For the latter case, a novel mechanism was identified, where each shock reflection gives rise to a significant enhancement of the gas reactivity and burnout of large portions of unreacted gas. This leads to a slow acceleration of the leading front, punctuated by smallscale local sudden reaccelerations. The resulting wave interactions give rise to a topologically complex reaction zone structure consisting of alternating layers of reacted and unreacted gas. The role of turbulent diffusive burning during this transient is discussed.
The laminar generalized Stokes layer and turbulent drag reduction
 MAURIZIO QUADRIO, PIERRE RICCO

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 16 November 2010, pp. 135157

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This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Both laminar and turbulent regimes for the streamwise flow are studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwisemodulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When the streamwise flow is turbulent, the laminar generalized Stokes layer solution describes well the spaceaveraged turbulent spanwise flow, provided that the phase speed of the waves is sufficiently different from the turbulent convection velocity, and that the time scale of the forcing is smaller than the life time of the nearwall turbulent structures. Under these conditions, the drag reduction is found to scale with the Stokes layer thickness, which renders the laminar solution instrumental for the analysis of the turbulent flow. A classification of the turbulent flow regimes induced by the waves is presented by comparing parameters related to the forcing conditions with the space and time scales of the turbulent flow.
Emergent order in rheoscopic swirls
 MICHAEL WILKINSON, VLAD BEZUGLYY, BERNHARD MEHLIG

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 29 November 2010, pp. 158187

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We consider the ordering of particles in a rheoscopic fluid (a suspension of microscopic rodlike particles) in a steady twodimensional flow, and discuss its consequences for the reflection of light. The ordering is described by an order parameter which is a nonoriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a onedimensional Schrödinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the longtime limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.
Cerebral water transport using multiplenetwork poroelastic theory: application to normal pressure hydrocephalus
 B. TULLY, Y. VENTIKOS

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 16 November 2010, pp. 188215

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The twentyfirst century is bearing witness to a drastic change in population demographics and diseases of old age, such as dementia, are placing an unprecedented burden on the global healthcare system. Normal pressure hydrocephalus may be the only curable form of dementia, yet its pathophysiology is paradoxical and a consistent treatment currently remains elusive. A novel application of multiplenetwork poroelastic theory (MPET) is proposed to investigate water transport in the cerebral environment. Specifically, MPET is modified to allow a detailed investigation of spatiotemporal transport of fluid between the cerebral blood, cerebrospinal fluid (CSF) and brain parenchyma across scales. This framework thus allows an exploration of hypotheses defining the initiation and progression of both acute and chronic hydrocephalus. Results show that a breakdown in the transport mechanisms between the arterial vascular network and interstitial space within the parenchyma may be a cause of accumulation of CSF in the ventricles. Specifically, there must be an increase in the compliance of the arteriole/capillary network, which may combine with a breakdown in the blood–CSF barrier to allow an increased flow from the arteriole/capillary blood to the CSF. The results of this study should prove useful to guide experimental exploration in areas that warrant further investigation and validation.
Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives
 JOHN F. BRADY

