Local Solvability of Problems with Free Boundaries in the Magnetic Hydrodynamics of an Ideal Compressible Fluid with and without Account for Surface Tension

2021 ◽  
Vol 62 (4) ◽  
pp. 181-190
Author(s):  
Yu. L. Trakhinin
Keyword(s):  
Surface Tension ◽  
Compressible Fluid ◽  
Local Solvability ◽  
Magnetic Hydrodynamics ◽  
Free Boundaries

The Critical Thickness of the Dislocation-Free Stranski Krastanow Pattern of Growth.

1992 ◽  
Vol 290 ◽  
Author(s):  
Michael A. Grinfeld
Keyword(s):  
Surface Tension ◽  
Critical Thickness ◽  
Epitaxial Films ◽  
Shear Stresses ◽  
Morphological Stability ◽  
Free Boundaries ◽  
Equilibrium Thermodynamics ◽  
Misfit Stress ◽  
Symmetry Change

AbstractIt was demonstrated earlier [1,2] in the framework of equilibrium thermodynamics that the morphological stability of the free boundaries and interfaces in crystals is extremely sensitive to the presence of shear stresses. Relying on that idea we have established the formula H = μσ/τ2 of a critical thickness of solidifying He4 films and of the dislocation-free Stranski-Krastanow growth of epitaxial films (where σ – the coefficient of surface tension, μ - the shear module of the crystal, τ - the external or misfit stress). In this report we present certain facts pertaining to possible patterns of the growing corrugations and introduce the second critical thickness at which a symmetry change in the patterns has to occur.

Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region

2000 ◽  
Vol 11 (3) ◽  
pp. 249-269 ◽  
Author(s):  
S. RICHARDSON
Keyword(s):  
Surface Tension ◽  
Stokes Flow ◽  
Flow Problem ◽  
Rotational Symmetry ◽  
Time Dependent ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Free Surfaces ◽  
Simply Connected ◽  
Surface Tension Model

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

Effect of Hall current and resistivity on the stability of a gas–liquid system

1970 ◽  
Vol 4 (3) ◽  
pp. 451-469 ◽  
Author(s):  
G. L. Kalra ◽  
S. N. Kathuria ◽  
R. J. Hosking ◽  
G. G. Lister
Keyword(s):  
Surface Tension ◽  
Compressible Fluid ◽  
Hall Current ◽  
Supersonic Flows ◽  
Liquid System ◽  
Conducting Fluid ◽  
The Stability

The stability of a non-conducting, compressible fluid (gas) flowing across the surface of incompressible conducting fluid (liquid) is discussed. Finite resistivity and Hall current are included in the hydromagnetic equations, together with surface tension. Both subsonic and supersonic flows are treated and some new instabilities are found, together with modifications to real and oscillatory modes obtained in earlier treatments.

Cusp development in free boundaries, and two-dimensional slow viscous flows

1995 ◽  
Vol 6 (5) ◽  
pp. 441-454 ◽  
Author(s):  
S. D. Howison ◽  
S. Richardson
Keyword(s):  
Surface Tension ◽  
Viscous Flows ◽  
Analytic Solutions ◽  
Time Dependent ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Mathematical Solution ◽  
Simply Connected Domain ◽  
Exact Analytic Solutions ◽  
Simply Connected

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.

Initiation of shape instabilities of free boundaries in planar Cauchy–Stefan problems

1993 ◽  
Vol 4 (4) ◽  
pp. 419-436 ◽  
Author(s):  
Qiang Zhu ◽  
Anthony Peirce ◽  
John Chadam
Keyword(s):  
Surface Tension ◽  
Large Time ◽  
Free Boundaries ◽  
Shape Stability ◽  
Stefan Problems ◽  
Moving Interfaces ◽  
Solidification Problem ◽  
Slow Moving ◽  
The Stability ◽  
Theoretical Results

The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the parameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in Chadam & Ortoleva [4] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension

1997 ◽  
Vol 8 (4) ◽  
pp. 311-329 ◽  
Author(s):  
S. RICHARDSON
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Stokes Flow ◽  
Moving Boundary ◽  
Time Dependent ◽  
Initial Configuration ◽  
Conformal Map ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Free Boundaries

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are such flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are N such disks in a general position, the evolution of the fluid region is described by a conformal map involving 2N−1 time-dependent parameters whose variation is governed by N invariants and N−1 first order differential equations. When N=2, or when the problem enjoys some special features of symmetry, the moving boundary of the fluid domain during the motion can be determined by solving purely algebraic equations, the solution of a single differential equation being needed only to link a particular boundary shape to a particular time. The analysis is aided by exploiting a connection with Hele-Shaw free boundary flows when the zero-surface-tension model is employed. If the initial configuration for the Stokes flow problem can be produced by injection (or suction) at N points into an initially empty Hele-Shaw cell, as can the N-disk configuration referred to above, then so can all later configurations; the points where the fluid must be injected move, but the amount to be injected at each of the N points remains invariant. The efficacy of our solution procedure is illustrated by a number of examples, and we exploit the method to show that the free boundary in such a Stokes flow driven by surface tension alone may pass through a cusped state.

scholarly journals Marangoni Convection In A Relatively Hotter Or Cooler Liquid Layer With Free Boundaries

2013 ◽  
Vol 18 (2) ◽  
pp. 581-588 ◽  
Author(s):  
A.K. Gupta ◽  
R.G. Shandil
Keyword(s):  
Surface Tension ◽  
Liquid Layer ◽  
Marangoni Convection ◽  
Fluid Layer ◽  
Slip Boundary Condition ◽  
Driving Mechanism ◽  
Free Boundaries ◽  
Slip Boundary ◽  
Thermally Conducting ◽  
Horizontal Fluid Layer

In this paper, we study the onset of cellular convection in a horizontal fluid layer heated from below, with a free-slip boundary condition at the bottom when the driving mechanism is surface tension at the upper free surface, in the light of the modified analysis of Banerjee et al. (Jour. Math. & Phys. Sci., 1983, 17, 603). This leads to a formulation of the problem which depends upon whether the liquid layer is relatively hotter or cooler. It is found that the phenomenon of surface tension driven instability problems should not only depend upon the Marangoni number which is proportional to the maintained temperature differences across the layer but also upon another parameter that arises due to variation in the specific heat at constant volume on account of the variations in temperature. Numerical results are obtained for the problem wherein the lower free boundary is perfectly thermally conducting.

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