On the instability of a free viscous rim

2010 ◽  
Vol 661 ◽  
pp. 206-228 ◽  
Author(s):  
ILIA V. ROISMAN
Keyword(s):  
Unstable Mode ◽  
Liquid Sheet ◽  
Theoretical Description ◽  
Momentum Balance ◽  
Governing Equations ◽  
Balance Equations ◽  
One Dimensional ◽  
Straight Line ◽  
Axisymmetric Disturbances ◽  
Viscous Stresses

This paper is devoted to the theoretical description of the dynamics of a rim formed by capillary forces at the edge of a free, thin liquid sheet. The rim dynamics are described using a quasi-one-dimensional approach accounting for the inertia of the liquid in the rim and for the liquid flow entering the rim from the sheet, surface tension and viscous stresses. The governing equations are derived from the mass, momentum and moment-of-momentum-balance equations of the rim. The theory provides a basis from which to analyse the linear stability of a straight line rim bounding a planar liquid sheet. The combined effect of the axisymmetric disturbances of the radius of the rim cross-section as well as of the transverse disturbances of the rim centreline is considered. The effect of the viscosity, relative film thickness and rim deceleration are investigated. The predicted wavelength of the most unstable mode is always very similar to the Rayleigh wavelength of the instability of an infinite cylindrical jet. This prediction is confirmed by various experimental data found in the literature. The maximum rate of growth of rim disturbances depends on all the parameters of the problem; however, the most pronounced effect can be attributed to the rim deceleration. This conclusion is confirmed by nonlinear simulations of rim deformation.

scholarly journals Sudden Pressurization of a Spherical Cavity in a Poroelastic Medium

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Mehmet Ozyazicioglu
Keyword(s):  
Frequency Domain ◽  
Momentum Balance ◽  
Governing Equations ◽  
Poroelastic Medium ◽  
Balance Equations ◽  
Classical Elasticity ◽  
Ode System ◽  
Type Boundary ◽  
System Time ◽  
Classical Elasticity Theory

Governing equations of poroelastodynamics in time and frequency domain are derived. The continuity equation complements the momentum balance equations. After reduction for spherical symmetry (geometry and loading), the governing equations in frequency domain are solved by introducing wave potentials. The wave propagation velocities are obtained as the real parts of the characteristic equation of the coupled ODE system. Time domain solution for Dirac type boundary pressure is obtained through numerical inversion of transformed solutions. The results are compared to the solution in classical elasticity theory found in the literature.

Analysis of shear bands in slow granular flows using a frictional Cosserat model

2000 ◽  
Vol 627 ◽  
Author(s):  
Prabhu R. Nott ◽  
K. Kesava Rao ◽  
L. Srinivasa Mohan
Keyword(s):  
Scaling Laws ◽  
Shear Bands ◽  
Shear Layers ◽  
Theoretical Description ◽  
Field Variable ◽  
Cosserat Continuum ◽  
Momentum Balance ◽  
Rigid Plastic ◽  
Independent Field ◽  
Cosserat Model

ABSTRACTThe slow flow of granular materials is often marked by the existence of narrow shear layers, adjacent to large regions that suffer little or no deformation. This behaviour, in the regime where shear stress is generated primarily by the frictional interactions between grains, has so far eluded theoretical description. In this paper, we present a rigid-plastic frictional Cosserat model that captures thin shear layers by incorporating a microscopic length scale. We treat the granular medium as a Cosserat continuum, which allows the existence of localised couple stresses and, therefore, the possibility of an asymmetric stress tensor. In addition, the local rotation is an independent field variable and is not necessarily equal to the vorticity. The angular momentum balance, which is implicitly satisfied for a classical continuum, must now be solved in conjunction with the linear momentum balances. We extend the critical state model, used in soil plasticity, for a Cosserat continuum and obtain predictions for flow in plane and cylindrical Couette devices. The velocity profile predicted by our model is in qualitative agreement with available experimental data. In addition, our model can predict scaling laws for the shear layer thickness as a function of the Couette gap, which must be verified in future experiments. Most significantly, our model can determine the velocity field in viscometric flows, which classical plasticity-based model cannot.

Design and Research of Flat Micro Heat Pipe with Glass Fiber Wick

2011 ◽  
Vol 483 ◽  
pp. 603-606
Author(s):  
Tian Han ◽  
Xiao Wei Liu ◽  
Chao Wang
Keyword(s):  
Glass Fiber ◽  
Heat Pipe ◽  
Experimental Testing ◽  
Governing Equations ◽  
Maximum Heat ◽  
One Dimensional ◽  
Micro Heat Pipe ◽  
Wick Structure

A kind of flat micro heat pipe with glass fiber wick structure is designed and fabricated. The structure of the wick is presented and also the excellence of the structure is described. For the glass fiber wick, the maximum heat transports is calculated by one-dimensional steady governing equations. Experimental testing is performed for the fabricated micro heat pipe in vacuum. The testing results is presented and analyzed.