 Published online by Cambridge University Press:
 03 December 2010, pp. 216259

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Diffusiophoresis, the motion of a particle in response to an externally imposed concentration gradient of a solute species, is analysed from both the traditional coarsegrained macroscopic (i.e. continuum) perspective and from a finegrained micromechanical level in which the particle and the solute are treated on the same footing as Brownian particles dispersed in a solvent. It is shown that although the two approaches agree when the solute is much smaller in size than the phoretic particle and is present at very dilute concentrations, the micromechanical colloidal perspective relaxes these restrictions and applies to any size ratio and any concentration of solute. The different descriptions also provide different mechanical analyses of phoretic motion. At the continuum level the macroscopic hydrodynamic stress and interactive force with the solute sum to give zero total force, a condition for phoretic motion. At the colloidal level, the particle's motion is shown to have two contributions: (i) a ‘backflow’ contribution composed of the motion of the particle due to the solute chemical potential gradient force acting on it and a compensating fluid motion driven by the longrange hydrodynamic velocity disturbance caused by the chemical potential gradient force acting on all the solute particles and (ii) an indirect contribution arising from the mutual interparticle and Brownian forces on the solute and phoretic particle, that contribution being nonzero because the distribution of solute about the phoretic particle is driven out of equilibrium by the chemical potential gradient of the solute. At the colloidal level the forces acting on the phoretic particle – both the statistical or ‘thermodynamic’ chemical potential gradient and Brownian forces and the interparticle force – are balanced by the Stokes drag of the solvent to give the net phoretic velocity.
For a particle undergoing selfphoresis or autonomous motion, as can result from chemical reactions occurring asymmetrically on a particle surface, e.g. catalytic nanomotors, there is no imposed chemical potential gradient and the backflow contribution is absent. Only the indirect Brownian and interparticle forces contribution is responsible for the motion. The velocity of the particle resulting from this contribution can be written in terms of a mobility times the integral of the local ‘solute pressure’ – the solute concentration times the thermal energy – over the surface of contact between the particle and the solute. This was the approach taken by CórdovaFigueroa & Brady (Phys. Rev. Lett., vol. 100, 2008, 158303) in their analysis of selfpropulsion. It is shown that full hydrodynamic interactions can be incorporated into their analysis by a simple scale factor.
An expression for droplet evaporation incorporating thermal effects
 K. SEFIANE, R. BENNACER

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 14 January 2011, pp. 260271

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We propose a general theoretical expression for sessile droplets' evaporation, incorporating thermal effects related to the thermal resistance of the substrate and liquid properties. We develop an expression which accounts for thermal effects associated with evaporative cooling; the latter leads to a reduction in the rate of evaporation, which is not accounted for in the current theories, i.e. ‘isothermal diffusion theories’. The threshold for transition to a regime in which thermal effects start to be significant is identified through a dimensionless number which includes substrate and liquid properties as well as the kinetics of evaporation. The proposed theory is validated against experimental data in a very wide range of conditions and for a variety of systems. The developed expression extends the domain of use of diffusionbased models for droplet evaporation and accurately describes some aspects of the phenomenon which, to the best of our knowledge, are highlighted for the first time.
Triggering in the horizontal Rijke tube: nonnormality, transient growth and bypass transition
 MATTHEW P. JUNIPER

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 25 November 2010, pp. 272308

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With a sufficiently large impulse, a thermoacoustic system can reach selfsustained oscillations even when it is linearly stable, a process known as triggering. In this paper, a procedure is developed to find the lowest initial energy that can trigger selfsustained oscillations, as well as the corresponding initial state. This is known as the ‘most dangerous’ initial state. The procedure is based on adjoint looping of the nonlinear governing equations, combined with an optimization routine. It is developed for a simple model of a thermoacoustic system, the horizontal Rijke tube, and can be extended to more sophisticated thermoacoustic models. It is observed that the most dangerous initial state grows transiently towards an unstable periodic solution before growing to a stable periodic solution. The initial energy required to trigger these selfsustained oscillations is much lower than the energy of the oscillations themselves and slightly lower than the lowest energy on the unstable periodic solution. It is shown that this transient growth arises due to nonnormality of the governing equations. This is analogous to the sequence of events observed in bypass transition to turbulence in fluid mechanical systems and has the same underlying cause. The most dangerous initial state is calculated as a function of the heatrelease parameter. It is found that selfsustained oscillations can be reached over approximately half the linearly stable domain. Transient growth in real thermoacoustic systems is 105–106 times greater than that in this simple model. One practical conclusion is that, even in the linearly stable regime, it may take very little initial energy for a real thermoacoustic system to trigger to highamplitude selfsustained oscillations through the mechanism described in this paper.
Hydrodynamic bound states of a lowReynoldsnumber swimmer near a gap in a wall
 DARREN CROWDY, OPHIR SAMSON

