The paper presents analytical estimates of the proximity of solutions to boundary value problems for elastic-creeping layered composite materials widely used in technology under long-term loading, and also gives the corresponding averaged model for such materials. The estimates show the possibility of using the averaged model over a long time interval for the problem of loading by a constantly acting force. Previously, this statement was confirmed by numerical experiments comparing solutions of boundary value problems for an effective (averaged) model and direct numerical calculation using the original model for a highly inhomogeneous layered material. Analytical estimates are based on previously obtained estimates of the proximity of solutions to stationary problems of elasticity theory. For the one-dimensional model considered in this work, the following property is established: if the constitutive relations for various phases of the composite material are written as dependences of deformations on stresses, then the coefficients for the same form of writing the constitutive relations of the averaged model are obtained as simple weighted average values of similar coefficients for individual phases.
Keywords: layered composite materials, creep theory, averaging method, evaluating the efficiency of the averaging method, asymptotically long time interval
The principles of constructing a mathematical model of water purification based on the use of a biologically active layer, the bacteria of which absorb harmful impurities contained in the water, are considered. A system of equations is given, on the basis of which a water treatment model is constructed in the simplest element, which is a rod covered with a biofilm. The system of equations is a system that includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on a part of the surface of the domain, connected to each other through the boundary condition and the potential in an equation of hyperbolic type. Next, an asymptotic analysis of this system is carried out, which makes it possible to reduce the model of an individual element to the solution of a simple ordinary differential equation. On this basis, a model is proposed for the operation of the entire water treatment device containing a large number of such elements.
Keywords: water treatment, biologically active layer, asymptotic analysis of solutions in a thin area, biofilm, mathematical model of pollution treatment, system of differential equations of mixed type, optimization of biofilter designs, urban wastewater