Viscous Sheet Research Articles

Instability study of a swirling annular liquid sheet of polymer produced by air-blast atomization

A temporal stability analysis was carried out to model the atomization of a swirling viscous annular liquid sheet emanating from an air-blast atomizer subject to inner and outer inviscid swirling air streams. The dimensionless dispersion equation that governs the instability of a viscous annular liquid sheet under swirling air streams was obtained. Numerical solutions to the dispersion equation under a wide range of flow conditions were obtained to investigate the effect of the liquid and gas flow on the maximum growth rate and its corresponding unstable wave number. The theoretical behaviour predicted by the dispersion diagrams was compared with the experimental results obtained by the same authors in previous works from the atomization of alginate solution using an air-blast atomizer. It was found that the instability model proposed justify the experimental effects found for the atomization of the fluid and under the work range for alginate flow rate and viscosity and air flow rate.

A study on the aerodynamic instability of attenuating liquid sheets

We describe the characteristics of a radially spreading unstable liquid sheet in quiescent air via optical measurement techniques and linear instability theory. A high speed CCD camera system and a complimentary laser refraction method were employed to measure the intact sheet diameter, unstable wave lengths, wave speed, wave frequency spectrum and spatial wave growth rates. Linear instability models for thinning, viscous and inviscid liquid sheets, which are available from the literature, allow for a comparison of experimental data and predicted sheet behaviour. The last section evaluates the differences and similarities between the current liquid sheet experiment and industrial spray applications such as fuel atomisation via pressure-swirl nozzles.

Influence of mantle dynamics on the topographic evolution of the Tibetan Plateau: Results from numerical modeling

We investigate numerically the evolution of crustal and lithospheric thickness, thermal structure, topography, and strain rate of the Tibetan Plateau through time, using the thin viscous sheet approach. We show that lithospheric mantle must have been removed from beneath Tibet to explain the present surface elevation and lack of regional surface slope. In the absence of this removal, the modeled topography reaches a maximum elevation of <4000 m (for weak rheology), or the surface slopes significantly northward (for strong rheology). The crust must have been warmed and weakened by an increase of radiogenic heat production at depth due to crustal thickening. In the absence of this warming, viscous stresses associated with plate convergence exceed stresses produced by topography, and the present pattern of vertical thinning and east‐west extension would not have developed. Continuous removal of lithosphere, by delamination or convection, does not allow sufficient crustal warming and fails to reproduce either the present topography or the pattern of active deformation on the plateau in a reasonable time. Geologically rapid removal of the lithospheric root beneath the thickened crust of Tibet successfully explains the current elevation of the plateau, its lack of surface slope, the steep south and north margins, and the pattern of the present deformation, including vertical thinning, E‐W extension, and extrusion and vertical axis rotation on the eastern margin. Our modeling suggests that this removal took place within the last 12 m.y.

Plate force due to bending at subduction zones

Bending of the lithosphere at subduction zones generates stresses that contribute to the force balance on plates. We determine the net horizontal force on subducting plates by imposing a simple kinematic description of strain in the equations that govern the mechanical equilibrium of a thin viscous sheet. The magnitude of the force depends on the velocity, thickness, and curvature of the subducting plate. Using representative values for old oceanic lithosphere, we obtain an effective horizontal force that is nearly 40% of the buoyancy force due to slabs in the upper mantle.

