Nikolay Kutev

Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein–Gordon equation with combined power-type nonlinearity is studied. Finite time blow-up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data. A new functional, invariant under the flow of the equation, is introduced and investigated.

Approximation of the oscillatory blood flow using the Carreau viscosity model

The analysis of non-Newtonian flows in tubes is very important when studying the blood flow in different types of arteries. Usually the blood viscosity is defined by shear-dependent models, for example by the Carreau model, which represents the viscosity as a non-linear function of the shear-rate. In this paper the unsteady (oscillatory) 2D model of the blood flow in a straight tube is discussed theoretically and numerically. The solution of the quasilinear parabolic equation for the velocity is constructed using appropriate analytical functions. Further the corresponding numerical solution is approximated by similar analytical functions.

Finite time blow up of the solutions to boussinesq equation with linear restoring force and arbitrary positive energy

Abstract Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

Application of the Carreau viscosity model to the oscillatory flow in blood vessels

When studying the oscillatory flow in different types of blood vessels it is very important to know what type of the blood viscosity model has to be used. In general the blood viscosity is defined as a shear-thinning liquid, for which there exist different shear-dependent models, for example the Carreau model, which represents the viscosity as a non-linear function of the shear-rate. In some cases, however, the blood viscosity could be regarded as constant, i.e., the blood is treated as Newtonian fluid. The aim of the present work is to show theoretically and numerically some approximate limits of the Newtonian model application, when the blood vessel is assumed as a 2D straight tube. The obtained results are in agreement with other authors’ numerical results based on similar blood viscosity models.

Asymptotic Study of the Nonlinear Velocity Problem for the Oscillatory Non-Newtonian Flow in a Straight Channel

The studies of non-Newtonian flows, such as blood flows in arteries and polymer flows in channels have very important applications. The non-Newtonian fluid viscosity is modelled by the Carreau model (nonlinear with respect to the viscosity dependence on the shear rate). In the present paper the oscillatory flow of Newtonian and non-Newtonian fluids in a straight channel is studied analytically and numerically. The flow in an infinite straight channel is considered, which leads to a parabolic non-linear equation for the longitudinal velocity. The Newtonian flow velocity is found analytically, while the non-Newtonian velocity is found numerically by the finite-difference Crank-Nicolson method. In parallel, the non-Newtonian (Carreau) velocity is developed in an asymptotic expansion with respect to a small parameter. The zero-th order term of this expansion is exactly the Newtonian velocity solution. The first order term of the velocity expansion is found analytically in terms of higher order harmonics in time. As an example, the polymer solution HEC 0.5\(\%\) is considered. It is shown that the obtained asymptotic solution and the numerical solution for the non-Newtonian (Carreau) velocity are close for different values of the small parameter.

Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.