scholarly journals Full regularity of the free boundary in a Bernoulli-type problem in two dimensions

2006 ◽  
Vol 13 (4) ◽  
pp. 667-681 ◽  
Author(s):  
Donatella Danielli ◽  
Arshak Petrosyan
Keyword(s):  
Free Boundary ◽  
Two Dimensions ◽  
Type Problem ◽  
Full Regularity

scholarly journals Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

On the Curvature of the Free Boundary for the Obstacle Problem in Two Dimensions

2004 ◽  
Vol 142 (1-2) ◽  
pp. 1-5 ◽  
Author(s):  
Bj�rn Gustafsson ◽  
Makoto Sakai
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Two Dimensions

Macroscopic models for melting derived from averaging microscopic Stefan problems I: Simple geometries with kinetic undercooling or surface tension

2000 ◽  
Vol 11 (2) ◽  
pp. 153-169 ◽  
Author(s):  
A. A. LACEY ◽  
L. A. HERRAIZ
Keyword(s):  
Free Boundary ◽  
Multiple Scales ◽  
Two Dimensions ◽  
Macroscopic Model ◽  
Mushy Region ◽  
Free Boundaries ◽  
Two Phase ◽  
Macroscopic Models ◽  
Stefan Problems ◽  
Kinetic Undercooling

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. For simplicity, the fine structure is here taken to be periodic, first in one dimension, and then a lattice of squares in two dimensions. A method of multiple scales is employed, with a classical free-boundary problem being used to model the evolution of the two-phase microstructure. Then a macroscopic model for the mush is obtained by an averaging procedure. The free-boundary temperature is taken to vary according to Gibbs–Thomson and/or kinetic-undercooling effects.

On free boundary problems in two dimensions

1985 ◽  
Vol 34 (3) ◽  
pp. 325-336
Author(s):  
Hans Lewy ◽  
Tang Zhiyuan
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Two Dimensions ◽  
Boundary Problems

On Free Boundary Рrоblems in Two Dimensions

2002 ◽  
pp. 414-425
Author(s):  
Hans Lewy ◽  
Tang Zhiyuan
Keyword(s):  
Free Boundary ◽  
Two Dimensions

Hydrodynamic jet incident on an uneven wall

2018 ◽  
Vol 28 (04) ◽  
pp. 771-827 ◽  
Author(s):  
Jianfeng Cheng ◽  
Lili Du
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Atmospheric Pressure ◽  
Free Boundary Problems ◽  
Applied Mathematics ◽  
Impinging Jet ◽  
Two Dimensions ◽  
Minimum Problem ◽  
Axially Symmetric ◽  
Optimal Decay

The axially symmetric free surface problem of an ideal incompressible jet issuing from a nozzle and impinging on an uneven wall is investigated in this paper. More precisely, we show that given a semi-infinitely long axially symmetric nozzle, a mass flux [Formula: see text] in the inlet and a constant atmospheric pressure, there exists a unique incompressible impinging jet whose free surface goes to infinity and is close to the impermeable wall at far field. Moreover, the free surface of the impinging jet initiates at the edge of the semi-infinitely long nozzle and the pressure remains the constant atmospheric pressure on the free surface. The main ingredient to show the existence and the uniqueness of the impinging jet is based on the variational method developed in a series of the celebrated works [Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981) 105–144; Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics (John Wiley & Sons, 1982)] by Alt, Caffarelli and Friedman. Furthermore, some important properties of the axially symmetric impinging jet, such as positivity of the radial velocity, asymptotic behavior of the impinging jet, and the optimal decay rate of the free surface and the impinging jet, are obtained. Moreover, the problem of the axially symmetric jet impinging on a hemispherical cup is also considered. Finally, we establish the well-posedness theory on the incompressible impinging jet in two dimensions.

scholarly journals Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient

2014 ◽  
Vol 24 (13) ◽  
pp. 2601-2626 ◽  
Author(s):  
Benoît Perthame ◽  
Min Tang ◽  
Nicolas Vauchelet
Keyword(s):  
Free Boundary ◽  
Tumor Growth ◽  
Traveling Wave ◽  
Wave Solution ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Necrotic Core ◽  
Traveling Wave Solution ◽  
Cancer Evolution

Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

scholarly journals Confined elasticae and the buckling of cylindrical shells

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Stephan Wojtowytsch
Keyword(s):  
Fourth Order ◽  
Cylindrical Shells ◽  
Two Dimensions ◽  
Integral Constraint ◽  
Type Problem ◽  
Material Parameters ◽  
First Order ◽  
Order Coefficient ◽  
Large Length ◽  
Small Excess

AbstractFor curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds {2\pi} and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion {2\pi+\Theta\delta^{\frac{1}{3}}+o(\delta^{\frac{1}{3}})} when a curve has length {2\pi+\delta} and determine first order coefficient {\Theta\approx 37}. We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.

Sign in / Sign up

Export Citation Format

Share Document