Nonlinear stability of a charged electrified viscous liquid sheet under the action of a horizontal electric field

In a recent paper [D. T. Papageorgiou and P. G. Petropoulos, J. Eng. Math. 50, 223 (2004)] we considered the linear stability of a two-dimensional incompressible leaky dielectric viscous liquid sheet surrounded by a hydrodynamically passive conducting medium, when an electric field is applied parallel to the initially flat bounding fluid interfaces. It was established that for order-one Reynolds numbers and when the dielectric permittivity ratio, εp=εin∕εout, and the electric conductivity ratio, σR=σout∕σin, satisfy εpσR>1, the flow is linearly stable in the absence of an electric field. When a field is present a band of unstable long waves emerges whose size increases as the field increases (the band remains finite, that is short waves are stable, for large fields). In the present study we consider the nonlinear dynamics in the vicinity of the zero electric field bifurcation. The scalings determined from the linear stability calculations are used to derive canonical strongly nonlinear evolution equations for the leading order shape of the sheet and the corresponding horizontal velocity. Numerical simulations indicate that for a wide class of initial conditions, a quasisteady state is reached in the long time when the layer organizes into a number of lobes connected by slowly draining threads whose height vanishes asymptotically in time. The number of lobes and their volumes depend on initial conditions. Using this insight, we construct an ordinary differential equation which describes the shape of the sheet in the limit t→∞.

A coupled multivalued model for ice streams and its numerical simulation

This paper deals with the numerical solution of a non-linear model describing a free-boundary problem arising in modern glaciology. Considering a shallow, viscous ice sheet flow along a soft, deformable bed, a coupled non-linear system of differential equations can be obtained. Particularly, an obstacle problem is then deduced and solved in the framework of its complementarity formulation. We present the numerical solution of the resulting multivalued system modelling the ice sheet non-Newtonian dynamics driven by the underlying drainage system. Our numerical results show the existence of fast ice streams when positive wave-like initial conditions are considered. The solutions are numerically computed with a decoupling iterative method and finite-element technique. A duality algorithm and a projected Gauss–Seidel method are the alternatives used to cope with the resulting variational inequality while the explicit treatment, Newton method or a duality method are proposed to deal with the non-linear source term. Finally, the numerical solutions are physically interpreted and some comparisons among the numerical methods are then discussed.

Spatial instability of a viscous liquid sheet

In the present study, the spatial instability for a two-dimensional viscous liquid sheet, which is thinning with time, has been analysed. The study includes the derivation of a spatial dispersion equation, numerical solutions for the growth rate of sinuous disturbances, and parameter sensitivity studies. For a given wave number, the growth rate of the disturbance is essentially a function of Weber number, Reynolds number, and gas/liquid density ratio. The analysis indicates that the cut-off wave number of the disturbance becomes larger with an increase in Weber number or gas/liquid density ratio. Thus, the liquid sheet should produce finer drops. When the Reynolds number decreases, the higher viscosity has a greater damping effect on shorter waves than longer waves. This could explain that only large drops and ligaments were observed in past measurements for the disintegration of a very viscous sheet. The spatial instability results of the present study were also compared with the temporal theory. The importance of spatial analysis was found and demonstrated for the cases of low Weber numbers. The temporal theory underestimates growth rates when the Weber number is less than 100. The discrepancy between the two theories increases as the Weber number further decreases. Copyright © 2005 John Wiley & Sons, Ltd.

EXPERIMENTAL STUDY ON EFFECT FACTORS OF VISCOUS ADHESIVE STRESS IN VISCOUS PRESSURE FORMING

Because of the coupled deformation characteristic between viscous medium and sheet metal, however, it is difficult to analyze the deformation behavior of sheet metal under the interaction of viscous adhesive stress. Based on the above situation, viscous medium tensile test is carried out by using a kind of polymethylvinylpolymer as viscous pressure-carrying medium. Using the viscous adhesive stress as tensile stress, sheet metal deformation can be measured directly. The influence of viscous adhesive stress on the deformation of sheet specimen is investigated and some effect factors of viscous adhesive stress are analyzed preliminarily. Experimental results show that the viscous adhesive stress increases with an increasing of pressure difference on the surface of sheet metal, flowing velocity and hydrostatic pressure of viscous medium. Research results provide a good foundation for utilizing viscous adhesive stress to improve the formability of sheet metal in viscous pressure forming.

Thermocapillary control of rupture in thin viscous fluid sheets

We consider the evolution of a thin viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions for rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location.

