Roll of Newtonian and Non-Newtonian Motion in Analysis of Two-Phase Hepatic Blood Flow in Artery during Jaundice

Singh, Abha; Khan, Rizwan Ahmad; Kushwaha, Sumit; Alshenqeeti, Tahani

doi:https://doi.org/10.1155/2022/7388096

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7388096 | https://doi.org/10.1155/2022/7388096

Roll of Newtonian and Non-Newtonian Motion in Analysis of Two-Phase Hepatic Blood Flow in Artery during Jaundice

Abha Singh,¹Rizwan Ahmad Khan,²Sumit Kushwaha,³and Tahani Alshenqeeti⁴

Academic Editor: Chin-Chia Wu

Received13 Apr 2022

Revised10 Jun 2022

Accepted13 Jun 2022

Published12 Jul 2022

Abstract

Biomathematics is an interdisciplinary subject consisting of mathematics and biology, which is widely applicable for the analysis of biological problems. In this paper, we provide a mathematical model of two-phase hepatic blood flow in a jaundice patient’s artery. The blood flow is thought to be a two-phased process. The clinical data of a jaundice patient (blood pressure and hemoglobin) is gathered. To begin, hemoglobin is transformed into hematocrit, and blood pressure is turned to a decline in blood pressure. For the examination of hepatic arteries in Newtonian and non-Newtonian movements, a mathematical model is constructed. The relationship between two-phase blood flow flux and blood pressure reduction in the hepatic artery is established. For various hematocrit levels, the blood pressure decrease is determined. The patient’s states are defined by the slope of the linear relationship between computed blood pressure decrease and hematocrit.

1. Introduction

Due to globalization and the multidisciplinary development of life science and mathematical science, mathematical modeling becomes a topic of interest for the scientific community. Mathematical modeling plays a significant role to understand the intricacies of infectious diseases [1–3]. It helps to study the mechanisms responsible for observed epidemiological patterns, assesses the efficiency of control policies, and predicts epidemiological inclinations. Nowadays, human beings suffer from many disorders due to global warming, pressure, and stress to lifestyle [4]. Some physiological elements that occur during exposure to weightlessness may include alteration in blood flow to the liver. So many disorders can be seen like diabetes, blood pressure, and other diseases estimation of hepatic blood flow.

Bio-mathematics is an emerging and dynamic field. The term bio-mathematics refers to the use of quantities and mathematical methods to solve biological problems [1, 2]. The bio-mathematics is an interdisciplinary subject to understand the mechanical properties of living tissues, anatomy and physiology in health and illness, and introductory biomechanics: from cell to organisms, blood flow, and microcirculation [5–7]. Bio-fluid mechanics is a well-established branch of bio-mathematics with the help of its normal functions, changed due to alternation via mathematical analysis [7].

It is well known that the heart circulates blood in the body using elastic tubes: the arteries, capillaries, and veins. A proper flow of blood is necessary for good health as it circulates oxygen and other essential nutrients to various parts of the body. Two main factors that affect the blood flow are known as the blood vessels and the properties of blood. Therefore, the modeling of blood flow is very complex as compared to other fluid flows. Blood is a non-Newtonian fluid; hence its viscosity is not constant, resulting in nonlinearity of stress and strain rate relation [8]. However, this fact can be approximated by the power law or Casson fluid model [9]. The walls of blood vessels are elastic and disobey Hooke’s law. The curvature of the wall can also affect the properties of blood flow. Due to the composite elastic materials of the wall, a nonlinear stress-strain relation exists. Blood flow is pulsatile resulting in the periodic nature of pressure gradient and the fluid velocity [10]. It is caused due to the beating of the heart. The radii of the tube are variable and can be further divided into multiple veins. Most of the flow is not fully developed due to the high Reynolds’s numbers and only inlet flow occurs. Blood is a nonhomogeneous, unsteady, and pulsatile flow that flows from the elastic tube which is branched repeatedly.

