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UNSTEADY TWO-LAYERED BLOOD FLOW THROUGH A $w$-SHAPED STENOSED ARTERY USING THE GENERALIZED OLDROYD-B FLUID MODEL

Part of: Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  21 July 2016

AKBAR ZAMAN ,
NASIR ALI ,
O. ANWAR BEG  and
M. SAJID
AKBAR ZAMAN*
Affiliation:
Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan email akbarzaman75@yahoo.com, nasir.ali@iiu.edu.pk, sajidqau2002@yahoo.com
NASIR ALI
Affiliation:
Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan email akbarzaman75@yahoo.com, nasir.ali@iiu.edu.pk, sajidqau2002@yahoo.com
O. ANWAR BEG
Affiliation:
Spray Research Group, Petroleum and Gas Engineering Division, School of Computing, Science and Engineering (CSE), University of Salford, M5 4WT, UK email O.A.Beg@salford.ac.uk
M. SAJID
Affiliation:
Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan email akbarzaman75@yahoo.com, nasir.ali@iiu.edu.pk, sajidqau2002@yahoo.com
*
✉akbarzaman75@yahoo.com
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Abstract

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A theoretical study of an unsteady two-layered blood flow through a stenosed artery is presented in this article. The geometry of a rigid stenosed artery is assumed to be $w$-shaped. The flow regime is assumed to be laminar, unsteady and uni-directional. The characteristics of blood are modelled by the generalized Oldroyd-B non-Newtonian fluid model in the core region and a Newtonian fluid model in the periphery region. The governing partial differential equations are derived for each region by using mass and momentum conservation equations. In order to facilitate numerical solutions, the derived differential equations are nondimensionalized. A well-tested explicit finite-difference method (FDM) which is forward in time and central in space is employed for the solution of a nonlinear initial boundary value problem corresponding to each region. Validation of the FDM computations is achieved with a variational finite element method algorithm. The influences of the emerging geometric and rheological parameters on axial velocity, resistance impedance and wall shear stress are displayed graphically. The instantaneous patterns of streamlines are also presented to illustrate the global behaviour of the blood flow. The simulations are relevant to haemodynamics of small blood vessels and capillary transport, wherein rheological effects are dominant.

Keywords

unsteady two-layered flowgeneralized Oldroyd-B fluidfinite-difference methodhaemodynamicsstenosisfinite-element methodstreamline visualization

MSC classification

Primary: 76A05: Non-Newtonian fluids
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 1 , July 2016 , pp. 96 - 118
Copyright
© 2016 Australian Mathematical Society 

