Some Hele Shaw flows with time-dependent free boundaries

1981 ◽  
Vol 102 ◽  
pp. 263-278 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Free Boundaries ◽  
Narrow Gap ◽  
Plan View ◽  
Image Domain ◽  
Simply Connected Domain ◽  
Straight Line ◽  
Quarter Plane ◽  
Geometric Character ◽  
Parallel Surfaces ◽  
Simply Connected

We consider a blob of Newtonian fluid sandwiched in the narrow gap between two plane parallel surfaces. At some initial instant, its plan-view occupies a given, simply connected domain, and its growth as further fluid is injected at a number of injection points in its interior is to be determined. It is shown that certain functionals of the domain of a purely geometric character, infinite in number, evolve in a predictable manner, and that these may be exploited in some cases of interest to yield a complete description of the motion.By invoking images, these results may be used to solve certain problems involving the growth of a blob in a gap containing barriers. Injection at a point in a half-plane bounded by a straight line, with an initially empty gap, is shown to lead to a blob whose outline is part of an elliptic lemniscate of Booth for which there is a simple geometrical construction. Injection into a quarter-plane is also considered in some detail when conditions are such that the image domain involved is simply connected.

Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel

1972 ◽  
Vol 56 (4) ◽  
pp. 609-618 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Approximation Scheme ◽  
Narrow Channel ◽  
Algebraic Equations ◽  
Similar System ◽  
Plan View ◽  
Simply Connected Domain ◽  
Subsequent Time ◽  
Geometric Character ◽  
Parallel Surfaces ◽  
Simply Connected

A blob of Newtonian fluid is sandwiched in the narrow gap between two plane parallel surfaces so that, a t some initial instant, its plan-view occupies a simply connected domain D0. Further fluid, with the same material properties, is injected into the gap at some fixed point within D0, so that the blob begins to grow in size. The domain D occupied by the fluid at some subsequent time is to be determined.It is shown that the growth is controlled by the existence of an infinite number of invariants of the motion, which are of a purely geometric character. For sufficiently simple initial domains D0 these allow the problem to be reduced to the solution of a finite system of algebraic equations. For more complex initial domains an approximation scheme leads to a similar system of equations to be solved.

scholarly journals Numerical solution of Hele-Shaw flows driven by a quadrupole

1997 ◽  
Vol 8 (6) ◽  
pp. 551-566 ◽  
Author(s):  
E. D. KELLY ◽  
E. J. HINCH
Keyword(s):  
Numerical Solution ◽  
Stable Equilibrium ◽  
Boundary Integral ◽  
Inviscid Fluid ◽  
Flow Strength ◽  
Plan View ◽  
Simply Connected Domain ◽  
Parallel Surfaces ◽  
Simply Connected ◽  
Steady State Solutions

A blob of viscous Newtonian fluid is surrounded by inviscid fluid and sandwiched in the narrow gap between two plane parallel surfaces, so that initially its plan view occupies a simply connected domain. Recently Entov, Etingoff & Kleinbock (1993) produced some steady-state solutions for the blob placed in a quadrupole driven flow, and including the effects of surface tension. Here a numerical solution of the time-dependent problem using a Boundary Integral algorithm finds that for low values of the flow rate there exist two solutions. We find that one, which is close in shape to a circle, is stable, while the other, more deformed equilibrium, is unstable. The analysis also reveals that for certain flow strengths stable non-convex shapes also exist. If the flow strength is too large no stable equilibrium is possible.

Hele–Shaw flows with free boundaries driven along infinite strips by a pressure difference

1996 ◽  
Vol 7 (4) ◽  
pp. 345-366 ◽  
Author(s):  
S. Richardson
Keyword(s):  
Free Boundary ◽  
Pressure Difference ◽  
Constant Pressure ◽  
Analytic Solutions ◽  
Infinite Strip ◽  
Free Boundaries ◽  
Plan View ◽  
Advancing Front ◽  
Simply Connected Region ◽  
Simply Connected

Consider the classical Hele–Shaw situation with two parallel planes separated by a narrow gap, and suppose the plan-view of the region occupied by fluid to be confined to an infinite strip by barriers in the form of two infinite parallel lines. With the fluid initially occupying a bounded, simply-connected region that touches both barriers along a single line segment, we seek to predict the evolution of the plan-view as the blob of fluid is driven along the strip by a pressure difference between its two free boundaries. Supposing the relevant free boundary condition to be one of constant pressure (but a different constant pressure on each free boundary), we show that the motion is characterized by (a) the existence of two functions, analytic in disjoint half-planes, that are invariants of the motion and (b) the centre of area of the plan-view of the blob has a component of velocity down the infinite strip that is simply related to the imposed pressure difference. These features allow explicit analytic solutions to be found; generically, the mathematical solution breaks down when cusps appear in the retreating free boundary. A rectangular blob, of course, moves down the strip unchanged, with no breakdown, but if it encounters stationary blobs of fluid placed within the strip then, modulo multiply-connected complications, these are first absorbed into the advancing front of the rectangular blob and then disgorged from its retreating rear, leaving behind stationary blobs of exactly the same form in exactly the same place as those originally present, but consisting of different fluid particles. This soliton-like interaction involves no phase change: with a given pressure difference driving the motion, the rectangular blob is in the same position at a given time after the interaction as it would have been had no intervening blobs been present.

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.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

scholarly journals On the Boundary Behavior of Conformal Maps

1967 ◽  
Vol 30 ◽  
pp. 83-101 ◽  
Author(s):  
S.E. Warschawski
Keyword(s):  
Boundary Behavior ◽  
Mapping Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conformal Maps ◽  
Simply Connected Domain ◽  
Definitive Solution ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Fundamental Paper

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

scholarly journals A Localization Principle for a Class of Analytic Functions

1963 ◽  
Vol 23 ◽  
pp. 207-212
Author(s):  
D. A. Storvick
Keyword(s):  
Analytic Function ◽  
Circular Disk ◽  
Inverse Image ◽  
Limit Values ◽  
Localization Principle ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Simply Connected Domain ◽  
Almost Everywhere ◽  
Simply Connected

It has been shown by Kiyoshi Noshiro [8; p. 35] that a bounded analytic function w = f(z) in |z| < 1 having radial limit values of modulus one almost everywhere satisfies a localization principle of the following type. Let (c) be any circular disk: | w − α | < ρ lying inside |w| < 1 whose periphery may be tangent to the circumference |w| = 1. Denote by Δ any component of the inverse image of (c) under w = f(z) and by z = z(ξ) a function which maps |ξ| < 1 onto the simply connected domain Δ in a one-to-one conformal manner. Then, the functionis also a bounded analytic function in | ξ | < 1 with radial limits of modulus one almost everywhere.

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