A note on Stokes' problem in dense granular media using the μ(I)\mu(I)--rheology

The classical Stokes' problem describing the fluid motion due to a steadily moving infinite wall is revisited in the context of dense granular flows of mono-dispersed beads using the recently proposed $\mu(I)$--rheology. In Newtonian fluids, molecular diffusion brings about a self-similar velocity profile and the boundary layer in which the fluid motion takes place increases indefinitely with time $t$ as $\sqrt{\nu t}$, where $\nu$ is the kinematic viscosity. For a dense granular visco-plastic liquid, it is shown that the local shear stress, when properly rescaled, exhibits self-similar behaviour at short-time scales and it then rapidly evolves towards a steady-state solution. The resulting shear layer increases in thickness as $\sqrt{\nu_g t}$ analogous to a Newtonian fluid where $\nu_g$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular media such as grain diameter $d$, density $\rho$ and friction coefficients but also on the applied pressure $p_w$ at the moving wall and the solid fraction $\phi$ (constant). In addition, the $\mu(I)$--rheology indicates that this growth continues until reaching the steady-state boundary layer thickness $\delta_s = \beta_w (p_w/\phi \rho g )$, independent of the grain size, at about a finite time proportional to $\beta_w^2 (p_w/\rho g d)^{3/2} \sqrt{d/g}$, where $g$ is the acceleration due to gravity and $\beta_w = (\tau_w - \tau_s)/\tau_s$ is the relative surplus of the steady-state wall shear-stress $\tau_w$ over the critical wall shear stress $\tau_s$ (yield stress) that is needed to bring the granular media into motion... (see article for a complete abstract).Comment: in press (Journal of Fluid Mechanics

Extended Squire's transformation and its consequences for transient growth in a confined shear flow

International audienceThe classical Squire transformation is extended to the entire eigenfunction structure of both Orr-Sommerfeld and Squire modes. For arbitrary Reynolds numbers Re, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number Re-2D. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times t with t/Re finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241-260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209-238) for arbitrary alpha Re, where alpha is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr-Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to 1 + (beta Re/Re-2D)(2) g where beta is the spanwise wavenumber and g is a known expression. For example, when the leading eigenmode is an Orr Sommerfeld mode, it is given by the product of respective gains from the 2D On mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient

Vortex-forced-oscillations of thin flexible plates

Fluid-structure interaction of a slender flexible cantilevered-element and vortices in an otherwise steady flow is considered here by investigating the dynamics of thin low-density polyethylene sheets subject to periodic forcing due to B\'enard-K\`arm\`an vortices in a $2$-meter long narrow water channel. The vortex shedding frequency $f_v$ is varied via the mean flow speed $U_0$ and the cylinder diameter $d_0 = 10$, $20$ and $40$ mm, while the structures' bending resistance is properly controlled via its Young's modulus $E$, thickness $e_b$ and length $L_b$. Thereby, it is first shown that the non-dimensional time-averaged sheet deflection, namely, the sheet \textit{reconfiguration} $\bar{h}_b/L_b \sim C_y^{\mathcal{V}/2}$ and also, the time-averaged \textit{drag force} $\bar{F}_d \propto U_0^{2+\mathcal{V}}$, where $\mathcal{V} \leq 0$ is the well-known Vogel number for flexible structures in a steady flow and $C_y = 12 \left({C_d \frac{1}{2} \rho U_0^2}/{E}\right) \left({ L_b^3/}{e_b^3} \right)$ is the Cauchy number comparing the relative magnitude of the profile drag force over a typical elastic restoring force, if the sheet were rigid. Measurements and a simple model based on torsional-spring-mounted flat plate illustrate that the tip amplitude $\delta_b$ is not only directly proportional to the characteristic size of the eddies, say $d_v$, but also to the sheet mechanical properties and the vortex flow characteristics such that $\delta_b/d_v \sim C_y^{(1+\mathcal{V})/2} \sqrt{U_0/f_v d_v}$. Furthermore, a rich phenomenology of structural dynamics including vortex-forced-vibration, lock-in with the sheet natural frequency, flow-induced vibration due to the sheet wake, multiple-frequency and modal response is reported