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

2009 ◽  
Vol 19 (02) ◽  
pp. 545-555 ◽  
Author(s):  
F. TRAMONTANA ◽  
L. GARDINI ◽  
D. FOURNIER-PRUNARET ◽  
P. CHARGE
Keyword(s):  
Focal Point ◽  
Singular Line ◽  
Two Dimensional ◽  
Focal Points ◽  
Singular Set ◽  
One Dimensional ◽  
Attracting Set ◽  
Straight Line ◽  
Special Character ◽  
Stable And Unstable Sets

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

scholarly journals Stability of a buoyancy-driven coastal current at the shelf break

2002 ◽  
Vol 452 ◽  
pp. 97-121 ◽  
Author(s):  
C. CENEDESE ◽  
P. F. LINDEN
Keyword(s):  
Continental Shelf ◽  
Continental Slope ◽  
Unstable Mode ◽  
Shelf Break ◽  
Coastal Current ◽  
Surface Currents ◽  
Flat Bottom ◽  
Vertical Boundary ◽  
Axisymmetric Disturbances ◽  
A Current

Buoyancy-driven surface currents were generated in the laboratory by releasing buoyant fluid from a source adjacent to a vertical boundary in a rotating container. Different bottom topographies that simulate both a continental slope and a continental ridge were introduced in the container. The topography modified the flow in comparison with the at bottom case where the current grew in width and depth until it became unstable once to non-axisymmetric disturbances. However, when topography was introduced a second instability of the buoyancy-driven current was observed. The most important parameter describing the flow is the ratio of continental shelf width W to the width L* of the current at the onset of the instability. The values of L* for the first instability, and L*−W for the second instability were not influenced by the topography and were 2–6 times the Rossby radius. Thus, the parameter describing the flow can be expressed as the ratio of the width of the continental shelf to the Rossby radius. When this ratio is larger than 2–6 the second instability was observed on the current front. A continental ridge allowed the disturbance to grow to larger amplitude with formation of eddies and fronts, while a gentle continental slope reduced the growth rate and amplitude of the most unstable mode, when compared to the continental ridge topography. When present, eddies did not separate from the main current, and remained near the shelf break. On the other hand, for the largest values of the Rossby radius the first instability was suppressed and the flow was observed to remain stable. A small but significant variation was found in the wavelength of the first instability, which was smaller for a current over topography than over a flat bottom.

scholarly journals A new method for solving a class of heat conduction equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1205-1210
Author(s):  
Yi Tian ◽  
Zai-Zai Yan ◽  
Zhi-Min Hong
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Dimensional Space ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Governing Equations ◽  
One Dimensional ◽  
Sparse System ◽  
Linear Algebraic Equations ◽  
Crank Nicolson

A numerical method for solving a class of heat conduction equations with variable coefficients in one dimensional space is demonstrated. This method combines the Crank-Nicolson and Monte Carlo methods. Using Crank-Nicolson method, the governing equations are discretized into a large sparse system of linear algebraic equations, which are solved by Monte Carlo method. To illustrate the usefulness of this technique, we apply it to two problems. Numerical results show the performance of the present work.

scholarly journals Mathematical analysis of hydrodynamics and tissue deformation inside an isolated solid tumor

2018 ◽  
Vol 45 (2) ◽  
pp. 253-278 ◽  
Author(s):  
Meraj Alam ◽  
Bibaswan Dey ◽  
Sekhar Raja
Keyword(s):  
Solid Tumor ◽  
Solid Phase ◽  
Fluid Motion ◽  
Extracellular Fluid ◽  
Mixture Theory ◽  
Coupled System ◽  
Weak Sense ◽  
Governing Equations ◽  
One Dimensional ◽  
2D And 3D

In this article, we present a biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. The tumor tissue considered here is an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are wet by a physiological extracellular fluid. Since the tumor is deformable in nature, the mass and momentum equations for both the phases are presented. The momentum equations are coupled due to the interaction (or drag) force term. These governing equations reduce to a one-way coupled system under the assumption of infinitesimal deformation of the solid phase. The well-posedness of this model is shown in the weak sense by using the inf-sup (Babuska?Brezzi) condition and Lax?Milgram theorem in 2D and 3D. Further, we discuss a one-dimensional spherical symmetry model and present some results on the stress fields and energy of the system based on ??2 and Sobolev norms. We discuss the so-called phenomena of ?necrosis? inside a solid tumor using the energy of the system.

scholarly journals A new class of discontinuous solar wind solutions

2020 ◽  
Vol 496 (2) ◽  
pp. 1023-1034
Author(s):  
Bidzina M Shergelashvili ◽  
Velentin N Melnik ◽  
Grigol Dididze ◽  
Horst Fichtner ◽  
Günter Brenn ◽  
...  
Keyword(s):  
Solar Wind ◽  
External Source ◽  
Analytic Solutions ◽  
Governing Equations ◽  
Discontinuous Solutions ◽  
One Dimensional ◽  
Heating Source ◽  
New Class ◽  
Localized Heating ◽  
Physical Quantities

ABSTRACT A new class of one-dimensional solar wind models is developed within the general polytropic, single-fluid hydrodynamic framework. The particular case of quasi-adiabatic radial expansion with a localized heating source is considered. We consider analytical solutions with continuous Mach number over the entire radial domain while allowing for jumps in the flow velocity, density, and temperature, provided that there exists an external source of energy in the vicinity of the critical point that supports such jumps in physical quantities. This is substantially distinct from both the standard Parker solar wind model and the original nozzle solutions, where such discontinuous solutions are not permissible. We obtain novel sample analytic solutions of the governing equations corresponding to both slow and fast winds.

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