scholarly journals Thermocapillary instabilities in a horizontal liquid layer under partial basal slip

2018 ◽  
Vol 855 ◽  
pp. 839-859 ◽  
Author(s):  
Katarzyna N. Kowal ◽  
Stephen H. Davis ◽  
Peter W. Voorhees
Keyword(s):  
Liquid Layer ◽  
Basal Slip ◽  
Marangoni Number ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Partial Slip ◽  
Two Dimensional ◽  
Slip Parameter ◽  
Rigid Surface ◽  
Selection Of

We investigate the onset of three-dimensional hydrothermal waves in a low-capillary-number liquid layer of arbitrary depth, bounded by a free liquid–gas interface from above and a partial slip, rigid surface from below. A selection of two- and three-dimensional hydrothermal waves, longitudinal rolls and longitudinal travelling waves, form the preferred mode of instability, which depends intricately on the magnitude of the basal slip. Partial slip is destabilizing for all modes of instability. Specifically, the minimal Marangoni number required for the onset of instability follows $M_{m}\sim a(\unicode[STIX]{x1D6FD}^{-1}+b)^{-c}$ for each mode, where $a,b,c>0$ and $\unicode[STIX]{x1D6FD}^{-1}$ is the slip parameter. In the limit of free slip, longitudinal travelling waves disappear in favour of longitudinal rolls. With increasing slip, it is common for two-dimensional hydrothermal waves to exchange stability in favour of longitudinal rolls and oblique hydrothermal waves. Two types of oblique hydrothermal waves appear under partial slip, which exchange stability with increasing slip. The oblique mode that is preferred under no slip persists and remains near longitudinal for small slip parameters.

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

scholarly journals Optimization of the multidimensional signal interpolator in a lower dimensional space

2019 ◽  
Vol 43 (4) ◽  
pp. 653-660 ◽  
Author(s):  
M.V. Gashnikov
Keyword(s):  
Experimental Study ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Multidimensional Signals ◽  
Multidimensional Signal ◽  
Signal Sample ◽  
Lower Dimensional Space ◽  
Lower Dimensional ◽  
Selection Of

Adaptive multidimensional signal interpolators are developed. These interpolators take into account the presence and direction of boundaries of flat signal regions in each local neighborhood based on the automatic selection of the interpolating function for each signal sample. The selection of the interpolating function is performed by a parameterized rule, which is optimized in a parametric lower dimensional space. The dimension reduction is performed using rank filtering of local differences in the neighborhood of each signal sample. The interpolating functions of adaptive interpolators are written for the multidimensional, three-dimensional and two-dimensional cases. The use of adaptive interpolators in the problem of compression of multidimensional signals is also considered. Results of an experimental study of adaptive interpolators for real multidimensional signals of various types are presented.

scholarly journals Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

2016 ◽  
Vol 802 ◽  
pp. 5-36 ◽  
Author(s):  
A. Kalogirou ◽  
D. T. Papageorgiou
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Type Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wide Range ◽  
Time Periodic ◽  
Two Fluid ◽  
Permanent Form

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

scholarly journals Modelling nonlinear hydroelastic waves

2011 ◽  
Vol 369 (1947) ◽  
pp. 2942-2956 ◽  
Author(s):  
P. I. Plotnikov ◽  
J. F. Toland
Keyword(s):  
Wave Speed ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Lagrangian Description ◽  
Two Dimensional ◽  
Cosserat Theory ◽  
Elastic Sheet ◽  
Eulerian Description ◽  
Eulerian Coordinates ◽  
Special Case

This paper uses the special Cosserat theory of hyperelastic shells satisfying Kirchoff’s hypothesis and irrotational flow theory to model the interaction between a heavy thin elastic sheet and an infinite ocean beneath it. From a general discussion of three-dimensional motions, involving an Eulerian description of the flow and a Lagrangian description of the elastic sheet, a special case of two-dimensional travelling waves with two wave speed parameters, one for the sheet and another for the fluid, is developed only in terms of Eulerian coordinates.