 Published online by Cambridge University Press:
 16 November 2010, pp. 309335

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The motion of an organism swimming at low Reynolds number near an infinite straight wall with a finitelength gap is studied theoretically within the framework of a twodimensional model. The swimmer is modelled as a point singularity of the Stokes equations dependent on a single real parameter. A dynamical system governing the position and orientation of the model swimmer is derived in analytical form. The dynamical system is studied in detail and a bifurcation analysis performed. The analysis reveals, inter alia, the presence of stable equilibrium points in the gap region as well as Hopf bifurcations to periodic bound states. The reducedmodel system also exhibits a global gluing bifurcation in which two symmetric periodic orbits merge at a saddle point into symmetric ‘figureofeight’ bound states having more complex spatiotemporal structure. The additional effect of a background shear is also studied and is found to introduce new types of bound state. The analysis allows us to make theoretical predictions as to the possible behaviour of a lowReynoldsnumber swimmer near a gap in a wall. It offers insights into the use of gaps or orifices as possible control devices for such swimmers in confined environments.
Modulated waves in a periodically driven annular cavity
 H. M. BLACKBURN, J. M. LOPEZ

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 25 November 2010, pp. 336357

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Timeperiodic flows with spatiotemporal symmetry Z2 × O(2) – invariance in the spanwise direction generating the O(2) symmetry group and a halfperiodreflection symmetry in the streamwise direction generating a spatiotemporal Z2 symmetry group – are of interest largely because this is the symmetry group of periodic laminar twodimensional wakes of symmetric bodies. Such flows are the base states for various threedimensional instabilities; the periodically shedding twodimensional circular cylinder wake with threedimensional modes A and B being the generic example. However, it is not easy to physically realize the ideal flows owing to the presence of end effects and finite spanwise geometries. Flows past rings are sometimes advanced as providing a relevant idealization, but in fact these have symmetry group O(2) and only approach Z2 × O(2) symmetry in the infinite aspect ratio limit. The present work examines physically realizable periodically driven annular cavity flows that possess Z2 × O(2) spatiotemporal symmetry. The flows have three distinct codimension1 instabilities: two synchronous modes (A and B), and two manifestations of a quasiperiodic (QP) mode, either as modulated standing waves or modulated travelling waves. It is found that the curvature of the system can determine which of these modes is the first to become unstable with increasing Reynolds number, and that even in the nonlinear regime near onset of threedimensional instabilities the dynamics are dominated by mixed modes with complicated spatiotemporal structure. Supplementary movies illustrating the spatiotemporal dynamics are available at journals.cambridge.org/flm.
The shape distribution of splashform tektites predicted by numerical simulations of rotating fluid drops
 S. L. BUTLER, M. R. STAUFFER, G. SINHA, A. LILLY, R. J. SPITERI

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 14 January 2011, pp. 358368

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Splashform tektites are glassy rocks ranging in size from roughly 1 to 100 mm that are believed to have formed from the splash of silicate liquid after a large terrestrial impact from which they are strewn over thousands of kilometres. They are found in an array of shapes including spheres, oblate ellipsoids, dumbbells, rods and possibly fragments of tori. It has recently become appreciated that surface tension and centrifugal forces associated with the rotation of fluid droplets are the main factors determining the shapes of these tektites. In this contribution, we compare the shape distribution of 1163 measured splashform tektites with the results of the time evolution of a 3D numerical model of a rotating fluid drop with surface tension. We demonstrate that many aspects of the measured shape distribution can be explained by the results of the dynamical model.
Gravityinduced collisions of spherical drops covered with compressible surfactant
 ALEXANDER Z. ZINCHENKO, MICHAEL A. ROTHER, ROBERT H. DAVIS

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 14 January 2011, pp. 369402