Gravitational and tectonic forces controlling postcollisional deformation and the present‐day stress field of the Alps: Constraints from numerical modeling

We perform numerical modeling to investigate the mechanisms leading to the postcollisional tectonic evolution of the Alps. We model the lithospheric deformation as a viscous thin sheet with vertically averaged rheology and coupled with surface mass transport. The applied kinematic boundary conditions simulate the convergence between the Adria indenter and the European foreland during the last 35 Myr. Model predictions of elevation, lithospheric structure, erosion/sedimentation pattern and vertical axis rotation are compared with observations of the planform shape of the chain, topography, crustal thickness, distribution of rock exhumation and sediment thickness, and paleomagnetic rotations. Thickening of the lithosphere in the Alpine region is shown to be highly sensitive to the assumed viscosity law, to the strength contrasts between the different regions and to the surface mass transport. Modeling results indicate that the large‐scale deformation of the Alps during the postcollisional phase is mainly controlled by accommodation of convergence in a weak orogenic zone caught between a nearly rigid Adria plate and a strong European foreland. Modeling of the present‐day stress field shows that (1) the present rotation of Adria is responsible for the change of extension direction from strike‐perpendicular in the western Alps to strike‐parallel in the east and (2) the strike‐perpendicular extension observed in the western Alps is likely due to lateral variations of gravitational potential energy. The results suggest a NNE shift of about 700 km of the Euler pole of Adria relative to Europe from its mean position during postcollisional deformation to the present day.

Thin-sheet modelling of lithospheric deformation and surface mass transport

We study the effects of incorporating surface mass transport and the gravitational potential energy of both crust and lithospheric mantle to the viscous thin sheet approach. Recent 2D (cross-section) numerical models show that surface erosion and sediment transport can play a major role in shaping the large-scale deformation of the crust. In order to study these effects in 3D (planform view), we develop a numerical model in which both the dynamics of lithospheric deformation and surface processes are fully coupled. Deformation is calculated as a thin viscous layer with a vertically-averaged rheology and subjected to plane stresses. The coupled system of equations for momentum and energy conservation is solved numerically. This model accounts for the isostatic and potential-energy effects due to crustal and lithospheric thickness variations. The results show that the variations of gravitational potential energy due to the lateral changes of the lithosphere–asthenosphere boundary can modify the mode of deformation of the lithosphere. Surface processes, incorporated to the model via a diffusive transport equation, rather than just passively reacting to changes in topography, play an active role in controlling the lateral variations of the effective viscosity and hence of the deformation of the lithosphere.

Propagation of tectonic waves

Mountain building depends on the disequilibrium between boundary stresses, either at the base of the deforming lithosphere or its lateral boundaries, and buoyancy stresses arising form lateral density variations within the lithosphere itself. On the basis of the thin viscous sheet approximation, we propose a model which accounts for both crustal and lithospheric thickness variations. The deformation is controlled by the sum of the moments of density anomalies (i.e. density anomalies times depth) of compositional and thermal origins. The transport of the compositional moment is obtained from the continuity equation while the transport of the thermal moment is obtained from the heat equation. The resulting set of equations controls the coupled behavior of the crust and lithosphere. It shows that various type of solutions can exist: unstable, stable and propagating. When propagation occurs, the crustal and the lithospheric thickness variations are out of phase. The tectonic waves propagate with velocities around 5 mm yr−1 that increase with the crustal thickness and decrease with the lithospheric viscosity. We discuss these solutions and argue that continents may in large part be in a domain of propagating tectonic waves.