The first attempt to understand the mechanism of blood flow through various types of vessels was made by Leonardo da Vinci and Descartes [11]. Later, Aird and Silverman explained the nature of the circulation of blood in the cardiovascular system [11, 12]. In 1899, the first mathematical model of propagation of arterial pulse waves is given by Frank [13]. Sharan and Popel [14] explained the execution of the two-phase model for the blood flow in slim tubes and explained how the powerful extended viscosity near the wall affects blood flow [14]. The viscosity of human blood concerning its performance ability in a high static magnetic field is discussed by Haik et al. [15]. Formaggia et al. [16] have studied the one-dimensional nonlinear system, which defines the blood as compliant with arteries [16]. They used the Navier–Stoke′s equation, which shows the algebraic relation between intramural pressure and vessel. Jiang et al. [17] concentrated on the physical demonstration and propelled reproductions of fuel-fluid two-stage stream streams in atomization and splashes [17]. It was proposed that the large blood vessels behave like a Newtonian. The proposed model calculates the parameters of blood flow have no restrictions on the vessel size [18]. The precarious and incompressible stream of non-Newtonian liquid through composite stenosis was also considered [19]. The two phases Bingham model was proposed by Ahnert et al. [20, 21]. They used the differential equation for solutions and solve the mathematical problems. They also used the drift-flux model which defines the behavior of the model like the resistance of the flow and also use the different types of parameters. Kumawat et al. [22] mathematically analyze two-phase blood flow through a stenosed curved artery with hematocrit and temperature dependent viscosity [22]. A patient-specific artery geometry in the presence of stenosis (plaque) was considered. In 2016, Achaba et al. [23] studied the blood flow in arteries through a non-Newtonian viscosity model [23]. They define the two major difficulties. The first is a constitutive equation. No one model accepted the behavior of blood viscosity, and the second is the highly nonlinear equation for blood motion. They used a two-dimensional equation. The study reveals that the power law model is better for non-Newtonian blood flow. Mekheimer and their research group did extensive work on the nanoparticles drug delivery to blood hemodynamics in diseased organs [24–28]. Sharma et al. [29] proposed entropy analysis of thermally radiating MHD slip flow of hybrid nanoparticles (Au-/Blood) through a tapered multistenosed artery [29]. Bhatti and Abdelsalam [30] studied the peristaltic propulsion of hybrid nanofluid flow with Tantalum and Gold nanoparticles under magnetic effects [30].

In this article, we consider the hepatic blood flow in arterioles with respect to the nature of the hepatic circulatory system in humans. The Herschel–Bulkley non-Newtonian model in Bio-fluid physiological is investigated.

2. Important Formulations

In this section, we discuss the meaning of Newtonian and non-Newtonian flow. The mathematical formulation can be found in our previous articles [31–35].

2.1. Covariant Vectors

A set of n function of n coordinates in a coordinate system are said to form the components of covariant vectors if they transform to another coordinate system according to the following rule [36]:

2.2. Contravariant Vectors

A set of n function of n coordinates in a coordinate system are said to form the components of covariant vectors if they transform to another coordinate system according to the following rule [36]:

2.3. Tensors of Second Order

2.3.1. Covariant Tensors of Order Two

A set of functions of coordinates in a coordinates system are said to form the components of a covariant tensor of order two if they transform to another coordinate system as follows [37]:

2.3.2. Contravariant Tensors of Order Two

A set of n² functions of n coordinates in a coordinates system are said to form the components of a contravariant tensor of order two, if they transform to another coordinate system as follows [37]:

2.3.3. Mixed Tensors of Order Two

A set of n² functions of n coordinates in a coordinates system are said to form the components of a mixed tensor of order two, if they transform to another coordinate system as follows [37]:

2.4. Christoffel’s Symbols

2.4.1. Christoffel Symbol of First Kind

The Christoffel 3 index symbol of the first kind are denoted by and defined by the following equation [37]:

2.4.2. Christoffel Symbol of Second Kind

The Christoffel 3 index symbol of the second kind is denoted by and defined by the following equation [37]:

3. Modeling of Blood Flow through Vessels

3.1. Numerical Analysis for Hepatic Arteries

The power law equation of continuity is written as follows [38]:

The motion equation is expressed as follows [39]:where is power law constitutive equation. The blood density equation iswhere is the volume ratio of blood cells and is hematocrit. The viscosity of a mixture of blood is expressed as follows:

In cylindrical form,

Tonsorial form in cylindrical coordinates,

The conjugate metric tensor is as follows:

The nonvanishing Christoffel’s symbols of 1^st kind are as follows:

And, all other Christoffel’s symbols of 1^st kind are zero. The nonvanishing Christoffel’s symbols of 2^nd kind are as follows:

And, all other Christoffel’s symbols of 2^nd kind are zero. Relation between contravariant and component of velocity of-blood flow is as follows:

The component of: . The components of shearing stress tensor,

The covariant derivative of is

3.2. Solution for Newtonian

The blood flow in the artery is symmetric with respect to the axis. Hence, , and do not depend upon , also . Since only one component of velocity, which is along the axis is effective. Now . Flow is steady, then obtain the following equation [39]:

Equation of continuity reduces to

Equation of motion reduces to

Z-component,

Let us assume that the pressure gradient of blood flow in arteries is .

Integration gives us,where are boundary conditions and is constant.

Integration of equation (26) is as follows:

Again, using the second boundary condition , then from equation (27), we get the constant of integration value B is

Putting the above-given value of B from equation (27), we obtain the following equation:where R is the radius of artery. If is the flux through the artery tube, then

Therefore,

Now,

Integration of both sides with limit initial to final,

3.3. Solution for Non-Newtonian

Now

The blood flow,

Equation of continuity reduces to,

Equation of motion reduces to,

Z-component,

Let us assume that the pressure gradient of blood flow in arteries is . Then,

We arrive at,

Let us consider boundary conditions; , and . We arrive at

After integration,

Under no slip boundary condition (, ), equation (41) can be written as follows:

By inserting the value of B in (41),

Equation (43) describes the velocity of blood flow in arteries. Total flux flow of blood () through a tube of arteries is defined as follows:

Thus,

Equation of motion,

Integration of equation (46) gives us the pressure drop of blood.

4. Analysis for Hepatic Arteries

4.1. For Newtonian Motion

If Q is the flux through the tube, then

Using (21), equation (48) can be written as follows:

Now,

Integration of both sides with limit initial to final,

Table 1 demonstrates the measured hemoglobin and standard BP of a jaundice patient (name: A, age: 43, and sex: F). Firstly, the hemoglobin is converted to hematocrit using the relation mentioned in Table 2. After that, the BP drop is assessed from standard BP using the following relation [31]:where is systolic and is diastolic. Later, the BP drop is converted to BP (in ). Let us consider that and blood pressure drop is 4332.9. Equation (11) gives us,

The relation between flow flux and blood pressure drop of two-phase blood flow in the hepatic artery is expressed as follows:

Using (10) & (63), the solution of equation (54) is obtained as follows:

Different values of gives us the blood pressure drop (Table 1). Figure 1 demonstrates the BP drop as a function of hematocrit. It is observed that the BP drop is increased as hematocrit enhances, which means that the hemoglobin of the patient is normal.

4.2. For Non-Newtonian Motion

The total flow-flux of blood through a tube of the arteries is Q defined by the following equation:

Both sides take power n,

We know that the for non-Newtonian motion. Hence,

Integration of both sides with limit initial to final,

Pressure drop of blood,

Let us consider that H = 0.02858 and BP drop = 4332.90 and

Using equation (11),

Again, by using equation (11), we arrive at,

Rearrangement of (59) provides the blood flow flux ,

Solution of equation (64) gives the value of ,

Inserting the values of , , , and in (60), we arrive at,

By inserting the value of in equation (66), we get a BP drop (Table 3).

