The Manhattan and Lorentz Mirror Models: A Result on the Cylinder with Low Density of Mirrors

We study the Manhattan and Lorentz mirror models on an infinite cylinder of finite even width n, with the mirror probability p satisfying $$p<Cn^{-1}$$ p < C n - 1 , C a constant. We show that the maximum height along the cylinder reached by a walker is order $$p^{-2}$$ p - 2 . We observe an algebraic structure, which helps organise our argument. The models on the cylinder can be thought of as Markov chains on the Brauer (in the Mirror case) or Walled Brauer (in the Manhattan case) algebra, with the transfer matrix given by multiplication by an element of the algebra.

Time-periodic Poiseuille-type solution with minimally regular flow rate

The nonstationary Navier–Stokes equations are studied in the infinite cylinder Π = {x = (x', xn) ∈ Rn: x' ∈ σ ∈ R n – 1: – ∞ < xn < ∞, n = 2, 3} under the additional condition of the prescribed time-periodic flow-rate (flux) F(t). It is assumed that the flow-rate F belongs to the space L2(0, 2π), only. The time-periodic Poiseuille solution has the form u(x, t) = (0, ... , 0, U(x', t)), p(x,t) = –q(t)xn + p0(t), where (U(x', t), q(t)) is a solution of an inverse problem for the time-periodic heat equation with a specific over-determination condition. The existence and uniqueness of a solution to this problem is proved.

Modeling of Fluid Flow in a Flexible Vessel with Elastic Walls

We exploit a two-dimensional model (Ghosh et al. in Q J Mech Appl Math 71(3):349–367, 2018; Kozlov and Nazarov in Dokl Phys 56(11):560–566, 2011, J Math Sci 207(2):249–269, 2015) describing the elastic behavior of the wall of a flexible blood vessel which takes interaction with surrounding muscle tissue and the 3D fluid flow into account. We study time periodic flows in an infinite cylinder with such intricate boundary conditions. The main result is that solutions of this problem do not depend on the period and they are nothing else but the time independent Poiseuille flow. Similar solutions of the Stokes equations for the rigid wall (the no-slip boundary condition) depend on the period and their profile depends on time.

Numerical Study of Crosshole Electromagnetic Tunnel Detection

Electromagnetic tunnel detection is studied numerically using a 3D analytic infinite lossy homogeneous space solution to magnetic dipole radiation and scattering from an infinite cylinder, in a crosshole context. At low frequencies this serves as a model for a transmit coil radiating a time-varying magnetic field that is then detected from the open-circuit voltage induced on a receive coil. Numerical simulations illustrate how various parameters influence the signal strength and the ability to discern the scattered signal. Tunnel detection is achieved at relatively high frequencies (but below typical GPR frequencies) for fresh water saturated sand and for weathered granite, which are lower loss media; for the coil and tunnel parameters used here, optimum frequencies appear to be between 100 kHz and 1 MHz. Tunnel detection for fresh water saturated clay, a much more lossy medium, can be achieved at a quite low frequency, with an optimum frequency between 1 and 10 kHz. These results suggest that, when a resonant coil system is employed, tunnel detection may be possible in a wider range of earth media than previously reported, when the best-suited choice of frequency is employed.

SOLUTION OF THE TWO-DIMENSIONAL WAVE DIFFRACTION PROBLEM ON FRACTAL BODIES BY THE PATTERN EQUATIONS METHOD

The solution of the two-dimensional wave diffraction problem for infinite cylinder of complex cross-section was considered by using the pattern equations method (PEM). A triangle and a Koch snowflake of first iteration were chosen as the geometry of the cross-sections of the cylinder. The numerical algorithms of the PEM for a single scatterer and for a group of bodies with the Dirichlet condition on their boundary are briefly presented, and the results of numerical calculations of the scattering characteristics for the above geometries are obtained using the PEM and the method of continued boundary conditions (MCBC). To check the convergence of the numerical algorithm in both methods, the optical theorem was used. The limits of applicability of the PEM for fractal scatterers are established. It is shown that for all convex bodies the algorithm of the PEM is sufficiently stable and allows obtaining calculation results with an accuracy acceptable in practice. In the case of a non-convex body, namely, a Koch snowflake, the algorithm of the PEM for a single scatterer turns out to be unstable and the acceptable accuracy can be obtained only if this geometry is considered as a group of bodies composed of convex geometries (for example, triangles).

CODE VERIFICATION OF MPACT USING GANAPOL CRITICAL ROD BENCHMARK

This work is dedicated to the code verification of MPACT, which is developed under the Consortium for Advanced Simulation of Light Water Reactors by the University of Michigan and Oak Ridge National Laboratory, where the numerical solution is compared to the reference solution of a benchmark problem with a known analytical solution. In this work, Benchmark Problem 3.4 in Barry Ganapol’s benchmark book was chosen as an MOC code verification test problem. Problem 3.4 is a bare cylinder of infinite height, which is an excellent benchmark problem for 2D MOC. To ensure that this benchmark problem exercised the same code as typically used by MPACT, the bare rod configuration was surrounded by a bounding box filled with a non-scattering material. To avoid implementing a critical rod search in the MPACT code, the MPACT analysis was performed using cross sections that yielded the given c, the average number of secondary neutrons per collision, and a rod radius that was the corresponding critical rod radius. MPACT agreed with all cases to within a few pcm. The convergence behavior was studied. The results show a 2nd order radial convergence, consistent with flat-source approximation. The convergence curves with respect to ray spacing and polar angle quadrature set order were also obtained. The other quantity of interest tabulated for Problem 3.4 was the radial distribution of the scalar flux. Two configurations were analyzed, and the resultant radial flux profiles agreed very well with the tabulated results. The verification of the production neutronics code MPACT has been augmented by the addition of the analytical solutions for an infinite cylinder from the Ganapol benchmark book. These test cases can be included in the regression suite for MPACT.