A physical model for the motion of the Sierra Block relative to North America

We explain the motion of the Sierra Block with respect to North America by treating it as a rigid inclusion within a thin sheet of non-Newtonian fluid, representing the continental lithosphere of North America. Deformation within North America is driven by a constant velocity applied to its western boundary by the Pacific plate. The deforming lithosphere applies forces to the edges of the block, and the requirement that these forces sum to zero leads to an approximate analytical solution in which the speed of the block, normalized by the rate of plate relative motion, depends on the dimensions of the block, its distance from the boundary, and on the power-law index, n, in the constitutive relation for the non-Newtonian fluid. Numerical solutions to the thin viscous sheet equation confirm the validity of this approximation. The speed of the Sierra Block with respect to North America is consistent with that of a rigid body, of the same shape and position, embedded within a thin viscous sheet of non-Newtonian fluid having a power-law index, n, between 2 and 6. A tighter constraint on n is provided by profiles of velocity, taken in the direction perpendicular to plate relative motion; comparison between observed profiles and those from a thin sheet model support a value of n ∼ 3 for the index.

Continental collision with finite indenter strength: 2. European Eastern Alps

Comparison of the results of a thin viscous sheet model with the velocity field determined by GPS stations in the Alpine‐Pannonian realm shows that the Adriatic plate is only slightly stronger than the Eastern Alps. The observed velocity field is best approximated with a Newtonian fluid. However, this scenario must be fairly recent, as the overall geometry of the orogen can only be reproduced if the rheology contrast was about 1:10 before the late Miocene. The boundary conditions for this model are tested against the orientation of the intraplate stress field and are robust for Argand numbers between 1 and 5. We conclude that the convergence rate between the Adriatic and European plates must have slowed since the Miocene and that the present‐day eastward extrusion of the Eastern Alps is largely due to lateral escape and only to a minor degree due to gravitational collapse. Scaled viscosities of the Eastern Alps are on the order of 1023 Pa s.

Continental collision with finite indenter strength: 1. Concept and model formulation

The partitioning of deformation during continental collision is modeled in plan view using a thin viscous sheet simulation. In contrast to many previous models, an indenter is defined here as a zone of finite viscosity contrast to the foreland and aspect ratio. Deformation is enforced by pushing the southern boundary into the model region. It is shown that deformation is equally partitioned between indenter and foreland if the aspect ratio of the indenter is about 1:2, and it is about twice as viscous as the foreland. For higher‐viscosity contrasts between the two plates, the partitioning of deformation between indenter and foreland depends more strongly on Argand number. The process of lateral extrusion is investigated by separating the lateral velocity field into a component that is due to lateral escape and a component that is due to gravitational extensional collapse. It is shown that lateral escape is roughly constant over time and is controlled by the convergence velocity and the obliquity of indentation.

Viscous liquid sheets and operability bounds in extrusion coating

The steady-state extrusion coating of an extremely viscous power-law liquid from a slot and onto a moving substrate across a narrow gap is examined. A key feature of this configuration is a two-dimensional liquid sheet that bridges the gap and might be subjected to an ambient pressure drop across its wide faces. The local thickness of this sheet may be viewed as gradually thinning from the slot to the substrate, and simplified equations governing its shape are derived. Analytical solutions for the shape of the liquid sheet, and the flow within, are obtained in the limit where viscous and pressure forces dominate all others. It is found that the centerline shape of the liquid sheet is circular when a pressure drop is applied, despite the fact that surface tension forces, which typically give rise to circular shapes in narrow gaps, are neglected. Nevertheless, it is analytically shown that the stretching of the viscous-dominated liquid sheet gives rise to an effective tension that plays an analogous role to surface tension. The range of possible applied pressures is thus deduced via geometrical considerations of a circular shape constrained in the narrow gap, as in the analogous surface-tension-dominated analysis (Ruschak, 1976. Chemical Engineering Science 31, 1057); extrusion coating fails when a liquid sheet cannot be constructed successfully.

