The weak solvability of the steady problem modelling the flow of a viscous incompressible heat-conductive fluid through the profile cascade

Purpose The paper aims to theoretically study the mathematical model of a steady flow of a heat-conductive incompressible viscous fluid through a spatially periodic plane profile cascade. Design/methodology/approach Reduction of the infinite periodical problem to one period. Leray-Schauder fixed point principle was used. Findings This study proves the existence of a weak solution for arbitrarily large given data (i.e. the inflow velocity and the acting specific body force). Practical implications The author proposed a special boundary condition on the outflow of the domain not only for the velocity and pressure but also for the temperature. Originality/value To the author’s knowledge, the problem has not been studied earlier. More detailed overview is given in the paper in the first part.

On the stationary vibrations of a rectangular plate subjected to stress prescribed partially at the circumference

The stationary periodical problem of a vibrating rectangular plate, stressed at a segment while fixed elsewhere at one of its edges, is considered. Using the finite Fourier transformation, the problem is converted to a singular integral equation that in turn can be reduced to an infinite system of algebraic equations. The truncation of the algebraic system is justified.