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Gravityinduced collisions of two spherical drops covered with an insoluble surfactant at low Reynolds numbers are considered. Unlike in previous collision studies, the present work accounts for nonlinear coupling between the surfactant distribution and drop hydrodynamics by solving the full unsteady convective–diffusion equation for the surfactant transport. Our method includes highorder threedimensional multipole expansions for hydrodynamics and a Galerkintype approach for the surfactant transport with implicit marching. The efficiency of the algorithm allows for calculating thousands of trajectories to very close contact and determining the collision efficiency (related to the critical initial horizontal offset) by trial and error. The solution is valid for arbitrary surface Péclet (Pes) and Marangoni (Ma) numbers and sets limitations on approximations used in prior work for collisionefficiency calculations. Two limiting cases are observed: at small Pes or large Ma, the variation in surfactant coverage is small, and the results for the incompressible surfactant model are recovered, while for large Pes and small Ma, the collision efficiency approaches the cleaninterface value. For moderate dropsize ratios (radius ratio k ≤ 0.5), the results generally fall between these limits. At larger size ratios, however, the collision efficiency may even exceed the geometrical Smoluchowski limit for both drops and bubbles. Moreover, with even moderate redistribution of the surfactant, equalsized drops can move relative to one another and collide. These novel effects do not exist for clean drops or drops covered with an incompressible surfactant, and they are due to the nonlinear coupling between surfactant dynamics and flow. This surfactantenhanced coalescence takes place, for example, in a physical system of air bubbles in water if the surfactant surface concentration is dilute (Γ ≈ 1×10−9 mol m−2, much smaller than the typical maximumpacking value of 10−5−10−6 mol m−2).
New bounds on the sedimentation velocity for hard, charged and adhesive hardsphere colloids
 W. TODD GILLELAND, SALVATORE TORQUATO, WILLIAM B. RUSSEL

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 14 January 2011, pp. 403425

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The sedimentation velocity of colloidal dispersions is known from experiment and theory at dilute concentrations to be quite sensitive to the interparticle potential with attractions/repulsions increasing/decreasing the rate significantly at intermediate volume fractions. Since the differences necessarily disappear at close packing, this implies a substantial maximum in the rate for attractions. This paper describes the derivation of a robust upper bound on the velocity that reflects these trends quantitatively and motivates wider application of a simple theory formulated for hard spheres. The treatment pertains to sedimentation velocities slow enough that Brownian motion sustains an equilibrium microstructure without largescale inhomogeneities in density.
Acoustic/steady streaming from a motionless boundary and related phenomena: generalized treatment of the inner streaming and examples
 A. Y. REDNIKOV, S. S. SADHAL

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 14 January 2011, pp. 426462

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As originally realized by Nyborg (J. Acoust. Soc. Am., vol. 30, 1958, p. 329), the problem of the inner acoustic/steady streaming can be analysed in quite general terms. The inner streaming is the one that develops in the highfrequency limit in a thin Stokes (shearwave) layer at a boundary, in contrast to the outer streaming in the main bulk of the fluid. The analysis provides relevant innerstreaming characteristics through a given distribution of the acoustic amplitude along the boundary. Here such a generalized treatment is revisited for a motionless boundary. By working in terms of surface vectors, though in elementary notations, new compact and easytouse expressions are obtained. The most important ones are those for the effective (apparent) slip velocity at the boundary as seen from a perspective of the main bulk of the fluid, which is often the sole driving factor behind the outer streaming, and for the induced (acoustic) steady tangential stress on the boundary. As another novel development, nonadiabatic effects in the Stokes layer are taken into account, which become apparent through the fluctuating density and viscosity perturbations, and whose contribution into the streaming is often ignored in the literature. Some important particular cases, such as the axisymmetric case and the incompressible case, are emphasized. As far as the application of the derived general innerstreaming expressions is concerned, a few examples provided here involve a plane acoustic standing wave, which either grazes a wall parallel to its direction (convenient for the estimation of the nonadiabatic effects), or into which a small (compared to the acoustic wavelength) rigid sphere is placed. If there are simultaneously two such waves, outofphase and, say, in mutually orthogonal directions, a disk placed coplanarly with them will undergo a steady torque, which is calculated here as another example. Two further examples deal with translational highfrequency harmonic vibrations of particles relative to an incompressible fluid medium, viz. of a rigid oblate spheroid (along its axis) and of a sphere (arbitrary three dimensional). The latter can be a fixed rigid sphere, one free to rotate or even a (viscous) spherical drop, for which the outer streaming and the internal circulation are also considered.
Infrared Reynolds number dependency of the twodimensional inverse energy cascade
 ANDREAS VALLGREN