Linear response of a viscous liquid sheet to external pressure perturbation. Long wave approximations

This paper is concerned with the linear signal response analysis of a thin viscous liquid sheet which is at rest in an appropriate frame of reference and in contact with passive external media, and on which localized external pressures act from the passive media as sources of perturbation. The frame of the analysis is provided by general formulae for the response signals of the sheet in the two excitation modes, sinuous and varicose, which result as the solution of the appropriate fluid dynamic initial-boundary value problems by the Fourier-Laplace transform technique. These formulae display how the signals depend on the nature of the perturbation and on the spectrum of the (linear) eigenmodes of the sheet. The signals can be evaluated either numerically or, as initiated in this paper, analytically, in long wave approximations. The long wave approximation will be seen in the sequel to concentrate on particular eigenmodes of the sheet spectrum, with small values k of the wave vector % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala % aaaa!36F0! $${\vec k}$$ along the sheet. The present paper is devoted mainly to a detailed analysis of the spectrum of eigenmodes of the sheet, and to the formulation of long wave approximations of (linear) response signals of the sheet in the context of this analysis. It turns out that the sheet spectrum of eigenmodes depends on only one characteristic number Γ, which depends on the (positive) fluid parameters: the surface tension γ at (both) the interfaces of the sheet, the density ρ of the fluid, its kinematic viscosity ν, and on the thickness h of the sheet: % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaOaaaeaadaWcaaqaaiabeo7aNjaadIgaaeaacaaIYaGaeqyW % diNaeqyVd42aaWbaaSqabeaacaaIYaaaaaaaaeqaaOGaaiOlaaaa!40E9! $$\Gamma = \sqrt {\frac{{\gamma h}} {{2\rho \nu ^2 }}} .$$ This number is closely related, by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaaaleqaaaaakiaa % d+eacaWGObWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiilaaaa!3E58! $$\Gamma = \frac{1} {{\sqrt 2 }}Oh^{ - 1} ,$$ to the Ohnesorge number % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaadI % gacqGH9aqpdaWcaaqaaiabeY7aTbqaamaakaaabaGaeq4SdCMaeqyW % diNaamiAaaWcbeaaaaGccaGGSaaaaa!3FA4! $$Oh = \frac{\mu } {{\sqrt {\gamma \rho h} }},$$ where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey % ypa0JaeqyWdiNaeqyVd4gaaa!3C22! $$\mu = \rho \nu $$ is the dynamic viscosity of the liquid. It will be shown in the sequel that of the infinitely many branches of the sheet spectrum only two (pairs of) ’soft’ branches, one sinuous and one varicose, will be relevant for a long wave approximation. For these branches asymptotic expansions of the dispersion relations ω(k) between the (complex) mode frequencies ω and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 % da9iaacYhaceWGRbGbaSaacaGG8baaaa!3AE6! $$k = |\vec k|$$ for long waves (i.e. for k → 0), which obey ω(k) → 0 with k → 0, will be derived. The analytic long wave dispersion relations for the (soft) sinuous and varicose spectral branches allow a very favourable insight into the qualitative mode behaviour, including analogies beyond liquid sheets. They show e.g. that to their lowest orders in k the soft sinuous modes are nondispersive, i.e. % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AaiaacYcaaaa!3C5A! $$\Im \omega \propto k,$$ and only weakly absorptive, i.e. ℜ ω ∝ k2 for low viscosity and ℜω ∝ k4 for high viscosity (ℜω < 0). Their signals are therefore expected to have some resemblance to signals of a flexible membrane on the one side and those of a (2-dimensional) diffusion process, or a hyperdiffusion process where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey % 4bIe9aaWbaaSqabeaacaqGYaaaaaaa!3943! $$ - \nabla ^{\text{2}} $$ is replaced by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaW % baaSqabeaacaaI0aaaaOGaaiOlaaaa!391B! $$\nabla ^4 .$$ The behaviour of the soft varicose modes will be seen to depend on the value of Γ: For Γ2 < 4 the asymptotic expansion for k → 0 gives $$ \Im \omega = {\text{0}} $$ and ℜω ∝ k2(ℜω 4 the situation is different: here % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AamaaCaaaleqabaGaaGOmaaaaaaa!3C93! $$\Im \omega \propto k^2 $$ and ℜω ∝ k2 (ℜω < 0). This behaviour bears resemblance to the vibration modes of a thin elastic plate with (diffusion-like) damping. The asymptotic insight into the behaviour of the individual branches of spectral modes allows the derivation of rather transparent approximate analytic expressions for the response signals of the sheet to external perturbations.