Figure 2 demonstrates the relation between the calculated BP drop and hematocrit for non-Newtonian motion. It is observed that the BP drop increases as the hematocrit increases.

Figures 3(a) and 3(b). Mathematical data of BP drop (calculated using (66)) and BP (in ) as a function of Hematocrit for non-Newtonian motion. Solid lines represent the linear fitting.

(a)

(b)

Rheological properties of the Reiner–Rivlin fluid model for blood flow through a tapered artery with stenosis have been studied by Akbar et al. [45]. Elogail and Mekheimer [46] implemented a numerical study that simulates blood flow through a microvessel involving oxytactic microorganisms and nanoparticles [46]. The oxytactic microorganisms exhibit negative chemotaxis to gradients of oxygen (oxygen repellents). These microorganisms are to batter infected hypoxic tumor cells as drug carriers [46]. Awad et al. [47] studied the flow of a non-Newtonian fluid with nonzero yield stress [47]. Navier stokes equation is used to simulate this subject mathematically. The elasticity on the stenosis arterial walls is simulated by Rubinow and Keller model and the Mazumdar model [47]. Kumawat et al. [22] mathematically analyze two-phase blood flow through a stenosed curved artery with hematocrit and temperature dependent viscosity [22].

Last but not least, we developed mathematical for blood flow in arteries during jaundice. We collected the blood pressure data of jaundice patients from the hospital and later applied a mathematical model to analyze the data.

5. Conclusion

We developed a deterministic model of the link between hematocrit and blood pressure fluctuations in jaundice patients in this study. The mathematical analysis is validated on the clinical data and calculated BP values of the jaundice patient (name: A, age: 43, and sex: F). The role of blood flow and hemoglobin malfunction is investigated in this research. It is feasible to propose a patient for better therapy based on the trend line of the association between hematocrit and BP decline. With the use of model Newtonian motion and non-Newtonian motion, if the trend line exhibits positive slope, then the patient’s hemoglobin is normal. If the trend line indicates a negative slope, however, the management of jaundice patients should be modified. This study is useful to predict the hemoglobin status of jaundice patients on the basis of blood pressure measurement.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Conceptualization was done by A. S. and R. A. K.; data curation was done by A. S., R. A. K., and T. A.; formal analysis was done by A. S. and R. A. K.; investigation was done by A. S., S. K., and R. A. K.; methodology was done by A. S. and R. A. K.; project administration and validation was done by A. S. and R. A. K; visualization was done by A. S., T. A., and R. A. K; writing-original draft was done by A. S., S. K., T. A., and R. A. K.; writing-review and editing was done by A. S., S. K., and R. A. K. All authors read and approved the final manuscript.