References

Akay, G. and Kaye, A., “Numerical solution of time dependent stratified two-phase flow of micropolar fluids and its application to flow of blood through fine capillaries”, Internat. J. Engrg. Sci. 23 (1985) 265–276 ; doi:10.1016/0020-7225(85)0047-3.Google Scholar
Ali, N., Zaman, A. and Sajid, M., “Unsteady blood flow through a tapered stenotic artery using Sisko model”, Comput. Fluids 101 (2014) 42–49 ; doi:10.1016/j.compfluid.2014.05.030.Google Scholar
Anand, M. and Rajagopal, K. R., “A shear-thinning viscoelastic fluid model for describing the flow of blood”, Int. J. Cardiovasc. Med. Sci. 4 (2004) 59–68 ;https://www.cs.cmu.edu/∼sangria/publications/MAKRR2004.pdf.Google Scholar
Anwar Beg, O., Beg, T. A., Bhargava, R., Rawat, S. and Tripathi, D., “Finite element study of pulsatile magneto-hemodynamic non-Newtonian flow and drug diffusion in a porous medium channel”, J. Mech. Med. Biol. 12 (2012) 1250081.1 ; doi:10.1142/S0219519412500819.Google Scholar
Anwar Beg, O., Bhargava, R., Sughanda, Rawat, S., Takhar, H. S. and Beg, T. A., “Computational simulation of biomagnetic micropolar blood flow in porous (tissue) medium”, in: 5th World Conf. Biomechanics, Munich, Germany (2006).Google Scholar
Arada, N. and Sequeira, A., “Steady flows of shear-dependent Oldroyd-B fluids around an obstacle”, J. Math. Fluid Mech. 7 (2005) 451–483 ; doi:10.1007/s00021-004-0133-7.Google Scholar
Bathe, K. J., Finite element procedures (Prentice-Hall, Upper Saddle River, NJ, 1996).Google Scholar
Bhargava, R., Sharma, S., Takhar, H. S., Beg, T. A., Anwar Beg, O. and Hung, T. K., “Peristaltic pumping of micropolar fluid in porous channel – model for stenosed arteries”, J. Biomech. 39 (2006) S649–S650 ; doi:10.1016/S0021-9290(06)85707-6.Google Scholar
Braun, D. B. and Rosen, M. R., Rheology modifiers handbook: practical use and application (William Andrew Publishers, Norwich, NY, 2000).Google Scholar
Bugliarello, G. and Sevilla, J., “Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes”, Biorheology 7 (1970) 85–107 ;http://www.ncbi.nlm.nih.gov/pubmed/5484335.Google Scholar
Burton, A. C., Physiology and biophysics of the circulation, introductory text (Year Book Medical Publishers, Chicago, IL, 1966).Google Scholar
Chaturani, P. and Upadhya, V. S., “On micropolar fluid model for blood flow through narrow tubes”, Biorheology 16 (1979) 419–428 ; http://www.ncbi.nlm.nih.gov/pubmed/534765.Google Scholar
Cho, Y. I. and Kensey, K. R., “Effects of the non-Newtonian viscosity of blood on hemodynamics of diseased arterial flows: part 1, steady flows”, Biorheology 28 (1991) 241–262 ;http://www.ncbi.nlm.nih.gov/pubmed/1932716.Google Scholar
Curiel Sosa, J. L., Anwar Beg, O. and Liebana Murillo, J. M., “Finite element analysis of structural instability using a switching implicit–explicit technique”, Int. J. Comput. Methods Eng. Sci. Mech. 14 (2013) 452–464 ; doi:10.1080/15502287.2013.784383.Google Scholar
Fahraeus, R. and Lindqvist, T., “The viscosity of blood in narrow capillary tubes”, Amer. J. Physiol. 96 (1931) 562–568 ; doi:10.1161/01.RES.22.1.28.Google Scholar
Fung, Y. C., Biomechanics: mechanical properties of living tissues (McGraw-Hill, New York, 1993).Google Scholar
Hoffmann, K. A. and Chiang, S. T., Computational fluid dynamics, 4th edn, (Engineering Education System, Wichita, KS, 2000).Google Scholar
Hourai, M. S. A., Tounsi, A. and Anwar Beg, O., “Thermoelastic bending analysis of functionally graded material sandwich plates using a new higher order shear and normal deformation theory”, Int. J. Mech. Sci. 76 (2013) 102–111 ; doi:10.1016/j.ijmecsci.2013.09.004.Google Scholar
Huang, C. R., Pan, W. D., Chen, H. Q. and Copley, A. L., “Thixotropic properties of whole blood from healthy human subjects”, Biorheology 24 (1987) 795–801 ;http://www.ncbi.nlm.nih.gov/pubmed/3502773.CrossRefGoogle ScholarPubMed
Ikbal, M. A., Chakravarty, S. and Mandal, P. K., “Two-layered micropolar fluid flow through stenosed artery: effect of peripheral layer thickness”, Comput. Math. Appl. 230 (2009) 243–259 ; doi:10.1016/j.camwa.2009.07.023.Google Scholar
Johnston, B. M., Johnston, P. R., Corney, S. and Kilpatrick, D., “Non-Newtonian blood flow in human right coronary arteries: steady state simulations”, J. Biomech. 37 (2004) 709–720 ; doi:10.1016/j.jbiomech.2003.09.016.Google Scholar
