Mathematical and Computational study of blood flow through diseased artery Abstract This paper presents the study of blood flow through a tapered stenosed artery. The fluid (blood) medium is assumed to be Power law fluid model. The governing equation for laminar, incompressible and non-newtonian fluid subject to the boundary conditions is solved by using a well known perturbation mathematical method. The analytical expressions for velocity component, volumetric flow rate, wall shear stress and pressure gradient are obtained. It is observed that the shape of artery have important impact on the velocity profile, pressure gradient and wall shear stress. It is also found that wall shear stress increases when peak is obtained then decreases for different values of tapering angle. Keywords: Tapered artery, Power Law fluid model, Blood flow, Stenosis, Volumetric flow rate, Wall shear stress. 1. INTRODUCTION Sapna Ratan Shah, Anamika School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067, (India) Cardiovascular diseases (CVDs) are a group of disorders of the heart and blood vessels and they include: CORONARY HEART DISEASE Disease of the blood vessels supplying the heart muscle. CEREBROVASCULAR DISEASE Disease of the blood vessels supplying blood to the brain. PERIPHERAL ARTERIAL DISEASE Disease of blood vessels supplying the blood to the arms and legs. RHEUMATIC HEART DISEASE Damage to the heart muscle and heart valves from rheumatic fever, caused by streptococcal bacteria. CONGENITAL HEART DISEASE Malformations of heart structure existing at birth. DEEP VEIN THROMBOSIS AND PULMONARY EMBOLISM Blood clots in the leg veins, which can dislodge and move to the heart and lungs. Fig.(1). Blood flow in a stenosed artery Fig.(2). Schematic diagram of blood flow in a stenosed tapered artery Heart attacks and strokes are usually acute events and are mainly caused by a blockage that prevents blood from flowing to the heart or brain [3, 7]. The most common reason for this is a build-up of fatty deposits on the inner walls of the blood vessels [fig. (1)]. Strokes can also be caused by bleeding from a blood vessel in the brain or from blood clots. The main cause of heart attacks and strokes are usually the presence of a combination of risk factors, such as Volume 5, Issue 6, June 2017 Page 1
tobacco use, unhealthy diet and obesity, physical inactivity and harmful use of alcohol, hypertension, diabetes and hyper lipidaemia etc. Various treatments are available to cure for heart attacks and strokes, like medication, bypass surgery, catheterization. Catheterization is the simple and frequently used approach as the procedure can open the narrowed heart valves, blocked arteries and repair the defects. Many researchers found that the study of blood flow through tapered tubes is important not only for an understanding of the behavior of the marvelous body fluid in arteries, but also for design of prosthetic blood vessels. Some researcher discussed heat and mass transfer effects on carreau fluid model for blood flow through a tapered artery with a stenosis. In (2011) [4], it is studied that the artificial neural network modeling for the system of blood flow through tapered artery with mild stenosis. In 2014 [2,6] A layered mathematical model for blood flow through tapering asymmetric stenosed artery with slip velocity at a interface under the effect of transverse magnetic field have been studied. Many researchers [1, 5, 9] studied the flow of blood as Newtonian and non- Newtonian fluid through tapered tubes. In this study pressure drop, pressure gradient and flux were measured in rigid wall model of tapered graphs under steady flow conditions. Both Newtonian and non- Newtonian fluids were examined. 2. FORMULATION OF THE PROBLEM Tapered blood arterial segment with stenosis in its lumen is modeled as a thin elastic tube with a circular cross-section containing an incompressible non-newtonian fluid characterized by Power law fluid model. The geometry of the timevariant stenosed arterial segment is given in Fig.(2) [8]. Consider an axially symmetric, laminar, and fully developed flow of blood in the z direction. It can be shown that the radial velocity is negligibly small in its magnitude and may be neglected for a low mean Reynolds number flow problem with stenosis. The momentum equation are where for a Power Law fluid model is given by where The boundary condition are Let us consider the pulsetile laminar flow of blood in the z direction through a compliant tube whose radius varies as (using non dimensional scheme) Volume 5, Issue 6, June 2017 Page 2
To solve the above system of equations following non-dimensional variables are introduced: where is the constant pressure gradient. The pressure gradient which is function of and, is represented as The pressure gradient which is function of and, is represented as In terms of these non-dimensional variables, eq. (1), (2), (7) reads The volumetric flow rate is given by u=0 at r=r, is finite at r=0. (10). (11) 3. SOLUTION OF THE PROBLEM Considering the Womersley parameter to be very small, the velocity u, shear stress as well as and can be expressed in the following form u(z, r, t) = u0(z, r, t) + α2u1(z, r, t) + (12) (z, r, t) = 0(z, r, t) + α2 1(z, r, t) + (13) Rp(z, t) = R0p(z, t) + α2r1p(z, t) + (14) up(z, t) = u0p(z, t) + α2u1p(z, t) + (15) Using (11) and (12) in (8) and boundary conditions. we have Volume 5, Issue 6, June 2017 Page 3
