Bubble growth in porous media and Hele–Shaw cells

1986 ◽  
Vol 102 (1-2) ◽  
pp. 141-148 ◽  
Author(s):  
S. D. Howison
Keyword(s):  
Free Boundary ◽  
Bubble Growth ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boundary Problems ◽  
Quadrature Domains ◽  
The Third ◽  
Whole Space

SynopsisWe consider the characterisation of a class of free boundary problems arising in the flow of a viscous liquid in a porous medium (or, in two dimensions, a Hele–Shaw cell). Injected air forms a bubble which grows as time increases; it is shown that three kinds of behaviour can occur. Firstly, the solution may cease to exist in finite time; secondly, the solution may exist for all time and the free boundary may have one or more limit points as t tends to infinity; and thirdly, the bubble may exist for all time and fill the whole space as t tends to infinity. Two-dimensional explicit examples arc given of all three types of behaviour, and it is proved that the only solutions of the third kind are those in which the bubble is always elliptical; the proof uses the theory of null quadrature domains. It is shown that solutions for ellipsoidal bubbles exist in three dimensions and it is conjectured that the only three-dimensional null quadrature domains with finite complement are those whose complement is an ellipsoid.

scholarly journals Free boundary methods and non-scattering phenomena

2021 ◽  
Vol 8 (4) ◽  
Author(s):  
Mikko Salo ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Incident Wave ◽  
Scattering Theory ◽  
Free Boundary Problems ◽  
Boundary Problems ◽  
Quadrature Domains ◽  
Real Analytic ◽  
Boundary Methods ◽  
Inverse Scattering Theory ◽  
Zero Frequency

AbstractWe study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

Developing a Better Understanding of the Engineer's Creative Role in Design

2002 ◽  
Vol 17 (2-3) ◽  
pp. 129-133
Author(s):  
Bill Addis
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Space Structures ◽  
The Third ◽  
Structural Problems ◽  
The World ◽  
Third Dimension ◽  
Dimension Space ◽  
Creative Role

Both architects and engineers are unconsciously drawn towards the two dimensional world – the ubiquity of the plan and elevation, and the ease of analysing 2-D structures. Yet the best architecture always exploits the three dimensional world, and the majority of structural problems and collapses occur when engineers have failed to think in the third dimension. Space structures offer an ideal learning environment for students of both architecture and engineering. They stimulate and challenge both the imagination and the intellect by forcing students out of the cosy, and often dull familiarity of two dimensions. They encourage students to conceive structures in three dimensions and drop down to two when necessary or convenient, rather than the other way round. In a world where form and forces so strongly interact, space structures force architects to step into the world of statics, and engineers into the world of geometry. An important result is a better understanding, for both architects and engineers, of the role engineers can play in helping create imaginative and practical structures.

ON THE EXISTENCE OF SPATIALLY PATTERNED DORMANT MALIGNANCIES IN A MODEL FOR THE GROWTH OF NON-NECROTIC VASCULAR TUMORS

2001 ◽  
Vol 11 (04) ◽  
pp. 601-625 ◽  
Author(s):  
AVNER FRIEDMAN ◽  
FERNANDO REITICH
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Radial Symmetry ◽  
Vascular Tumors ◽  
Free Boundaries ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Equilibrium Configurations ◽  
Growth Equilibrium

Despite their great importance in determining the dynamic evolution of solutions to mathematical models of tumor growth, equilibrium configurations within such models have remained largely unexplored. This was due, in part, to the complexity of the relevant free boundary problems, which is enhanced when the process deviates from radial symmetry. In this paper, we present the results of our investigation on the existence of non-spherical dormant states for a model of non-necrotic vascularized tumors. For the sake of clarity we perform the analysis on two-dimensional geometries, though our techniques are evidently applicable to the full three-dimensional problem. We rigorously show that there is, indeed, an abundance of steady states that are not radially symmetric. More precisely, we prove that at any radially symmetric stationary state with free boundary r=R0 (which we first show to exist), there begin infinitely many branches of equilibria that bifurcate from and break the symmetry of that radial state. The free boundaries along the bifurcation branches are of the form [Formula: see text], where ℓ=2,3,… and |ε|<ε0; each choice of ℓ and ε determines a non-radial steady configuration.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

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