Mathematical Modelling of Blood Flow in an Artery with an Unsteady Stenosis using Power Law Fluid Model (Research Paper Sample)

Mathematical Modelling of Blood Flow in an Artery with an Unsteady Stenosis using Power Law Fluid Model
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Abstract
The simulation of blood flow in the arteries using the mathematical models that are not considered as Newtonian have always been accompanied with significant challenges. For instance, choosing a constitutive equation which is both appropriate and a representative of the exact behaviour of viscosity of the blood is a challenge since no such equations have ever been satisfactorily accepted in the field of research. Secondly, the motion of the blood is very complex to an extent that it is not easy to determine the numerical convergence of the complex non-linear equations used in controlling the motion of blood. This paper presents an examination of the pulsatile blood motion through a stenosis of the arteries. This is achieved through mathematical and numerical modelling involving unsteady blood flow in arteries. The study makes use of Power-law fluid model, where time-dependent stenosis is used. Computational and mathematical models assists in determining the gradient of blood , pressure, velocity, flow, impedance, the stenosis height/length (critical in this case), as well as the throat shear and stress. The relationship between these attributes are then investigated on the basis of spatial and temporary variable, with time and frequency of oscillation being the basic parameters. The main observation made here that an increase in the magnitude of stenosis is directly proportional to time, velocity size, impedance, frequency, as well as stress and shear within the stenosis zone. Furthermore, there is a difference in the values of the same quantities when Newtonian fluid (fluid that obeys the Newtonian equation) and non-Newtonian fluids are used.
Introduction
Death rates have risen among both the young and the old in the recent past, in the whole world. Among the major causes of such deaths are arteriosclerosis, heart attacks, and strokes. Heart attack and stroke are majorly a consequence of deposit of large amount of fatty acids, fibrin and cholesterol in the inner linings of the major blood vessels, among them being the artery CITATION Ran14 \l 2057 [1]. As a result of the deposits, some kind of obstruction to blood flow is formed in the blood vessels, thus preventing blood from flowing smoothly to all the organs of the human body. Sufficient research has not yet been done to establish the flow patterns in the artery with a stenosis CITATION Ran14 \l 2057 [1]. Through stenosis, blood flow is modelled in such a way that the above diseases can be identified and treated while still in their early stages of development.
Researchers have made several attempts under various conditions to find the relationship between the dynamically induced stresses with how stenosis is formed during the process of circulation of blood in the human body CITATION Ran14 \l 2057 [1]. Some of these studies were specifically meant to study the relationship between early stages atherosclerotic lesion growth and various attributes of blood flow. While most studies involved a lot of intensive numerical and experimental approaches of steady blood flow stenosis, while taking blood as a Newtonian fluid, other methods involved the roles played by hydrodynamic factors in determining the formation of stenosis during blood circulation.
Through similar investigations, studies have also revealed that human blood has the ability to behave like a non-Newtonian fluid with a very significant shear rate. Such behaviour is evident in diseased conditions where the flow is pulsatile, with blood flowing with low shear rates. The studies that were done by various scholars have provided knowledge other studies can be built up, to understand that blood flow rate actually depend on the geometrical patterns and upstream numbers. Two and three dimensional cases of the study of the study of the pulsatile flow effects in geometrical stenosis behaviour has also been studied.
A full understanding of hemodynamic behaviour of blood together with the use of power-law modelling technique provides a more comprehensive evaluation and examination of the blood flow via the steno tic artery from the physical view point CITATION Ran14 \l 2057 [1]. Mathematical models that captures rheological response of blood over a relatively larger shear rates are always the best in understanding the flow of blood with the artery. The blood viscosity models like Power-law and Quemada models have been found to be relatively more efficient in describing the complex behaviour of the non-Newtonian flow nature of blood. In order to effectively apply the models, the results from the models have to be compared to those obtained from the fluids that are Newtonian in nature.
In general, all the studies that have been done previously on the same subject have indicated that blood has a non-Newtonian behaviour, which affects the characteristics of flow via the stenotic artery. The biggest challenge posed in these studies are majorly as a result of the complexity of the chemical structure of the blood. Indeed, blood contains molecules that collides at a very higher rates, the modelling equations used so far have not fully exploited the topic in order to come up with a conclusive knowledge about the flow of blood.
This paper considers turbulent flow blood flow through an artery, together with composite sine stenosis which is time dependent CITATION Ran14 \l 2057 [1]. Modelling is done in such a way that the stud...