Linear and nonlinear stability of floating viscous sheets

2011 ◽  
Vol 683 ◽  
pp. 112-148 ◽  
Author(s):  
G. Pfingstag ◽  
B. Audoly ◽  
A. Boudaoud
Keyword(s):  
Surface Tension ◽  
Surface Forces ◽  
Two Dimensional ◽  
Buckling Mode ◽  
Geometrical Nonlinearities ◽  
Flat Sheet ◽  
Long Time ◽  
Classical Models ◽  
Linear And Nonlinear ◽  
The Stability

AbstractWe study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.

scholarly journals Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids in a horizontal magnetic field

2008 ◽  
Vol 12 (3) ◽  
pp. 103-110 ◽  
Author(s):  
Aiyub Khan ◽  
Neha Sharma ◽  
P.K. Bhatia
Keyword(s):  
Magnetic Field ◽  
Surface Tension ◽  
Growth Rate ◽  
Unstable Mode ◽  
Two Dimensional ◽  
Horizontal Magnetic Field ◽  
Streaming Velocity ◽  
The Stability ◽  
Conducting Fluids ◽  
Streaming Motion

The Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids has been investigated in the taking account of effects of surface tension, when the whole system is immersed in a uniform horizontal magnetic field. The streaming motion is assumed to be two-dimensional. The stability analysis has been carried out for two highly viscous fluid of uniform densities. The dispersion relation has been derived and solved numerically. It is found that the effect of viscosity, porosity and surface tension have stabilizing influence on the growth rate of the unstable mode, while streaming velocity has a destabilizing influence on the system.

Evolution and Shape of Two-Dimensional Stokesian Drops under the Action of Surface Tension and Electric Field: Linear and Nonlinear Theory and Experiment

Langmuir ◽  
2021 ◽  
Vol 37 (39) ◽  
pp. 11429-11446
Author(s):  
Rafael Granda ◽  
Jevon Plog ◽  
Gen Li ◽  
Vitaliy Yurkiv ◽  
Farzad Mashayek ◽  
...  
Keyword(s):  
Surface Tension ◽  
Electric Field ◽  
Nonlinear Theory ◽  
Two Dimensional ◽  
Linear And Nonlinear ◽  
Theory And Experiment

scholarly journals A Stability Analysis of Finite-Volume Advection Schemes Permitting Long Time Steps

2007 ◽  
Vol 135 (7) ◽  
pp. 2658-2673 ◽  
Author(s):  
Peter Hjort Lauritzen
Keyword(s):  
Stability Analysis ◽  
Finite Volume ◽  
Two Dimensional ◽  
Finite Volume Schemes ◽  
Courant Number ◽  
Von Neumann ◽  
Dispersion Properties ◽  
Advection Schemes ◽  
Long Time ◽  
The Stability

Abstract Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

Absolute nodal coordinate finite element approach to the two-dimensional liquid sloshing problems

2020 ◽  
Vol 234 (2) ◽  
pp. 322-346
Author(s):  
Kai Pan ◽  
Dengqing Cao
Keyword(s):  
Finite Element ◽  
Large Amplitude ◽  
Kinetic Equations ◽  
Liquid Sloshing ◽  
Penalty Function Method ◽  
Two Dimensional ◽  
Absolute Nodal Coordinate ◽  
Long Time ◽  
Integral Scheme ◽  
The Stability

Two-dimensional large-amplitude liquid sloshing in the rectangular rigid container is numerically simulated through absolute nodal coordinate finite element method, which can describe the large deformation of continuum by using a small number of elements. The incompressible constraint of Newtonian fluid is imposed by the penalty function method. Furthermore, the motion of rigid container is described by absolute nodal coordinate reference node and the liquid kinetic equations are derived in the total Lagrangian formulation, which can easily be combined with the solid nonlinear finite element and the multi-body system algorithms. The free sliding and non-penetrating boundary constraint equations for rectangular tank are derived. To ensure the stability and the conservation of the solution in long time simulations, the system dynamic equations are solved by Bathe integral scheme. Three numerical examples are used to verify the effectiveness of the proposed method, including the free spreading of a square liquid column and the large amplitude sloshing of liquid under rotational and horizontal excitations. A good consistency is obtained by comparing the calculated results with experimental and other numerical results reported in the literature.

scholarly journals Linear stability analysis and direct numerical simulation of two-layer channel flow

2016 ◽  
Vol 798 ◽  
pp. 889-909 ◽  
Author(s):  
Kirti Chandra Sahu ◽  
Rama Govindarajan
Keyword(s):  
Surface Tension ◽  
Mixed Layer ◽  
Finite Thickness ◽  
Plane Channel ◽  
Reynolds Numbers ◽  
Two Fluids ◽  
Miscible Flow ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Nonlinear Regimes

We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.

