Natalya Kizilova

A review on constitutive equations proposed for mathematical modeling of laminar and turbulent flows of blood as a concentrated suspension of soft particles is given. The rheological models of blood as a uniform Newtonian fluid, non-Newtonian shear-thinning, viscoplastic, viscoelastic, tixotropic and micromorphic fluids are discussed. According to the experimental data presented, the adequate rheological model must describe shear-thinning tixotropic behavior with concentration-dependent viscoelastic properties which are proper to healthy human blood. Those properties can be studied on the corresponding mathematical problem formulations for the blood flows through the tudes or ducts. The corresponding systems of equations and boundary conditions for each of the proposed rheological models are discussed. Exact solutions for steady laminar flows between the parallel plates and through the circular tubes have been obtained and analyzed for the Ostwald, Hershel-Bulkley, and Bingham shear-thinning fluids. The influence of the model parameters on the velocity profiles has been studied for each model. It is shown, certain sets of fluid parameters lead to flattening of the velocity profile while others produce its sharpening around the axis of the channel. It is shown, the second-order terms in the viscoelastic models give the partial derivative differential equations with high orders in time and mixed space-time derivatives. The corresponding problem formulations for the generalized rhelogical laws are derived. Their analytical solutions in the form of a normal mode are obtained. It is shown, the dispersion equations produce an additional set for the speed of sound (so called second sound) in the fluid. It is concluded, the most general rheological model must include shear-thinning, concentration and second sound phenomena

The fractal structure of the blade-type lines of shells of extinct forms of ammonites of various types is investigated. These lines correspond to the connecting seams between the previous and new chambers that the mollusc builds during its gradual growth. Evolutionarily older forms have nearly straight or wavy seams with low tortuosity (the ratio of line length to end-point distance). Over the course of evolution, the shape of these lines has become more complex and acquired fractal properties, so that evolutionarily new types have connecting seams of a very complex shape. The fractal dimensions Df of the blade lines were measured using the box-counting method. It was shown that the fractal dimensions of the lobed lines of shells of different evolutionary ages increase from Df= 1.1 to Df = 1.9. In addition, with an increase in the chamber number, the Df values increase and remain larger on the outer surface of the chamber than on the inner one. The relationship between the fractal structure and an increase in the mechanical strength of the weld is discussed.