Kang, C. K. and Eringen, A. C., “The effect of microstructure on the rheological properties of blood”, Bull. Math. Biol. 38 (1976) 135–158 ; doi:10.1016/S0092-8240(76)80030-4.Google Scholar
Ling, S. C. and Atabek, H. B., “A nonlinear analysis of pulsatile flow in arteries”, J. Fluid Mech. 55 (1972) 493–511 ; doi:10.1017/S0022112072001971.Google Scholar
Majhi, S. N. and Nair, V. R., “Pulsatile flow of third grade fluids under body acceleration – modeling blood flow”, Internat. J. Engrg. Sci. 32 (1994) 839–846 ; doi:10.1016/0020-7225(94)90064-7.Google Scholar
Majhi, S. N. and Usha, L., “Modeling the Fahraeus–Lindqvist effect through fluids of differential type”, Internat. J. Engrg. Sci. 26 (1988) 503–508 ; doi:10.1016/0020-7225(88)90008-0.Google Scholar
Mandal, P. K., “An unsteady analysis of non-Newtonian blood through tapered arteries with a stenosis”, Int. J. Non-Linear Mech. 40 (2005) 151–164 ; doi:10.1016/j.ijnonlinmec.2004.07.007.Google Scholar
Mandal, P. K., Chakravarty, S., Mandal, A. and Amin, N., “Effect of body acceleration on unsteady pulsatile flow of non-Newtonian fluid through a stenosed artery”, Appl. Math. Comput. 189 (2007) 766–779 ; doi:10.1016/j.amc.2006.11.139.Google Scholar
Massoudi, M. and Phuoc, T. X., “Pulsatile flow of a blood using second grade fluid model”, Comput. Math. Appl. 16 (2007) 199–211 ; doi:10.1016/j.camwa.2007.07.018.Google Scholar
Mohamed, H. B. H. and Reddy, B. D., “Some properties of models for generalized Oldroyd-B fluids”, Internat. J. Engrg. Sci. 48 (2010) 1470–1480 ; doi:10.1016/j.ijengsci.2010.09.014.Google Scholar
Nithiarasu, P., Hassan, O., Morgan, K., Weatherill, N. P., Fielder, C., Whittet, H., Ebden, P. and Lewis, K. R., “Steady flow through a realistic human upper airway geometry”, Int. J. Numer. Methods Fluids 57 (2008) 631–651 ; doi:10.1002/fld.1805.Google Scholar
Oldroyd, J. G., “On the formulation of rheological equations of state”, Proc. R. Soc. Lond. A 200 (1950) 523–541 ; doi:10.1098/rspa.1950.0035.Google Scholar
Pires, M. and Sequeira, A., “Flows of generalized Oldroyd-B fluids in curved pipes”, Prog. Nonlinear Differential Equations Appl. 80 (2011) 21–43 ; doi:10.1007/978-3-0348-0075-4-2.Google Scholar
Pontrelli, G., “Pulsatile blood flow in a pipe”, Comput. Fluids 27 (1998) 367–380 ; doi:10.1016/S0045-7930(97)00041-8.Google Scholar
Pontrelli, G., “Blood flow through a circular pipe with an impulsive pressure gradient”, Math. Models Methods Appl. Sci. 10 (2000) 187–202 ; doi:10.1142/S0218202500000124.Google Scholar
Rajagopal, K. R. and Srinivasa, A. R., “A thermodynamic frame work for rate type fluid models”, J. Non-Newtonian Fluid Mech. 88 (2000) 207–227 ; doi:10.1016/S0377-0257(99)00023-3.Google Scholar
Rao, M. A., Rheology of fluids and semisolid foods: principles and applications (Springer, New York, 2007) 1–481; doi:10.1007/978-0-387-70930-7.Google Scholar
Reddy, J. N., An introduction to the finite element method (McGraw-Hill, New York, 1985).Google Scholar
Sajid, M., Zaman, A., Ali, N. and Siddiqui, A. M., “Pulsatile flow of blood in a vessel using an Oldroyd-B fluid”, Int. J. Nonlinear Sci. Numer. Simul. 16 (2015) 197–206 ; doi:10.1515/ijnsns-2013-0133.Google Scholar
Sankar, D. S. and Lee, U., “Two-fluid non-linear model for flow in catheterized blood vessels”, Int. J. Non-Linear Mech. 43 (2008) 622–631 ; doi:10.1016/j.ijnonlinmec.2008.02.007.CrossRefGoogle Scholar
Sugiura, Y., “A method for analyzing non Newtonian blood viscosity data in low shear rates”, Biorheology 25 (1988) 107–112 ; http://www.ncbi.nlm.nih.gov/pubmed/3196806.Google Scholar
Thurston, G. B., “Rheological parameters for the viscosity, viscoelasticity, and thixotropy of blood”, Biorheology 16 (1979) 149–162 ; http://www.ncbi.nlm.nih.gov/pubmed/508925.Google Scholar
Yeleswarapu, K. K., “Evaluation of continuum models for characterizing the constitutive behavior of blood”, Ph.D. Dissertation, University of Pittsburgh, PA, 1996.Google Scholar
Yilmaz, F. and Gundogdu, M. Y., “A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions”, Korea–Australia Rheol. J. 20 (2008) 197–211 ;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.462.7926&rep=rep1&type=pdf.Google Scholar
Zaman, A., Ali, N., Sajid, M. and Hayat, T., “Effects of unsteadiness and non-Newtonian rheology on blood flow through a tapered time-variant stenotic artery”, AIP Adv. 5 (2015) 037129 ; doi:10.1063/1.4916043.Google Scholar