thin domains
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2021 ◽  
Vol 923 (2) ◽  
pp. 158
Author(s):  
David Ruffolo ◽  
Nawin Ngampoopun ◽  
Yash R. Bhora ◽  
Panisara Thepthong ◽  
Peera Pongkitiwanichakul ◽  
...  

Abstract The Parker Solar Probe (PSP) spacecraft is performing the first in situ exploration of the solar wind within 0.2 au of the Sun. Initial observations confirmed the Alfvénic nature of aligned fluctuations of the magnetic field B and velocity V in solar wind plasma close to the Sun, in domains of nearly constant magnetic field magnitude ∣ B ∣, i.e., approximate magnetic pressure balance. Such domains are interrupted by particularly strong fluctuations, including but not limited to radial field (polarity) reversals, known as switchbacks. It has been proposed that nonlinear Kelvin–Helmholtz instabilities form near magnetic boundaries in the nascent solar wind leading to extensive shear-driven dynamics, strong turbulent fluctuations including switchbacks, and mixing layers that involve domains of approximate magnetic pressure balance. In this work we identify and analyze various aspects of such domains using data from the first five PSP solar encounters. The filling fraction of domains, a measure of Alfvénicity, varies from median values of 90% within 0.2 au to 38% outside 0.9 au, with strong fluctuations. We find an inverse association between the mean domain duration and plasma β. We examine whether the mean domain duration is also related to the crossing time of spatial structures frozen into the solar wind flow for extreme cases of the aspect ratio. Our results are inconsistent with long, thin domains aligned along the radial or Parker spiral direction, and compatible with isotropic domains, which is consistent with prior observations of isotropic density fluctuations or flocculae in the solar wind.


Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3056
Author(s):  
Shai Gul ◽  
Reuven Cohen

We present efficient strategies for covering classes of thin domains in the plane using unit discs. We start with efficient covering of narrow domains using a single row of covering discs. We then move to efficient covering of general rectangles by discs centered at the lattice points of an irregular hexagonal lattice. This optimization uses a lattice that leads to a covering using a small number of discs. We compare the bounds on the covering using the presented strategies to the bounds obtained from the standard honeycomb covering, which is asymptotically optimal for fat domains, and show the improvement for thin domains.


2021 ◽  
pp. 1-31
Author(s):  
Bruna C. dos Santos ◽  
Sergio M. Oliva ◽  
Julio D. Rossi

In this paper, we analyze a model composed by coupled local and nonlocal diffusion equations acting in different subdomains. We consider the limit case when one of the subdomains is thin in one direction (it is concentrated to a domain of smaller dimension) and as a limit problem we obtain coupling between local and nonlocal equations acting in domains of different dimension. We find existence and uniqueness of solutions and we prove several qualitative properties (like conservation of mass and convergence to the mean value of the initial condition as time goes to infinity).


2021 ◽  
Vol 47 (5) ◽  
Author(s):  
Mehdi Elasmi ◽  
Christoph Erath ◽  
Stefan Kurz

AbstractWe present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform h- and p-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.


2021 ◽  
Vol 184 (2) ◽  
Author(s):  
Jory Griffin ◽  
Jens Marklof

AbstractWe study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann–Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry–Tabor conjecture in quantum chaos.


2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Giuseppe Cardone ◽  
Carmen Perugia ◽  
Manuel Villanueva Pesqueira

Author(s):  
Alaa Armiti-Juber ◽  
Tim Ricken

AbstractWe study fluid-saturated porous materials that undergo poro-elastic deformations in thin domains. The mechanics in such materials are described using a biphasic model based on the theory of porous media (TPM) and consisting of a system of differential equations for material’s displacement and fluid’s pressure. These equations are in general strongly coupled and nonlinear, such that exact solutions are hard to obtain and numerical solutions are computationally expensive. This paper reduces the complexity of the biphasic model in thin domains with a scale separation between domain’s width and length. Based on standard asymptotic analysis, we derive a reduced model that combines two sub-models. Firstly, a limit model consists of averaged equations that describe the fluid pore pressure and displacement in the longitudinal direction of the domain. Secondly, a corrector model re-captures the mechanics in the transverse direction. The validity of the reduced model is finally tested using a set of numerical examples. These demonstrate the computational efficiency of the reduced model, while maintaining reliable solutions in comparison with original biphasic TPM model in thin domain.


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