Vortices catapult droplets in atomization

International audienceA droplet ejection mechanism in planar two-phase mixing layers is examined. Any disturbance on the gas-liquid interface grows into a Kelvin-Helmholtz wave, and the wave crest forms a thin liquid film that flaps as the wave grows downstream. Increasing the gas speed, it is observed that the film breaks up into droplets which are eventually thrown into the gas stream at large angles. In a flow where most of the momentum is in the horizontal direction, it is surprising to observe these large ejection angles. Our experiments and simulations show that a recirculation region grows downstream of the wave and leads to vortex shedding similar to the wake of a backward-facing step. The ejection mechanism results from the interaction between the liquid film and the vortex shedding sequence: a recirculation zone appears in the wake of the wave and a liquid film emerges from the wave crest; the recirculation region detaches into a vortex and the gas flow over the wave momentarily reattaches due to the departure of the vortex; this reattached flow pushes the liquid film down; by now, a new recirculation vortex is being created in the wake of the wave--just where the liquid film is now located; the liquid film is blown up from below by the newly formed recirculation vortex in a manner similar to a bag-breakup event; the resulting droplets are catapulted by the recirculation vortex

LA CROISSANCE TRANSITOIRE DANS LES ÉCOULEMENTS DE RAYLEIGH-BÉNARD-POISEUILLE/COUETTE

OPTIMAL GROWTH MECHANISMS IN WALL-BOUNDED SHEAR FLOWS, IN PARTICULAR, PLANE COUETTE AND PLANE POISEUILLE FLOW, WITH AND WITHOUT A DESTABILIZING WALL-NORMAL TEMPERATURE GRADIENT ARE STUDIED EXTENSIVELY. IN THE CASE WITH A CROSS-STREAM TEMPERATURE GRADIENT IN A BOUSSINESQ FLUID, A COMPREHENSIVE NON-MODAL STABILITY ANALYSIS IS PERFORMED OVER VARIOUS REYNOLDS, RAYLEIGH AND PRANDTL NUMBERS. THE SCALING LAWS PERTAINING TO TRANSIENT GROWTH IN PURE SHEAR FLOWS ARE SHOWN TO HOLD EVEN IN THE PRESENCE OF A DESTABILIZING TEMPERATURE GRADIENT. THE LIFT-UP EFFECT REMAINS THE PREDOMINANT TRANSIENT GROWTH MECHANISM. THE CLASSICAL INVISCID LIFT-UP MECHANISM CHARACTERIZES THE SHORT-TIME BEHAVIOR WHEREAS THE RAYLEIGH-BÉNARD EIGENMODE WITHOUT ITS STREAMWISE VELOCITY COMPONENT CHARACTERIZES THE LONG-TIME BEHAVIOR. THE SQUIRE TRANSFORMATION IS EXTENDED TO PROVIDE NEW INSIGHTS ON THE OPTIMAL GROWTH OF ARBITRARY 3D DISTURBANCES IN PARALLEL SHEAR FLOWS BOUNDED IN THE CROSS-STREAM DIRECTION. IT ALSO PERMITS TO DEMONSTRATE THAT THE LONG-TIME OPTIMAL GROWTH FOR PERTURBATIONS OF ARBITRARY WAVENUMBERS MAY BE DECOMPOSED AS A PRODUCT OF THE RESPECTIVE GAINS ARISING FROM THE 2D ORR-MECHANISM AND THE LIFT-UP MECHANISM. THIS ASYMPTOTIC SOLUTION IS SHOWN TO DESCRIBE THE LONG-TIME AND EVEN THE INTERMEDIATE-TIME DYNAMICS OF THE OPTIMAL DISTURBANCES AND PROVIDES A GOOD ESTIMATE OF THE MAXIMUM OPTIMAL GAIN AT ALL TIME.LES MÉCANISMES DE CROISSANCE OPTIMALE DANS DES