The plug core velocity can be obtain from Eq. (15) as Here R0p is the first approximation plug core radius. Neglecting the term with α2 and higher powers of α in Eq. (13), the expression for R0p can be obtain from Eq. (14) as Similarly, the solution for τ1, u1, and u1p can be obtained as The volumetric flow rate is given by It may be noted that if we write u = u0 + α2nu1. From Eq. (22) for small, we have 4. RESULTS AND DISCUSSION The volumetric flow rate and the wall shear stress are the two important characteristics in the study of fluid flow through a stenosed artery. Using appropriate boundary conditions, analytical expressions for the velocity profile, volumetric flow rate and shear stress have been obtained. The expressions for volumetric flow rate and wall shear stress, given by (26) and (27) respectively have been numerically evaluated using MATLAB software for different Volume 5, Issue 6, June 2017 Page 4
values of relevant parameters. For the purpose of numerical computation of the quantities of interest, we have performed a thorough quantitative analysis, by taking the following values of the different parameters involved in the present study. a = 0.5mm, L = 30, L0 = 10, d = 10, θ = 0.05, A = 0.7, δ = 0.1, α2 = 0.049, m = 2.0, T = 1.0. Fig. (3) depicts the effect of volumetric flow rate with z for different values of φ. It is shown in this figure that the volumetric blood flow will decreases as axial distance increases and it is increases with tapered angle (φ) as well.. This result is similar to the [5]. Fig.(4) shows the variation of volumetric flow rate with axial distance (z) for different values of time (t). it is observed in this figure that the volumetric flow rate decreases when the axial distance (z) increases [10]. Fig. (6) depicts the variation of wall shear stress with axial distance (z) for different values of time (t). This shows that the wall shear stress increases in the starting when z varies from14 to 16 then decreases from 16 to 18 and decreases for decreasing value of time (t) and also for decreasing values of tapered angle [3]. 5. CONCLUSION In this study it is obtained that the blood velocity decreases with the radial distribution for any given value of φ. It is also found that the velocity distribution of the two-fluid non- Newtonian Power law model are considerably higher than those of the Newtonian fluid models. It is also observed that the volumetric flow rate and wall shear stress are very low for the non- Newtonian power law fluid model than those of the Newtonian fluid model. Hence, the non- Newtonian Power Law fluid model would be more useful than the Newtonian model to analyze the blood flow through stenosed tapered arteries. References [1] D. Ssrikanth, J. V. Ramana Reddy and Vssnvg Krishna Murthy, (2015). Shear stress distribution at the wall of omega (ω) shaped stenotic tapered artery in the presence of catheter and velocity slip-effects of polar fluid, International Conference on Frontiers in Mathematics. [2] G. C. Hazarika, Barnali Sharma, (2014). Two Layered Mathematical Model for Blood Flow through Tapering Asymmetric Stenosed Artery with Velocity Slip at the Interface under the Effect of Transverse Magnetic Field, International Journal of Computer Applications, vol 105, pp. 1-8. [3] Jain, N., Singh, S., and M. Gupta, Steady flow of blood through an atherosclerotic artery: A non-newtonian model, International Journal of Applied Mathematics and Mechanics, (2012), Vol.8, pp. 52-63. [4] Jyoti Kumar Arora, (2011). Artificial Neural Network modelling for the System of blood flow through tapered artery with mild stenosis, International Journal of Mathematics Trends and Technology. [5] Kumar, S. and Kumar, S. (2006): Numerical study of the axisymmetric blood flow in a constricted rigid tube, Inter. Review of pure and Applied Mathematics, vol. 2(2), pp. 99-109. [6] Noreen Sher Akbar, S. Nadeem, (2014). Carreau fluid model for blood flow through a tapered artery with a stenosis. Ain Shams Engineering Journal,vol. 5, pp. 1307-1316. [7] Shah, S. R., An innovative solution for the problem of blood flow through Stenosed artery using generalized Bingham plastic fluid model. IMPACT, IJRANSS, (2013), Vol. 1(3), pp. 97-104. [8] Srivastava V.P. and Mishra S., "Non-newtonian arterial blood flow through an overlapping stenosis", Appl. and Appl. Math.: An Int. J. (AAM), (2010), Vol. 5, Issue 1, pp. 225 238. Volume 5, Issue 6, June 2017 Page 5
[9] Surendra Kumar, M. K. Sharma, Kuldip Singh and N. R. Garg, (2011) Mhd Two-Phase Blood Flow Through An Artery With Axially Non-Symmetric Stenosis, Int. J. of Math. Sci. & Engg. Appls, (2011), Vol. 5 No. II, pp. 63-74. [10] W. X. Chen, P. Barlis, (2015). Cfd analyses on incomplete stent apposition: from tapered to curved artery. Australian biomedical engineering conference. AUTHORS Dr. Sapna Ratan Shah, Associate Professor, School of Computational and Integrative Sciences, Jawaharlal Nehru, New Delhi, received M.Sc. and Pd.D degree in Mathematics from Christ Church P. G. College, and Harcourt Butler Technical University, Kanpur respectively. She has been doing research work in Applied Mathematics, Biomechanics, Biomathematics since 2002. She has published more than seventy research papers in various reputed international journals. Mrs. Anamika, Systems Analyst, School of Computational and Integrative Sciences, Jawaharlal Nehru, New Delhi, received B.E. degree in Computer Sciences from M.D.U Rohtak,, Post graduation diploma in advance design and development from CDAC Noida and M.S. degrees in Software Systems from BITS Pilani, Rajesthan. She is perusing her research work in applied Mathematics. Volume 5, Issue 6, June 2017 Page 6