Thin Viscous Layer Research Articles

Molecular dispersion in pre-turbulent and sustained turbulent flow of carbon dioxide

The average dispersion coefficients, Da¯, of two small molecules (acetonitrile and coronene) were measured under laminar, transient, and sustained turbulent flow regimes along fused silica open tubular capillary (OTC) columns (180 μm inner diameter by 20 m length). Carbon dioxide was used as the mobile phase at room temperature (296 K) and at average pressures in the range from 1500 to 2700 psi. The Reynolds number (Re) was increased from 600 to 5000. The measurement of Da¯ is based on the observed plate height of the non-retained analytes as a function of the applied Reynolds number. Da¯ values are directly estimated from the best fit of the general Golay HETP equation to the experimental plate height curves.The experimental data revealed that under a pre-turbulent flow regime (Re < 2000), Da¯ is 2–6 times larger (3.5 × 10−4 cm2/s) than the bulk diffusion coefficients Dm of the analyte (1.6 × 10−4 and 5.8 × 10−5 cm2/s for acetonitrile and coronene, respectively). This result was explained by the random formation of decaying or vanishing turbulent puffs under pre-turbulent flow regime. Yet, the peak width remains controlled exclusively by the slow mass transfer in the mobile phase across the inner diameter (i.d.) of the OTC. Under sustained turbulent flow regime (Re > 2500), Da¯ is about four to five orders of magnitude larger than Dm. The experimental data slightly overestimated the turbulent dispersion coefficients predicted by Flint-Eisenklam model (Da¯=4 cm2/s). The discrepancy is explained by the approximate nature of the general Golay equation, which assumes that Da¯ is strictly uniform across the entire i.d. of the OTC. In fact, both the viscous and buffer wall layers, in which viscous effects dominate inertial effects, cannot be considered as fully developed turbulent regions. Remarkably, the mass transfer mechanism in OTC under sustained turbulent flow regime is not only controlled by longitudinal dispersion but also by a slow mass transfer in the mobile phase across the thick buffer layer and the thin viscous layer. Altogether, these layers occupy as much as 35% of the OTC volume at Re = 4000. From a theoretical viewpoint, the general Golay HETP equation is only an approximate model which should be refined based on the actual profile of the analyte dispersion coefficient across the OTC i.d. In practice, the measured plate height of non-retained analytes under sustained turbulent flow of carbon dioxide are two orders of magnitude smaller than those expected under hypothetical laminar flow regime.

How the presence of a salt décollement in the sedimentary cover influences the behavior of subsalt thrusts in fold-and-thrust belts

We conducted a series of analogue experiments on shortening of a brittle cover (dry sand) above a deep, thin, frictional detachment (glass microbeads). In some experiments, the cover was homogeneous, entirely brittle. In others, there was a thin viscous silicone layer (representing salt) embedded at mid height into the cover, and initially located in the foreland of the fold-and-thrust belt. Our goal was to determine whether or not the presence of such a décollement in the cover could have an impact on the mechanics and kinematics of the underlying subsalt thrusts. Results confirm that, once the front of the foldbelt reached the hinterland salt pinch out, the kinematics of the deeper belt changed drastically: its front stopped propagating forward, and most of the subsequent shortening was accommodated by a larger-than-normal slip along the foremost and youngest deep thrust, while, above the salt décollement, the deformation front propagated very fast, creating a very low surface slope. We hypothesize that it is the gentle sub-critical surface slope associated with rocksalt’s low viscosity that prevents the build-up of an overall surface slope steep enough to allow the underlying, deep foldbelt to continue propagating forward. Finally, one experiment in which only one half of the width of the model comprised an interbedded viscous décollement has shown that the kinematics of the deep thrust was affected even in the adjacent salt-free domain.

Symmetry in the problem of wave modes of thin viscous liquid layer flow

The equations in conservative form for modelling nonlinear waves on a liquid film flowing down a vertical plane have been investigated. It has been found out that the equations with boundary conditions are invariant under parity transformation in the extended computational domain. The steady-state travelling solutions are numerically shown to have the detected symmetry for moderate Reynolds numbers. The use of this symmetry for the numerical solution of the problem by Galerkin methods significantly increases the efficiency of calculations.

Precessing cylinders at the second and third resonance: turbulence controlled by geostrophic flow.