References

R. Cassidy, N. S. Singh, P. R. Schiratti et al., “Mathematical modelling for health systems research: a systematic review of system dynamics and agent-based models,” BMC Health Services Research, vol. 19, p. 845, 2019.
View at: Publisher Site | Google Scholar
B. Divya and K. Kavitha, “A review on mathematical modelling in biology and medicine,” Adv Math Sci J, vol. 9, no. 8, pp. 5869–5879, 2020.
View at: Publisher Site | Google Scholar
G. Kaiser, “Mathematical modelling and applications in education,” Encyclopedia of mathematics education, Springer, Cham, pp. 553–561, 2020.
View at: Google Scholar
P. Kumar Rai and J. S. Singh, “Invasive alien plant species: their impact on environment, ecosystem services and human health,” Ecological Indicators, vol. 111, Article ID 106020, 2020.
View at: Publisher Site | Google Scholar
A. Saleem, S. Akhtar, and S. Nadeem, “Bio-mathematical analysis of electro-osmotically modulated hemodynamic blood flow inside a symmetric and nonsymmetric stenosed artery with joule heating,” International Journal of Biomathematics, vol. 15, no. 02, 2021.
View at: Publisher Site | Google Scholar
Y. Liu and W. Liu, “Blood flow analysis in tapered stenosed arteries with the influence of heat and mass transfer,” J. Appl. Math. Comput., vol. 63, no. 1-2, pp. 523–541, 2020.
View at: Publisher Site | Google Scholar
H. Sadaf, “Bio-fluid flow analysis based on heat transfer and variable viscosity,” Applied Mathematics and Mechanics, vol. 40, no. 7, pp. 1029–1040, 2019.
View at: Publisher Site | Google Scholar
W. I. Gabr, H. T. Dorrah, and M. S. Elsayed, “A new symbolic-based continuous (infinite) modal approach for systems control and operation using computational mathematics,” Ain Shams Engineering Journal, vol. 11, no. 3, pp. 575–586, 2020.
View at: Publisher Site | Google Scholar
R. Ahmed, N. Ali, K. Al-Khaled, S. U. Khan, and I. Tlili, “Finite difference simulations for non-isothermal hydromagnetic peristaltic flow of a bio-fluid in a curved channel: applications to physiological systems,” Computer Methods and Programs in Biomedicine, vol. 195, Article ID 105672, 2020.
View at: Publisher Site | Google Scholar
I. Sazonov, S. Y. Yeo, R. L. T. Bevan, X. Xie, R. Van Loon, and P. Nithiarasu, “Modelling pipeline for subject-specific arterial blood flow-A review,” International Journal for Numerical Methods in Biomedical Engineering, vol. 27, no. 12, pp. 1868–1910, 2011.
View at: Publisher Site | Google Scholar
W. C. Aird, “Discovery of the cardiovascular system: from galen to william Harvey,” Journal of Thrombosis and Haemostasis, vol. 9, pp. 118–129, 2011.
View at: Publisher Site | Google Scholar
M. E. Silverman, “William Harvey and the discovery of the circulation of blood,” Clinical Cardiology, vol. 8, no. 4, pp. 244–246, 1985.
View at: Publisher Site | Google Scholar
K. H. Parker, “A brief history of arterial wave mechanics,” Medical, & Biological Engineering & Computing, vol. 47, no. 2, pp. 111–118, 2009.
View at: Publisher Site | Google Scholar
M. Sharan and A. S. Popel, “A two phase model for flow of blood in narrow tubes with increased effective viscosity near the wall,” Biorheology, vol. 38, no. 5-6, pp. 415–428, 2001.
View at: Google Scholar
Y. Haik, V. Pai, and C. J. Chen, “Apparent viscosity of human blood in a high static magnetic field,” Journal of Magnetism and Magnetic Materials, vol. 225, no. 1-2, pp. 180–186, 2001.
View at: Publisher Site | Google Scholar
L. Formaggia, D. Lamponi, and A. Quarteroni, “One-dimensional models for blood flow in arteries,” Journal of Engineering Mathematics, vol. 47, no. 3/4, pp. 251–276, 2003.
View at: Publisher Site | Google Scholar