A predictive approach to simulating the forming of viscous textile composite sheet

A commercial Finite Element (FE) code is used to simulate the forming of a thermoplastic viscous textile composite sheet. The main success of this work is in combining two distinct models. The first is a rate and temperature dependent unit cell energy model, designed to predict the shear force - shear angle - shear rate response of viscous textile composites. The second is a non-orthogonal rate-independent constitutive model that has been implemented previously in the code. Both models are reviewed briefly and the method of combining these models in the code is described. Preliminary results of Picture Frame test simulations together with complex forming simulations are presented and discussed.

GROWTH OF LONGITUDINAL WAVES IN PLANE LIQUID SHEETS HAVING LATERAL WAVE MODES WHEN EXPOSED TO TWO GAS STREAMS OF UNEQUAL VELOCITIES

The growth of waves along the direction of motion of thin viscous liquid sheets is investigated when lateral waves are additionally present. Gas streams are considered to move at different relative velocities over the gas—liquid interfaces. A dispersion relation is derived using spatial growth of the longitudinal waves. Para-antisymmetric waves are shown to have significant growth rates in the presence of lateral wave motion at larger values of Weber number, gas-to-liquid density ratio, and relative velocity ratio. The presence of lateral wave motion does not affect the growth rates when gas Weber number exceeds 10. The droplet sizes, formed from the disintegration of the liquid sheets, are also determined from the longitudinal and lateral wave numbers. The distribution of droplet sizes is modeled and it is shown that the droplet sizes, formed in the presence of lateral wave motion, are restricted over a smaller range of diameters as compared to the wider distribution obtained in the absence of the lateral wave modes.

Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method

A numerical investigation of the long-time dynamics of two immiscible two-dimensional fluids shearing past one another is presented. The fluids are incompressible and the interface between the bulk phases is subjected to surface tension. The simple case of density and viscosity matched fluids is considered. The two-dimensional Navier–Stokes equations are solved numerically with a fully adaptive nonstiff strategy based on the immersed boundary method. Dynamically adaptive mesh refinements are used to cover at all times the separately tracked fluid interface at the finest grid level. In addition, by combining adaptive front tracking, in the form of continuous interface marker equidistribution, with a predictor–corrector discretization an efficient method is introduced to successfully treat the well-known numerical difficulties associated with surface tension. The resulting numerical method can be used to compute stably and with high resolution the flow for wide-ranging Weber numbers but this study focuses on the computationally challenging cases for which elongated fingering and interface roll-up are observed. To assess the importance of the viscous and vortical effects in the interfacial dynamics the full viscous flow simulations are compared with inviscid counterparts computed with a state-of-the-art boundary integral method. In the examined cases of roll-up, it is found that in contrast to the inviscid flow in which the interface undergoes a topological reconfiguration, the viscous interface remarkably escapes self-intersection and rich long-time dynamics due to separation, transport, and diffusion of vorticity is observed. An even more striking motion occurs at an intermediate Weber number for which elongated interpenetrating fingers of fluid develop. In this case, it is found that the Kelvin–Helmholtz instability weakens due to shedding of vorticity and unlike the inviscid counterpart in which there is indefinite finger growth the viscous interface is pulled back by surface tension. As the interface recedes, thin necks connecting pockets of fluid with the rest of the fingers form. Narrow jets are observed at the necking regions but the vorticity there ultimately appears to be insufficient to drain all the fluid and cause reconnection. However, at another point, two disparate portions of the interface come in close proximity as the interface continues to contract. Large curvature points and an intense concentration of vorticity are observed in this region and then the motion is abruptly terminated by the collapse of the interface.