We investigate, via both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in fluid-filled circular cylinders that rotate rapidly about their symmetry axis and precess about a different axis and that are marked by radius-height aspect ratios Γ=1.045945 and Γ=1.611089. At these radius-height aspect ratios, the Poincaré force resonates directly with the two special inertial modes that have the simplest vertical structure. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers, showing quantitative agreement with the result of direct nonlinear numerical simulation. Our numerical simulation makes use of a finite-element method with the three-dimensional tetrahedralization of a cylindrical cavity that allows the construction of dense nodes in the vicinity of the bounding surface of the cavity for resolving the thin viscous boundary layer. It is found that axisymmetric geostrophic flow in the alternating eastward and westward direction can be generated and maintained by nonlinear and viscous effects in the viscous boundary layer. It is also found that, when the precessing rate is moderate and, consequently, the geostrophic flow is weak, nonlinear interaction between the resonant inertial mode and the nonesonant inertial modes driven by the Poincaré force and the boundary-layer influx leads to strongly turbulent flow with irregular temporal-spatial fluctuation. When the cylinders are strongly precessing such that the geostrophic flow becomes predominant, however, the effect of the geostrophic flow controls/stabilizes its nonlinear dynamics, leading to weakly turbulent flow that can be largely described by a dominant quasisteady geostrophic component and a weak nonaxisymmetric component localized in the region where the geostrophic flow is weak.

Boundary-Layer Effects on Acoustic Transmission Through Narrow Slit Cavities.

We explore the slit-width dependence of the resonant transmission of sound in air through both a slit array formed of aluminum slats and a single open-ended slit cavity in an aluminum plate. Our experimental results accord well with Lord Rayleigh's theory concerning how thin viscous and thermal boundary layers at a slit's walls affect the acoustic wave across the whole slit cavity. By measuring accurately the frequencies of the Fabry-Perot-like cavity resonances, we find a significant 5% reduction in the effective speed of sound through the slits when an individual viscous boundary layer occupies only 5% of the total slit width. Importantly, this effect is true for any airborne slit cavity, with the reduction being achieved despite the slit width being on a far larger scale than an individual boundary layer's thickness. This work demonstrates that the recent prevalent loss-free treatment of narrow slit cavities within acoustic metamaterials is unrealistic.

Big Cells Cleave as Fast as Small Ones: the Physics of Cytokinesis

Big Cells Cleave as Fast as Small Ones: the Physics of Cytokinesis

Features of the behavior of a liquid film heated from above and placed over deep liquid

The features of rupture occurring in a liquid film lying on deep liquid are studied. A thermal load applied to the film free surface leads to its deformation and rupture. The process is studied using a mathematical model describing the motion of a thin viscous nonisothermal liquid layer. The model is based on the two-dimensional Navier-Stokes equations. To solve them, the boundary conditions on the film-gas and film-liquid interfaces are written in explicit form. The effect of the thermal load on the film rupture is analyzed numerically depending on the Marangoni number and the stresses on the film-liquid interface. The results of solving model problems are presented.

Small Divisor Problems in Fluid Mechanics

Several small divisor problems occuring in Fluid Mechanics are presented. Two of them come from water waves: 3D periodic travelling gravity waves, and 2D standing gravity waves. The last example comes from quasipatterns observed for thin viscous horizontal fluid layers periodically vertically shaked (Faraday type experiment).

Including fluid shear viscosity in a structural acoustic finite element model using a scalar fluid representation

An approximate boundary condition is developed in this paper to model fluid shear viscosity at boundaries of coupled fluid–structure system. The effect of shear viscosity is approximated by a correction term to the inviscid boundary condition, written in terms of second order in-plane derivatives of pressure. Both thin and thick viscous boundary layer approximations are formulated; the latter subsumes the former. These approximations are used to develop a variational formation, upon which a viscous finite element method (FEM) model is based, requiring only minor modifications to the boundary integral contributions of an existing inviscid FEM model. Since this FEM formulation has only one degree of freedom for pressure, it holds a great computational advantage over the conventional viscous FEM formulation which requires discretization of the full set of linearized Navier–Stokes equations. The results from thick viscous boundary layer approximation are found to be in good agreement with the prediction from a Navier–Stokes model. When applicable, thin viscous boundary layer approximation also gives accurate results with computational simplicity compared to the thick boundary layer formulation. Direct comparison of simulation results using the boundary layer approximations and a full, linearized Navier–Stokes model are made and used to evaluate the accuracy of the approximate technique. Guidelines are given for the parameter ranges over which the accurate application of the thick and thin boundary approximations can be used for a fluid–structure interaction problem.