X. Jiang, G. A. Siamas, K. Jagus, and T. G. Karayiannis, “Physical modelling and advanced simulations of gas–liquid two-phase jet flows in atomization and sprays,” Progress in Energy and Combustion Science, vol. 36, no. 2, pp. 131–167, 2010.
View at: Publisher Site | Google Scholar
A. E. Medvedev and V. M. Fomin, “Two-phase blood-flow model in large and small vessels,” Doklady Physics, vol. 56, no. 12, pp. 610–613, 2011.
View at: Publisher Site | Google Scholar
M. H. Mansour, A. Kawahara, and M. Sadatomi, “Experimental investigation of gas–non-Newtonian liquid two-phase flows from T-junction mixer in rectangular microchannel,” International Journal of Multiphase Flow, vol. 72, pp. 263–274, 2015.
View at: Publisher Site | Google Scholar
T. Ahnert, A. Münch, and B. Wagner, “Models for the two-phase flow of concentrated suspensions,” European Journal of Applied Mathematics, vol. 30, no. 3, pp. 585–617, 2019.
View at: Publisher Site | Google Scholar
T. Ahnert, A. Münch, B. Niethammer, and B. Wagner, “Stability of concentrated suspensions under Couette and Poiseuille flow,” Journal of Engineering Mathematics, vol. 111, no. 1, pp. 51–77, 2018.
View at: Publisher Site | Google Scholar
C. Kumawat, B. K. Sharma, and K. S. Mekheimer, “Mathematical analysis of two-phase blood flow through a stenosed curved artery with hematocrit and temperature dependent viscosity,” Physica Scripta, vol. 96, no. 12, Article ID 125277, 2021.
View at: Publisher Site | Google Scholar
L. Achaba, M. Mahfouda, and S. Benhadida, “Numerical study of the non-Newtonian blood flow in a stenosed artery using two rheological models,” Thermal Science, vol. 20, no. 2, pp. 449–460, 2016.
View at: Publisher Site | Google Scholar
K. S. Mekheimer, R. E. Abo-Elkhair, S. I. Abdelsalam, K. K. Ali, and A. M. Moawad, “Biomedical simulations of nanoparticles drug delivery to blood hemodynamics in diseased organs: synovitis problem,” International Communications in Heat and Mass Transfer, vol. 130, Article ID 105756, 2022.
View at: Publisher Site | Google Scholar
A. M. Abdelwahab, K. S. Mekheimer, K. K. Ali, A. El-Kholy, and N. S. Sweed, “Numerical simulation of electroosmotic force on micropolar pulsatile bloodstream through aneurysm and stenosis of carotid,” Waves in Random and Complex Media, vol. 21, pp. 1–32, 2021.
View at: Publisher Site | Google Scholar
N. S. Sweed, K. S. Mekheimer, A. El‐Kholy, and A. M. Abdelwahab, “Alterations in pulsatile bloodstream with the heat and mass transfer through asymmetric stenosis artery: erythrocytes suspension model,” Heat Transfer, vol. 50, no. 3, pp. 2259–2287, 2021.
View at: Publisher Site | Google Scholar
S. I. Abdelsalam, K. S. Mekheimer, and A. Z. Zaher, “Alterations in blood stream by electroosmotic forces of hybrid nanofluid through diseased artery: aneurysmal/stenosed segment,” Chinese Journal of Physics, vol. 67, pp. 314–329, 2020.
View at: Publisher Site | Google Scholar
K. S. Mekheimer, A. Z. Zaher, and A. I. Abdellateef, “Entropy hemodynamics particle-fluid suspension model through eccentric catheterization for time-variant stenotic arterial wall: catheter injection,” International Journal of Geometric Methods in Modern Physics, vol. 16, no. 11, Article ID 1950164, 2019.
View at: Publisher Site | Google Scholar
B. K. Sharma, R. Gandhi, and M. M. Bhatti, “Entropy analysis of thermally radiating MHD slip flow of hybrid nanoparticles (Au-Al₂O₃/Blood) through a tapered multi-stenosed artery,” Chemical Physics Letters, vol. 790, Article ID 139348, 2022.
View at: Publisher Site | Google Scholar
M. M. Bhatti and S. I. Abdelsalam, “Bio-inspired peristaltic propulsion of hybrid nanofluid flow with Tantalum (Ta) and Gold (Au) nanoparticles under magnetic effects,” Waves in Random and Complex Media, vol. 9, pp. 1–26, 2021.
