Method for estimating parameters of coupled problem of interaction of gas flows loaded by solid particles with solids

The present paper outlines a method of processing and analysing experimental data based on the methodology of the solution of inverse heat transfer problems. An algorithm and the results of the computational and experimental study of heat transfer in the vicinity of the critical point of a specimen in a high-enthalpy particles-loaded flow are presented. One of the main difficulties here is how to determine coefficients of the mathematical model, which provide its adequacy to real processes. Direct measurement of most characteristics of heat transfer is usually impossible, and their theoretical estimates are often far from being true and often contradictory. That is why, the unknown parameters of the heat-balance equation at the external moving boundary of the specimen are determined from the inverse problem of heat transfer, which is solved by the method of iterative regularization. The results of experimental data processing for the interaction of particles-loaded flows with plane surfaces of the cylindrical specimen are also presented as well as the optimal experiment design problems for corresponded experiments

On complex-valued 2D eikonals. Part four: continuation past a caustic

Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -- in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension $2.$ We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions.Comment: 48 pages, 15 figure