The solution of problems on an self-similar electrochemical shaping by means of hydrodynamic analogy

2008 ◽  
Vol 6 ◽  
pp. 150-155
Author(s):  
A.R. Urakov ◽  
A.A. Gordeev ◽  
S.S. Porechny
Keyword(s):  
Boundary Condition ◽  
Geometrical Similarity ◽  
Hydrodynamic Analogy ◽  
Self Similar ◽  
Electrochemical Shaping

Self-similar (at preservation of geometrical similarity of borders) solutions of non-stationary Hele-Shaw problems in connection to an electrochemical shaping are considered. The problem of a flow about an arch of a circle on which the border condition for Zhukovsky’s function has the form similar to a boundary condition of a self-similar problem is used for the solution.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

2004 ◽  
Vol 47 (1) ◽  
pp. 30-37
Author(s):  
Xinyu He
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Local Existence ◽  
Similar Solution ◽  
Navier Stokes ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

scholarly journals On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Zhilei Liang
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Newtonian Equation ◽  
Comparison Methods ◽  
Self Similar

We identify the blow-up set of solutions to the problem , , , , , and , , where . We obtain that the blow up set satisfies . The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

scholarly journals Recursive evolution of spin-wave multiplets in magnonic crystals of antidot-lattice fractals

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Gyuyoung Park ◽  
Jaehak Yang ◽  
Sang-Koog Kim
Keyword(s):  
Boundary Condition ◽  
Hausdorff Dimension ◽  
Magnetic Fields ◽  
Spin Wave ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Micromagnetic Simulations ◽  
Self Similar ◽  
Magnonic Crystals ◽  
Magnonic Crystal

AbstractWe explored spin-wave multiplets excited in a different type of magnonic crystal composed of ferromagnetic antidot-lattice fractals, by means of micromagnetic simulations with a periodic boundary condition. The modeling of antidot-lattice fractals was designed with a series of self-similar antidot-lattices in an integer Hausdorff dimension. As the iteration level increased, multiple splits of the edge and center modes of quantized spin-waves in the antidot-lattices were excited due to the fractals’ inhomogeneous and asymmetric internal magnetic fields. It was found that a recursive development (Fn = Fn−1 + Gn−1) of geometrical fractals gives rise to the same recursive evolution of spin-wave multiplets.

scholarly journals Convergence to diffusion waves for solutions of 1D Keller-Segel model

2021 ◽  
Author(s):  
Fengling Liu ◽  
Nangao Zhang ◽  
Changjiang Zhu
Keyword(s):  
Boundary Condition ◽  
Cauchy Problem ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Initial Boundary Value

In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initial-boundary value problem) of one-dimensional Keller-Segel model. For the Cauchy problem, we prove that the solutions time-asymptotically converge to the nonlinear diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which is derived by Darcy’s law, as in [11, 28]. For the initial-boundary value problem, we consider two cases: Dirichlet boundary condition and null-Neumann boundary condition on (u, ρ). In the case of Dirichlet boundary condition, similar to the Cauchy problem, the asymptotic profile is still the self similar solution of the corresponding parabolic equation, which is derived by Darcy’s law, thus we only need to deal with boundary effect. In the case of null-Neumann boundary condition, the global existence and asymptotic behavior of solutions near constant steady states are established. The proof is based on the elementary energy method and some delicate analysis of the corresponding asymptotic profiles.

One-Dimensional Fin-Tip Boundary Condition Correction II

2001 ◽  
Vol 22 (5) ◽  
pp. 35-40 ◽  
Author(s):  
D. C. Look Jr ◽  
Arvind Krishnan
Keyword(s):  
Boundary Condition ◽  
One Dimensional

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar
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