Numerical modelling of generalized Newtonian fluids in bypass tube
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Numerical modelling of generalized Newtonian fluids in bypass tube Radka Keslerova´ 1
ˇ ´ s Padelek ˇ 2 · Hynek Rezn´ ıcˇ ek1 · Tomaˇ
Received: 4 October 2018 / Accepted: 8 March 2019 / Published online: 27 May 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The following paper describes a numerical simulation of a complete bypass of a stenosed human artery. The considered geometry consists of the narrowed host tube and the bypass graft with a 45-degree angle of connection. Different diameters of the narrowing are tested. Blood is the fluid with shear rate–dependent viscosity; therefore, various rheology mathematical models for generalized Newtonian fluids are considered, namely Cross model, modified Cross model, Carreau model, and Carreau-Yasuda model. The fundamental system of equations is based on the system of generalized Navier-Stokes equations. Generalized Newtonian fluids flow in a bypass tube is numerically simulated by using a SIMPLE algorithm included in the open-source CFD tool, OpenFOAM. The aim of this work is to compare the numerical results for the different mathematical models of the viscosity with the changing diameter of the narrowed channel. Keywords Generalized Newtonian fluids · Generalized Navier-Stokes equations · OpenFOAM · Bypass Mathematics Subject Classification (2010) 65L06 · 65N08 · 76A05 · 76A10 · 76D05
Communicated by: Pavel Solin Radka Keslerov´a
[email protected] ˇ Hynek Rezn´ ıcˇ ek [email protected] Tom´asˇ Padˇelek [email protected] 1
Department of Technical Mathematics, FME CTU in Prague, Karlovo Nam. 13, 121 35 Praha 2, Czech Republic
2
Department of Transport Systems, FTS CTU in Prague, Konviktsk´a 20, 110 00 Praha 1, Czech Republic
2048
R. Keslerov´a et al.
1 Introduction A blood is a body fluid in humans which is composed of blood cells (red blood cells, white blood cells, and platelets) suspended in a plasma. The blood cells comprise 45% of the blood fluid. Other 55% of the blood fluid is the plasma which is a yellow liquid consisted of mostly water (90% of volume) and the rest are organic substances. The blood cells are produced in a bone marrow by hematopoiesis and they are decomposed in a liver. The red, white blood cells, and platelets in the scanning electron microscope image are presented in Fig. 1. Description of the blood cells, see, e.g., [10]: Red blood cells or erythrocytes (96%) contain hemoglobin (red blood coloring) and carry oxygen and collect carbon dioxide. Hemoglobin is a protein that makes easy transportation of oxygen from the lungs to tissues and carbon dioxide from tissues to lungs to be exhaled. A lifetime of red blood cells is 120 days. Red blood cells in mammals are non full-bodied they do not have a nucleus and organelle. The condition when blood contains too few red blood cells is known as anemia, while the other condition when blood contains too many red blood cells is known as polycythemia. White blood cells or leukocytes (3%) are a part of an immune system and defend the body again
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