ÉCOULEMENTS DE CISAILLEMENT CONFINES, EN PARTICULIER LES ÉCOULEMENTS DE COUETTE PLAN ET POISEUILLE PLAN, LORSQU'ILS SONT SOUMIS OU NON À UN GRADIENT DE TEMPÉRATURE DÉSTABILISANT NORMAL À LA PAROI SONT ÉTUDIÉS EN DÉTAIL. DANS LE CAS D'UN FLUIDE DE BOUSSINESQ SOUMIS À UN GRADIENT DE TEMPÉRATURE TRANSVERSE, UNE ANALYSE EXHAUSTIVE DE STABILITÉ NON MODALE EST EFFECTUÉE POUR DIFFÉRENTS NOMBRES DE REYNOLDS, DE RAYLEIGH ET DE PRANDTL. ON MONTRE QUE LES LOIS D'ÉCHELLE RELATIVES À LA CROISSANCE TRANSITOIRE DANS DES ÉCOULEMENTS CISAILLES PURS SONT ROBUSTES, Y COMPRIS EN PRÉSENCE D'UN GRADIENT DE TEMPÉRATURE DÉSTABILISANT. L'EFFET DE ''LIFT-UP" RESTE LE MÉCANISME PRÉDOMINANT DE CROISSANCE TRANSITOIRE. LE MÉCANISME DE ''LIFT-UP" NON VISQUEUX CLASSIQUE CARACTÉRISE LE COMPORTEMENT AUX TEMPS COURTS ALORS QUE LE MODE PROPRE DE RAYLEIGH-BÉNARD SANS SA COMPOSANTE DE VITESSE LONGITUDINALE CARACTÉRISE LE COMPORTEMENT AUX TEMPS LONGS. LA COURBE DE GAIN OPTIMAL EST AINSI DÉCRITE ET INTERPRÉTÉE ENTIÈREMENT. DANS LE CAS D'ÉCOULEMENTS CISAILLES PURS, LE RÔLE DE TRANSFORMATION DE SQUIRE EST ÉTENDUE À LA CROISSANCE TRANSITOIRE OPTIMALE D'UNE PERTURBATION ARBITRAIRE 3D DANS LE CAS D'ÉCOULEMENTS CISAILLES PARALLÈLES D'EXTENSION TRANSVERSE FINIE. CELA PERMET AUSSI DE DÉMONTRER QUE LES CROISSANCES OPTIMALES AUX TEMPS LONGS POUR DES PERTURBATIONS DE NOMBRE D'ONDE ARBITRAIRES PEUVENT ÊTRE DÉCOMPOSÉES COMME UN PRODUIT DES GAINS RESPECTIFS RÉSULTANT DU MÉCANISME D'ORR 2D ET DU MÉCANISME DE " LIFT-UP "

Nurse Selection Project - digest of work

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Inertial Landau-Levich problem : sheets, films and drops on a rotating drum

International audienc

Transient growth in Rayleigh-Benard-Poiseuille/Couette convection

International audienceAn investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth G(max) of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbers Re, the supremum of G(max) is always attained for streamwise-uniform perturbations (or nearly streamwise-uniform perturbations, in the case of plane Couette flow) which produce large streamwise streaks and Rayleigh-Beacutenard convection rolls (RB). The optimal growth curves retain the same large-Reynolds-number scaling as in pure shear flow. A 3D vector model of the governing equations demonstrates that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this mechanism is secondary and negligible. The optimal input for the largest long-time response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the RB eigenmode without its streamwise velocity component. These short-time and long-time responses depict, to leading order, the optimal transient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. Over a general class of norms that can be considered as growth functions, the results remain qualitatively similar, for example, the dominant adjoint eigenmode still approximates the maximum optimal response. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4704642