View at: Publisher Site | Google Scholar
J. P. Singh, A. Agrawal, V. Upadhayay, and A. Kumar, “Mathematical model and analysis of two phase hepatic blood flow through arterioles with the special reference of hepatitis A,” American Journal of Modeling and Optimization, vol. 3, no. 1, pp. 22–25, 2015.
View at: Google Scholar
R. A. Khan, A. K. Agrawal, and V. Upadhyay, “Mathematical modeling and graphical presentation of two phase hepatic blood flow in capillary during jaundice,” Vigyan Grima Shindhu (CSTT, MHRD), vol. 112, pp. 174–185, 2020.
View at: Google Scholar
R. A. Khan, A. K. Agrawal, and D. Kumar, “A mathematical analysis of artery with special reference to jaundice”,” International Journal of Advanced Scientific Research and Management, vol. 5, pp. 72–76, 2019.
View at: Google Scholar
C. M. Ionescu, I. R. Birs, D. Copot, C. I. Muresan, and R. Caponetto, “Mathematical modelling with experimental validation of viscoelastic properties in non-Newtonian fluids,” Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, vol. 378, no. 2172, Article ID 20190284, 2020.
View at: Publisher Site | Google Scholar
D. W. Dodge and A. B. Metzner, “Turbulent flow of non‐Newtonian systems,” AIChE Journal, vol. 5, no. 2, pp. 189–204, 1959.
View at: Publisher Site | Google Scholar
B. Spain, Tensor Calculus: a concise course, Dover Publication, IL, USA, 2003.
S. Laue, M. Mitterreiter, and J. Giesen, “A simple and efficient tensor calculus,” in Proceedings of the AAAI Conference on Artificial Intelligence, pp. 4527–4534, NY, USA, February 2020.
View at: Publisher Site | Google Scholar
M. J. Kirkby, “Hillslope process-response models based on the continuity equation,” Inst Br Geogr Spec Publ, vol. 3, pp. 5–30, 1971.
View at: Google Scholar
M. A. J. Chaplain, “Multiscale mathematical modelling in biology and medicine,” IMA Journal of Applied Mathematics, vol. 76, no. 3, pp. 371–388, 2011.
View at: Publisher Site | Google Scholar
Y. C. Fung, A First Course in Continuum mechanics: for physical and biological scientists and engineers, Prentice-Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1994.
G. Elert, The physics hypertext book, Glen Elert, 2019.
D. R. Gustafson, Physics health and human body, Wadsworth Publishing Company, Belmont, California, USA, 1980.
R. Hasan, V. Abeysuriya, S. Kumarage, and J. A. A. S. Wijesinghe, “Morphological characteristics of the arterial supply of the extra- hepatic biliary system,” IOSR Journal of Dental and Medical Science, vol. 15, no. 09, pp. 74–76, 2016.
View at: Publisher Site | Google Scholar
L. A. D. Silveira, F. B. C. Silveira, and V. P. S. Fazan, “Arterial diameter of the celiac trunk and its branches: anatomical study,” Acta Cirurgica Brasileira, vol. 24, no. 1, pp. 43–47, 2009.
View at: Publisher Site | Google Scholar
N. S. Akbar, S. Nadeem, and K. S. Mekheimer, “Rheological properties of Reiner-Rivlin fluid model for blood flow through tapered artery with stenosis,” Journal of the Egyptian Mathematical Society, vol. 24, no. 1, pp. 138–142, 2016.
View at: Publisher Site | Google Scholar
M. A. Elogail and K. S. Mekheimer, “Modulated viscosity-dependent parameters for MHD blood flow in microvessels containing oxytactic microorganisms and nanoparticles,” Symmetry, vol. 12, p. 2114, 2020.
View at: Publisher Site | Google Scholar
A. M. Awad, K. S. Mekheimer, S. A. Elkilany, and A. Z. Zaher, “Leveraging elasticity of blood stenosis to detect the role of a non-Newtonian flow midst an arterial tube: Mazumdar and Keller models,” Chinese Journal of Physics, vol. 77, pp. 2520–2540, 2022.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Abha Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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