Quasipatterns in a parametrically forced horizontal fluid film

We shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω, only one harmonic), to reproduce the Faraday experiment and using the system derived in Rojas et al. (2010) [34] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω/2, and the mode with frequency Ω. Moreover, at criticality the corresponding wave lengths kc and kc′ are such that if we define the family of 2q equally spaced (horizontal) wave vectors kj on the circle of radius kc, then kj+kl=kn′, with |kj|=|kl|=kc,|kn′|=kc′.It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequency Ω/2) having a spatially quasiperiodic pattern if q≥4. Restricting our study to solutions invariant under rotations of angle2π/q, gives a kernel of dimension 4.In the spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω/2) quasipatterns. In particular, we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angleπ/qof the pattern.

Turbulence dynamics near a turbulent/non-turbulent interface

AbstractThe characteristics of the boundary layer separating a turbulence region from an irrotational (or non-turbulent) flow region are investigated using rapid distortion theory (RDT). The turbulence region is approximated as homogeneous and isotropic far away from the bounding turbulent/non-turbulent (T/NT) interface, which is assumed to remain approximately flat. Inviscid effects resulting from the continuity of the normal velocity and pressure at the interface, in addition to viscous effects resulting from the continuity of the tangential velocity and shear stress, are taken into account by considering a sudden insertion of the T/NT interface, in the absence of mean shear. Profiles of the velocity variances, turbulent kinetic energy (TKE), viscous dissipation rate ($\varepsilon $), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer),$\varepsilon $decays as${z}^{\ensuremath{-} 6} $, where$z$is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and$\varepsilon $display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness$\delta $of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that$\delta \ensuremath{\sim} \eta $(where$\eta $is the Kolmogorov microscale), which is consistent with the scaling law identified in a very recent DNS study for shear-free T/NT interfaces.

The 3D circulation and cyclone-anticyclone asymmetry in the shallow layers of viscous rotating fluid

A modified von Kármán problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flow is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the numerical model is given.

The effect of an electrostatic field on the rupture of a thin viscous static film by van der Waals attractions

This research examines rupture phenomena of a horizontal static thin viscous layer on a solid plate under an electrostatic field generating from a charged foil above the film. The dynamics of the electrified liquid film is formulated to derive a long-wave evolution equation of local film thickness. It determines two-dimensional nonlinear behavior of the film subject to surface tension, viscous, electrically induced forces, and van der Waals attractions. Linear stability analysis is used to obtain the maximum growth rate of a periodic disturbance and its corresponding wavenumber. To see the development of film rupture the strongly nonlinear partial differential equation is numerically solved for the unlimited or limited foil length as part of an initial-value problem with spatially periodic boundary conditions. The stronger electric forces make the thin layer more unstable and speed up its rupture.