New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

scholarly journals Instant Stent-Accentuated Three-Dimensional Optical Coherence Tomography Guided Selection of Proper Distal Cell for Side Branch Dilatation in Bifurcation Stenting

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Fumiaki Nakao
Keyword(s):  
Optical Coherence Tomography ◽  
Three Dimensional ◽  
Side Branch ◽  
Optical Coherence ◽  
Two Dimensional ◽  
Bifurcation Stenting ◽  
Distal Cell ◽  
Selection Of

In the bifurcation stenting, the distal rewiring for the side branch postdilatation confirmed by two-dimensional modalities may not lead to favorable results in some cases. If there are two distal cells divided by the link bridging from the carina, the rewiring through the larger distal cell may be recommended for the side branch postdilatation. Detailed confirmation of the rewired cell by the intraprocedural instant stent-accentuated three-dimensional optical coherence tomography is important.

scholarly journals Modulated waves in a periodically driven annular cavity

2010 ◽  
Vol 667 ◽  
pp. 336-357 ◽  
Author(s):  
H. M. BLACKBURN ◽  
J. M. LOPEZ
Keyword(s):  
Symmetry Group ◽  
Temporal Dynamics ◽  
Travelling Waves ◽  
Temporal Structure ◽  
Three Dimensional ◽  
Spanwise Direction ◽  
Two Dimensional ◽  
Periodically Driven ◽  
Spatio Temporal ◽  
Temporal Symmetry

Time-periodic flows with spatio-temporal symmetry Z2 × O(2) – invariance in the spanwise direction generating the O(2) symmetry group and a half-period-reflection symmetry in the streamwise direction generating a spatio-temporal Z2 symmetry group – are of interest largely because this is the symmetry group of periodic laminar two-dimensional wakes of symmetric bodies. Such flows are the base states for various three-dimensional instabilities; the periodically shedding two-dimensional circular cylinder wake with three-dimensional 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) spatio-temporal symmetry. The flows have three distinct codimension-1 instabilities: two synchronous modes (A and B), and two manifestations of a quasi-periodic (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 three-dimensional instabilities the dynamics are dominated by mixed modes with complicated spatio-temporal structure. Supplementary movies illustrating the spatio-temporal dynamics are available at journals.cambridge.org/flm.

Spontaneous breaking of reflexional symmetry in real quasi-two-dimensional turbulence: stochastic travelling waves and helical solitons in atmosphere and laboratory

1993 ◽  
Vol 441 (1911) ◽  
pp. 147-155 ◽  
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Laboratory Experiments ◽  
Travelling Waves ◽  
Qualitative Difference ◽  
Three Dimensional ◽  
Spontaneous Breaking ◽  
Two Dimensional ◽  
Boundary Layer Turbulence ◽  
The Difference

The theme of this note is the qualitative difference between strictly two-dimensional (2D) and quasi-two-dimensional (Q2D) turbulence in spite of the ‘smallness’ of the difference in their geometry. It is argued that the Q2D régime arises as a result of a spontaneous breaking of reflexional symmetry, which in turn is a consequence of the instability of 2D turbulence to three-dimensional helical travelling waves and solitons (through super-critical and sub-critical bifurcations). The difference between 2D and Q2D turbulence, which is primarily of a topological nature (related to helicity and super helicity) is manifested in different spectral and diffusive properties. The arguments are supported by a large number of experimental data from laboratory experiments (stably stratified, rotating, magnetohydrodynamic and boundary layer turbulence) and from observations in the stratosphere.

Bénard–Marangoni convection at low Prandtl number

1999 ◽  
Vol 399 ◽  
pp. 251-275 ◽  
Author(s):  
THOMAS BOECK ◽  
ANDRÉ THESS
Keyword(s):  
Free Surface ◽  
Prandtl Number ◽  
Marangoni Number ◽  
Marangoni Convection ◽  
Three Dimensional ◽  
Bottom Wall ◽  
Liquid Sodium ◽  
Small Scale ◽  
Quiescent State ◽  
Two Dimensional

Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of spatio-temporal chaos. No qualitative difference is observed between the zero-Prandtl-number limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

Overview, Classification and Selection of Map Projections for Geospatial Applications

2010 ◽  
pp. 89-98
Author(s):  
Eric Delmelle ◽  
Raymond Dezzani
Keyword(s):  
Three Dimensional ◽  
Research Trends ◽  
Geometric Features ◽  
Two Dimensional ◽  
Geospatial Information ◽  
Map Projection ◽  
Map Projections ◽  
The Earth ◽  
Dimensional Surface ◽  
Selection Of

There has been a dramatic increase in the handling of geospatial information, and also in the production of maps. However, because the Earth is three-dimensional, geo-referenced data must be projected on a two-dimensional surface. Depending on the area being mapped, the projection process generates a varying amount of distortion, especially for continental and world maps. Geospatial users have a wide variety of projections too choose from; it is therefore important to understand distortion characteristics for each of them. This chapter reviews foundations of map projection, such as map projection families, distortion characteristics (areal, angular, shape and distance), geometric features and special properties. The chapter ends by a discussion on projection selection and current research trends."

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