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 14 January 2011, pp. 463473

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Highresolution simulations of forced twodimensional turbulence reveal that the inverse cascade range is sensitive to an infrared Reynolds number, Reα = kf/kα, where kf is the forcing wavenumber and kα is a frictional wavenumber based on linear friction. In the limit of high Reα, the classic k−5/3 scaling is lost and we obtain steeper energy spectra. The sensitivity is traced to the formation of vortices in the inverse energy cascade range. Thus, it is hypothesized that the dual limit Reα → ∞ and Reν = kd/kf → ∞, where kd is the smallscale dissipation wavenumber, will lead to a steeper energy spectrum than k−5/3 in the inverse energy cascade range. It is also found that the inverse energy cascade is maintained by nonlocal triad interactions.
Shallowwater sloshing in vessels undergoing prescribed rigidbody motion in three dimensions
 HAMID ALEMI ARDAKANI, THOMAS J. BRIDGES

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 14 January 2011, pp. 474519

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New shallowwater equations (SWEs), for sloshing in three dimensions (two horizontal and one vertical) in a vessel which is undergoing rigidbody motion in 3space, are derived. The rigidbody motion of the vessel (roll–pitch–yaw and/or surge–sway–heave) is modelled exactly and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallowwater models. The model also extends to three dimensions the essence of the Penney–Price–Taylor theory for the highest standing wave. The surface SWEs are simulated using a splitstep alternating direction implicit finitedifference scheme. Numerical experiments are reported, including comparisons with existing results in the literature, and simulations with vessels undergoing full 3D rotations.
Planar channel flow in Braginskii magnetohydrodynamics
 PAUL J. DELLAR

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 14 January 2011, pp. 520543

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Braginskii magnetohydrodynamics (MHD) is a singlefluid description of largescale motions in strongly magnetised plasmas. The ion Larmor radius in these plasmas is much shorter than the mean free path between collisions, so momentum transport across magnetic field lines is strongly suppressed. The relation between the strain rate and the viscous stress becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field. We present an analytical study of the steady planar flow across an imposed uniform magnetic field driven by a uniform pressure gradient along a straight channel, the configuration known as Hartmann flow, in Braginskii MHD. The global momentum balance cannot be satisfied by just the parallel viscous stress, so we include the viscous stress perpendicular to magnetic field lines as well. The ratio of perpendicular to parallel viscosities is the key small parameter in our analysis. When another parameter, the Hartmann number, is large the flow is uniform across most of the channel, with boundary layers on either wall that are modifications of the Hartmann layers in standard isotropic MHD. However, the Hartmann layer solution predicts an infinite current and infinite shear at the wall, consistent with a local series solution of the underlying differential equation that is valid for all Hartmann numbers. These singularities are resolved by inner boundary layers whose width scales as the threequarters power of the viscosity ratio, while the maximum velocity scales as the inverse onequarter power of the viscosity ratio. The inner wall layers fit between the Hartmann layers, if present, and the walls. The solution thus does not approach a limit as the viscosity ratio tends to zero. Essential features of the solution, such as the maximum current and maximum velocity, are determined by the size of the viscosity ratio, which is the regularising small parameter.