scholarly journals Time-evolving bubbles in two-dimensional Stokes flow

1995 ◽  
Vol 301 ◽  
pp. 325-344 ◽  
Author(s):  
Saleh Tanveer ◽  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Shear Flow ◽  
Exact Solutions ◽  
Stokes Flow ◽  
Mathematical Structure ◽  
The Other ◽  
Two Dimensional ◽  
Slow Viscous Flow ◽  
Expanding Circle ◽  
Linear Shear

A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.

The bursting of two-dimensional drops in slow viscous flow

1973 ◽  
Vol 60 (4) ◽  
pp. 625-639 ◽  
Author(s):  
J. D. Buckmaster ◽  
J. E. Flaherty
Keyword(s):  
Differential Equation ◽  
Surface Tension ◽  
Analytic Function ◽  
Function Theory ◽  
Equilibrium Shape ◽  
The Other ◽  
Two Dimensional ◽  
Approximate Analysis ◽  
Slow Viscous Flow ◽  
Corner Flow

We consider the deformation of two-dimensional drops when immersed in a slow viscous corner flow. The problem is formulated as one of analytic function theory and simplified by assuming that both the drop and the exterior fluid have the same viscosity. An approximate analysis is carried out, in which the conditions at the interface are satisfied in an average sense, and this reveals the following features of the solution. A drop of given physical properties (volume, surface tension and viscosity), when immersed in a corner flow, has no steady equilibrium shape if the rate of strain of the applied flow is too large. On the other hand, if the rate of strain is small enough for a steady solution to exist, then in general there are two possible solutions. These features are confirmed by formulating the exact problem in terms of a nonlinear integro-differential equation, which is solved numerically.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

THE CONCENTRATIONS OF OIL IN SEA WATER RESULTING FROM NATURAL AND CHEMICALLY INDUCED DISPERSION OF OIL SLICKS

1977 ◽  
Vol 1977 (1) ◽  
pp. 381-385 ◽  
Author(s):  
D. Cormack ◽  
J.A. Nichols
Keyword(s):  
Surface Tension ◽  
Oil Recovery ◽  
Oil Spills ◽  
Sea Water ◽  
The Other ◽  
Viscous Drag ◽  
Two Dimensional ◽  
Drag Forces ◽  
Chemically Induced ◽  
Low Toxicity

ABSTRACT Results are presented on the factors relating to the dissipation of oil spills at sea, including evaporation, emulsion formation, spreading, and natural dispersion into the water column. For Ekofisk oil, 20% evaporates in about 7.5 hours and, while emulsion formation is as rapid as for Kuwait crude, the resulting viscosity is low and insufficient to allow interference with the natural spreading and dispersion rates. Spreading has two components. One is controlled by surface tension-viscous drag forces and the other is wind-induced. Together they contribute to the two dimensional dissipation of the oil so that subsequent oil concentrations in the sea are of necessity, low. These concentrations were measured for naturally dispersing and chemically dispersed slicks. The chemically-dispersed slicks were of two kinds. One was previously weathered for three hours, the other was of controlled thickness and was dispersed immediately upon being laid. Resulting concentrations of oil in the sea are low and of short duration compared with those required to give observable effects in laboratory toxicity studies. No significant deleterious effects were found to result from the dispersion of oil slicks at sea using low toxicity dispersant chemicals; also it was noted that, in any case, substantial quantities of oil can be expected to enter the sea before oil recovery operations can be mounted.

scholarly journals Two-dimensional fluctuating vesicles in linear shear flow

2008 ◽  
Vol 25 (3) ◽  
pp. 309-321 ◽  
Author(s):  
R. Finken ◽  
A. Lamura ◽  
U. Seifert ◽  
G. Gompper
Keyword(s):  
Shear Flow ◽  
Two Dimensional ◽  
Linear Shear

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.

The evolution of thermocline waves from an oscillatory disturbance

1993 ◽  
Vol 254 ◽  
pp. 401-416 ◽  
Author(s):  
D. Nicolaou ◽  
R. Liu ◽  
T. N. Stevenson
Keyword(s):  
Finite Difference ◽  
Shear Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Ray Theory ◽  
Navier Stokes Equations ◽  
Linear Shear ◽  
Wave Shapes ◽  
The Way

The way in which energy propagates away from a two-dimensional oscillatory disturbance in a thermocline is considered theoretically and experimentally. It is shown how the St. Andrew's-cross-wave is modified by reflections and how the cross-wave can develop into thermocline waves. A linear shear flow is then superimposed on the thermocline. Ray theory is used to evaluate the wave shapes and these are compared to finite-difference solutions of the full Navier–Stokes equations.

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