Generalized Viscoelastic Fluids with a Free Boundary without Surface Tension

2014 ◽  
Vol 46 (1) ◽  
pp. 428-458 ◽  
Author(s):  
Manuel Nesensohn
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Viscoelastic Fluids

Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

2015 ◽  
Vol 26 (4) ◽  
pp. 401-425 ◽  
Author(s):  
FUJUN ZHOU ◽  
JUNDE WU
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Stationary Solution ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Surface Tension Coefficient ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
A Free Boundary Problem

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Hele–Shaw flow with a point sink: generic solution breakdown

2004 ◽  
Vol 15 (1) ◽  
pp. 1-37 ◽  
Author(s):  
L. J. CUMMINGS ◽  
J. R. KING
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Complex Plane ◽  
Arbitrary Number ◽  
Blow Up ◽  
Arbitrary Angle ◽  
Small Surface ◽  
Singularity Structure ◽  
Numerical Evidence ◽  
Special Complex

Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.

Hele–Shaw flows with a free boundary produced by multipoles

1993 ◽  
Vol 4 (2) ◽  
pp. 97-120 ◽  
Author(s):  
Vladimir M. Entov ◽  
Pavel I. Etingof ◽  
Dmitry Ya. Kleinbock
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Complex Variable ◽  
Finite Time ◽  
Moving Boundary ◽  
Stationary Solutions ◽  
Time Interval ◽  
Explicit Solutions ◽  
Finite Time Interval ◽  
Variable Approach

We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

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