Instability, coalescence and fission of finite-area vortex structures

1973 ◽  
Vol 61 (2) ◽  
pp. 219-243 ◽  
Author(s):  
J. P. Christiansen ◽  
N. J. Zabusky
Keyword(s):  
Growth Rate ◽  
Small Scale ◽  
Two Dimensional ◽  
Large Area ◽  
Finite Area ◽  
Inviscid Fluids ◽  
Von Karman ◽  
Long Time ◽  
The Stability ◽  
Von Kármán

We have made computational experiments to study the stability and long-time evolution of two-dimensional wakes. We have used the VORTEX code, a finite-difference realization of two-dimensional motions in incompressible inviscid fluids. In the first experiment an initial shear-unstable triangular velocity profile evolves into a non-homogeneous, finite-area, asymmetric vortex array and like-signed regions attract andfuse(or coalesce). Enhanced transport across the profile is due to ‘capture’ and convection of small-scale vortex regions by larger opposite-signed vortex regions. In the following experiments we study the stability of an asymmetric four-vortexfinite-areasystem corresponding to a von Kármán street of point vortices. Here the critical parameter isb/a, the initial transverse-to-longitudinal separation ratio of vortex centres. At\[ b/a = 0.281 \]the four-vortex system is stable and we observe that large-area vortex regions develop elliptical (m= 2), triangular (m= 3), etc. surface modes owing to mutual interactions. Atb/a= 0 the measured growth rate is smaller than that for the corresponding von Kármán system and atb/a= 0·6 the measured growth rate is larger. Atb/a= 0 one vortex undergoes fission in the high-shear field produced by two nearest-neighbour opposite-signed vortex regions. Heuristic comparisons are made with the two-dimensional tunnel experiments of Taneda and others.

scholarly journals The effect of surface tension on the shape of fingers in a Hele Shaw cell

1981 ◽  
Vol 102 ◽  
pp. 455-469 ◽  
Author(s):  
J. W. McLean ◽  
P. G. Saffman
Keyword(s):  
Surface Tension ◽  
Experimental Results ◽  
Two Dimensional ◽  
Singular Perturbation Analysis ◽  
Linearized Stability ◽  
Finger Width ◽  
The Stability ◽  
Hele Shaw Cell ◽  
Effect Of Surface ◽  
Small Disturbances

The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.

Stability of an elastico-viscous liquid film flowing down an inclined plane

1967 ◽  
Vol 63 (2) ◽  
pp. 527-536 ◽  
Author(s):  
A. S. Gupta ◽  
Lajpat Rai
Keyword(s):  
Surface Tension ◽  
Stress Relaxation ◽  
Viscous Liquid ◽  
Perturbation Technique ◽  
Two Dimensional ◽  
Regular Perturbation ◽  
Inclined Plane ◽  
Viscous Liquid Film ◽  
The Stability ◽  
Sommerfeld Equation

AbstractAn analysis is made of the stability of a layer of an elastico-viscous liquid flowing down an inclined plane in the presence of two-dimensional disturbances. The modified Orr-Sommerfeld equation is solved by a regular perturbation technique for disturbances of large wavelengths. It is shown that in the absence of surface tension, the layer is more unstable as compared with that for an ordinary viscous liquid if Q1 > Q2, Q1 and Q2 being stress relaxation and strain retardation parameters respectively.

Sustainability and adaptability of regional development in the conditions of digitalization

2020 ◽  
Vol 19 (9) ◽  
pp. 1550-1613
Author(s):  
O.E. Akimova ◽  
S.K. Volkov ◽  
E.A. Gladkaya ◽  
I.M. Kuzlaeva
Keyword(s):  
Sustainable Development ◽  
Regional Development ◽  
Economic Behavior ◽  
Regional Inequality ◽  
Environmental Damage ◽  
Economic Potential ◽  
Human Welfare ◽  
Long Time ◽  
The Stability ◽  
Smart Technologies

Subject. The article discusses the sustainability of regional economy development, its definition, and the substance of sustainable development. Objectives. We aim at performing a comprehensive analysis of indicators of sustainability and adaptability of regional development in the context of digitalization, formulating a strategy for economic behavior that takes into account the multidimensional nature of regional inequality and is focused on boosting the economic potential of regions. Methods. The study draws on dialectic and systems approaches, general scientific methods of retrospective, situational, economic and statistical, and comparative analysis. Results. The sustainability of the region focuses on improving the human welfare over long time horizon. This happens in three areas, i.e. maximizing the efficiency of resource use; ensuring justice and democracy; minimizing resource consumption and environmental damage. The stability of the region can be assessed by using one parameter, or by combining the parameters in accordance with the type of region and expected results. Conclusions. The adaptation of a region to changing conditions depends on its type (‘adapted’, ‘adaptive’, and ‘non-adapted’). Regional inequality has two main components: difference in economic potential and social satisfaction of residents. Another component, affecting the stability and adaptability of regions, is the level of their digitalization. However, some regions have only formally embarked on the path of digitalization. Moreover, a focus on smart technologies, solutions and digitalization often leads to ignoring the goals of sustainable development. Smart technologies should be aimed at ensuring sustainability within the framework of the smart sustainable city concept.

Sign in / Sign up

Export Citation Format

Share Document