TY - JOUR

T1 - Analysis of stationary solutions of the equations of one-dimensional hemodynamics

AU - Krivovichev, G. V.

AU - Egorov, N. V.

N1 - Publisher Copyright: © 2022 Saint-Petersburg State University. All rights reserved.

PY - 2022

Y1 - 2022

N2 - In last years, one-dimensional (1D) models of hemodynamics are widely used for the diagnostics of cardiovascular diseases, surgical operations, and for the analysis of the vascular pathologies eects. Models of this type are constructed by the averaging of the equations of the hydrodynamics of a viscous incompressible uid on the vessel cross-section, taking some simpli cations into account. The paper presents 1D models of blood ow, where the non-Newtonian properties of blood are considered. For the construction of models, the rheological relations for generalized Newtonian uids are used. The following models, applied in 2D and 3D simulations are considered: the power law model, the Carreau, Carreau Yasuda and Cross models, the simpli ed Cross model, the Yeleswarapu model, and Quemada and the modi ed Yeleswarapu models, which are dependent on hematocrit. For the closure of models, a model power-law representation of the dimensionless velocity pro le is used. The parameter of this dependence is varied during the calculations. The steady ow regime leads to the consideration of the nonlinear ordinary dierential equation on the cross-sectional area. For the power law model, the simpli ed Cross model, and the Quemada model, integrals of this equation are obtained. Conditions for the existence and uniqueness of the solution of the initial problem are obtained. During the calculations, the parameters for the iliac artery are considered. The in uence of the velocity pro le and hematocrit on the obtained solutions is investigated. It is shown, that the attening of the velocity pro le leads to a decrease in the length of the interval, where the stationary solutions exist. A similar situation occurs with an increase of hematocrit. The case of a vessel with stenosis, with the shape described by a model function, is considered. It is shown that a change in the geometric parameters aects the length of the interval of existence of the solution. The solutions obtained can be useful for the comparison of dierent 1D models of blood as a viscous uid and for testing programs that implement algorithms of numerical methods.

AB - In last years, one-dimensional (1D) models of hemodynamics are widely used for the diagnostics of cardiovascular diseases, surgical operations, and for the analysis of the vascular pathologies eects. Models of this type are constructed by the averaging of the equations of the hydrodynamics of a viscous incompressible uid on the vessel cross-section, taking some simpli cations into account. The paper presents 1D models of blood ow, where the non-Newtonian properties of blood are considered. For the construction of models, the rheological relations for generalized Newtonian uids are used. The following models, applied in 2D and 3D simulations are considered: the power law model, the Carreau, Carreau Yasuda and Cross models, the simpli ed Cross model, the Yeleswarapu model, and Quemada and the modi ed Yeleswarapu models, which are dependent on hematocrit. For the closure of models, a model power-law representation of the dimensionless velocity pro le is used. The parameter of this dependence is varied during the calculations. The steady ow regime leads to the consideration of the nonlinear ordinary dierential equation on the cross-sectional area. For the power law model, the simpli ed Cross model, and the Quemada model, integrals of this equation are obtained. Conditions for the existence and uniqueness of the solution of the initial problem are obtained. During the calculations, the parameters for the iliac artery are considered. The in uence of the velocity pro le and hematocrit on the obtained solutions is investigated. It is shown, that the attening of the velocity pro le leads to a decrease in the length of the interval, where the stationary solutions exist. A similar situation occurs with an increase of hematocrit. The case of a vessel with stenosis, with the shape described by a model function, is considered. It is shown that a change in the geometric parameters aects the length of the interval of existence of the solution. The solutions obtained can be useful for the comparison of dierent 1D models of blood as a viscous uid and for testing programs that implement algorithms of numerical methods.

KW - hemodynamics

KW - One-dimensional models

KW - stationary solutions

UR - http://www.scopus.com/inward/record.url?scp=85129612517&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85129612517

VL - 2022

SP - 64

EP - 87

JO - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1817-2172

IS - 1

ER -