Viscous coupling of shear-free turbulence across nearly flat fluid interfaces

The interactions between shear-free turbulence in two regions (denoted as + and − on either side of a nearly flat horizontal interface are shown here to be controlled by several mechanisms, which depend on the magnitudes of the ratios of the densities, ρ+/ρ−, and kinematic viscosities of the fluids, μ+/μ−, and the root mean square (r.m.s.) velocities of the turbulence, u0+/u0−, above and below the interface. This study focuses on gas–liquid interfaces so that ρ+/ρ− ≪ 1 and also on where turbulence is generated either above or below the interface so that u0+/u0− is either very large or very small. It is assumed that vertical buoyancy forces across the interface are much larger than internal forces so that the interface is nearly flat, and coupling between turbulence on either side of the interface is determined by viscous stresses. A formal linearized rapid-distortion analysis with viscous effects is developed by extending the previous study by Hunt &amp; Graham (J. Fluid Mech., vol. 84, 1978, pp. 209–235) of shear-free turbulence near rigid plane boundaries. The physical processes accounted for in our model include both the blocking effect of the interface on normal components of the turbulence and the viscous coupling of the horizontal field across thin interfacial viscous boundary layers. The horizontal divergence in the perturbation velocity field in the viscous layer drives weak inviscid irrotational velocity fluctuations outside the viscous boundary layers in a mechanism analogous to Ekman pumping. The analysis shows the following. (i) The blocking effects are similar to those near rigid boundaries on each side of the interface, but through the action of the thin viscous layers above and below the interface, the horizontal and vertical velocity components differ from those near a rigid surface and are correlated or anti-correlated respectively. (ii) Because of the growth of the viscous layers on either side of the interface, the ratio uI/u0, where uI is the r.m.s. of the interfacial velocity fluctuations and u0 the r.m.s. of the homogeneous turbulence far from the interface, does not vary with time. If the turbulence is driven in the lower layer with ρ+/ρ− ≪ 1 and u0+/u0− ≪ 1, then uI/u0− ~ 1 when Re (=u0−L−/ν−) ≫ 1 and R = (ρ−/ρ+)(v−/v+)1/2 ≫ 1. If the turbulence is driven in the upper layer with ρ+/ρ− ≪ 1 and u0+/u0− ≫ 1, then uI/u0+ ~ 1/(1 + R). (iii) Nonlinear effects become significant over periods greater than Lagrangian time scales. When turbulence is generated in the lower layer, and the Reynolds number is high enough, motions in the upper viscous layer are turbulent. The horizontal vorticity tends to decrease, and the vertical vorticity of the eddies dominates their asymptotic structure. When turbulence is generated in the upper layer, and the Reynolds number is less than about 106–107, the fluctuations in the viscous layer do not become turbulent. Nonlinear processes at the interface increase the ratio uI/u0+ for sheared or shear-free turbulence in the gas above its linear value of uI/u0+ ~ 1/(1 + R) to (ρ+/ρ−)1/2 ~ 1/30 for air–water interfaces. This estimate agrees with the direct numerical simulation results from Lombardi, De Angelis &amp; Bannerjee (Phys. Fluids, vol. 8, no. 6, 1996, pp. 1643–1665). Because the linear viscous–inertial coupling mechanism is still significant, the eddy motions on either side of the interface have a similar horizontal structure, although their vertical structure differs.

Vortex patterns in quasi-two-dimensional flows of a viscous rotating fluid

A modified von Karman problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flows is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the model problem is given.

粘着層付き薄膜を有する制振機構の提案

Passive damping augmentation is one of attractive methods for vibration suppression to various kinds of structures because it is definitely stable and generally simple. Using viscous adhesive indicated a remarkable effect to the vibration suppression in the practical application to a satellite. In this paper, mathematical model of thin viscous adhesive layer using non-linear elements has been proposed. In this model, the characteristics of elements are correlated with non-linear internal phenomena of polymer. The simulation results using this proposal model are good agreement with experimental ones.

Morphological instabilities of stressed and reactive geological interfaces

Interfaces between contacting rocks of the Earth's crust are shown to be unstable and corrugating and develop roughness at various scales when submitted to nonhydrostatic stress. This instability may occur in various geological settings as long as a coherent deformation of the interface is allowed and the bodies that the interface separates have different material properties (i.e., viscosity, density, or elastic moduli). Relevant examples include fault planes, dissolution interfaces, or grain boundaries. Performing a two‐dimensional linear stability analysis, we consider two cases: one solid in contact with a viscous layer and two solids separated by a thin viscous layer. In both cases either shear and/or normal loads are imposed on the interface and thermodynamical conditions for the initiation of roughening are established. Applied on several geological systems such as grain contacts and fault planes, we propose that our analysis can explain how complex patterns may emerge at rock‐rock interfaces. Finally, we provide an analysis of the evolution of the static friction coefficient along sheared interfaces. The evolution is shown to depend solely on Poisson's ratio of the solid and the ratio of the shear and compressional stresses along the interface.

Thin shear layers – The key to turbulence structure?

Sharply sheared interfaces determine the structure of turbulent motions both on large and small scales, as recent experiments and simulations have demonstrated. Shear layers form along contours defined by very large gradients in turbulence intensity and/or large gradients in the large scale velocity field. Inhomogeneous eddy motions adjacent to these layers stretch and distort the larger scale vorticity so as to counter the tendency of the interface to thicken by small scale eddy diffusion. These concepts are applied to the structure of high Reynolds number turbulence by focusing on the dynamics of the flow within and near the shear layers that are observed to exist between large eddies. The numerical simulations of Ishihara et al. (2009) show that on the edges of the layers and in their interiors thin viscous layers form, with thickness approximately equal to the Taylor microscale. Intermittent small-scale vortices within these viscous layers are stretched and amplified, as their thickness reduces to a limiting value determined by viscous diffusion, which is equal to the Kolmogorov micro length scale. But the root-mean -square values of vorticity and velocity, and also their flatness factors, are given by statistical averaging across these layers, taking into account the degree to which the thin layers are sufficiently convoluted to be ‘space-filling’. Through the blocking of the external scale eddies outside the layers as they impinge onto the interfaces, upscale and downscale energy transfer processes which are of comparable magnitude lead to distorted inertial-range motions with a wide range of length scales and characteristic self-similar power-law spectra. Here the skewness of the velocity derivatives becomes negative. The upscale motions are essential for maintaining the large structures and the thin shear layers between them, as two-dimensional experiments demonstrate. A significant implication of this ‘interface dynamics’ mechanism is that small scale turbulence is produced as much by larger eddy structures, as by interactions between progressively smaller, and more numerous, self-similar eddies – the ‘cascade’ mechanism proposed by L.F. Richardson. The latter mechanism, which is dynamically equivalent to the Kolmogorov–Obukhov statistical-physics models for homogeneously distributed fluctuations, implies that turbulence structure evolves more slowly than is observed. By contrast the model presented here for the statistics and dynamics of high Reynolds number turbulence is based on a quasi-deterministic zonal analysis within and adjacent to intermittently distributed thin shear layers, which occupy a finite fraction of the flow and cause the eddy motions to have different ranges of length scales in separate parts of the local flow fields. The structure does not change qualitatively as the Reynolds number varies, but is significantly changed if the turbulence is forced at small as well as large scales.

Energy dissipation in microfluidic beam resonators

The fluid–structure interaction of resonating microcantilevers immersed in fluid has been widely studied and is a cornerstone in nanomechanical sensor development. In many applications, fluid damping imposes severe limitations by strongly degrading the signal-to-noise ratio of measurements. Recently, Burget al. (Nature, vol. 446, 2007, pp. 1066–1069) proposed an alternative type of microcantilever device whereby a microfluidic channel was embedded inside the cantilever with vacuum outside. Remarkably, it was observed that energy dissipation in these systems was almost identical when air or liquid was passed through the channel and was 4 orders of magnitude lower than that in conventional microcantilever systems. Here, we study the fluid dynamics of these devices and present a rigorous theoretical model corroborated by experimental measurements to explain these observations. In so doing, we elucidate the dominant physical mechanisms giving rise to the unique features of these devices. Significantly, it is found that energy dissipation is not a monotonic function of fluid viscosity, but exhibits oscillatory behaviour, as fluid viscosity is increased/decreased. In the regime of low viscosity, inertia dominates the fluid motion inside the cantilever, resulting in thin viscous boundary layers – this leads to an increase in energy dissipation with increasing viscosity. In the high-viscosity regime, the boundary layers on all surfaces merge, leading to a decrease in dissipation with increasing viscosity. Effects of fluid compressibility also become significant in this latter regime and lead to rich flow behaviour. A direct consequence of these findings is that miniaturization does not necessarily result in degradation in the quality factor, which may indeed be enhanced. This highly desirable feature is unprecedented in current nanomechanical devices and permits direct miniaturization to enhance sensitivity to environmental changes, such as mass variations, in liquid.

The Ekman Layer and Why Tea Leaves Go to the Center of the Cup

Consider a transparent, cylindrical container filled with water and sitting in the center of a record player turntable. When the turntable is started suddenly, the container rotates with the turntable, but the bulk of the fluid initially remains at rest. A thin (∼1 mm) viscous boundary layer (Ekman layer) forms almost immediately at the bottom and top (if there is a lid) of the fluid. Here we describe a laboratory or demonstration exercise in which we use dye crystals1 and fine particles2 as tracers to study the flow in the Ekman layer. We also give a general method for using fine particles to measure the time it takes for the bulk of the fluid to rotate with the turntable (spin-up time) for different rates of rotation and fluid depths.