## Abstract

The unsteady two-fluid blood flow model in an artery with mild stenosis is considered by taking into account of effects of both heat transfer and magnetic field. Such a combination has not been reported in the literature of blood flow. The effects of plasma layer thickness, magnetic field, radiation parameter, thermal conductivity and viscosity ratio on flow variables are discussed and depicted graphically. The phase lag between pressure gradient and flow variables has been predicted and the effects of magnetic and radiation parameters, thermal conductivity, plasma layer thickness and Grashof number on the phase lag are brought out which form new information that are, for the first time, added to the literature. It has been pointed out here that the temperature and shear stress (or wall shear stress) decrease with increasing of plasma layer thickness. The flow resistance decreases with the increase in Grashof number and plasma layer thickness. Hence, the existence of the peripheral plasma layer and the pivotal role of Grashof number could be useful for the functions of the diseased arterial system and hence it is concluded that the present study is believed to yield some good improvement over two-fluid models discussed in the literature.

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## References

- 1.
Ambethkar V, Singh PK (2011) Effect of magnetic field on an oscillatory flow of a viscoelastic fluid with thermal radiation. Appl Math Sci 5:935–946

- 2.
Anwar Beg O, Bhargava R, Rawat S, Halim K, Takhar HS (2008) Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium. Meccanica 43:391–410

- 3.
Bhargava R, Rawat S, Takhar HS, Beg OA (2007) Pulsatile magneto-biofluid flow and mass transfer in a non-Darcian porous medium channel. Meccanica 42:247–262

- 4.
Bird RB, Stewart WE, Lightfoot EN (1960) Transport phenomena. Wiley, New York

- 5.
Bugliarello G, Hayden JW (1963) Detailed characteristics of the flow of blood in vitro. Trans Soc Rheol 7:209–230

- 6.
Bugliarello G, Sevilla J (1970) Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7:85–107

- 7.
Caro CG, Fitz-Gerald JM, Schroter RC (1971) Atheroma and arterial wall shear observation, correlation and proposal of a shear dependent mass transfer mechanism of atherogenesis. Proc R Soc Lond B Biol Sci 177:109–133

- 8.
Caro CG (1981) Arterial fluid mechanics and atherogenesis. Recent Adv Cardiovasc Dis 2:6–11

- 9.
Charm S, Paltiel B, Kurland GS (1968) Heat transfer coefficients in blood flow. Biorheology 5:133–145

- 10.
Chaturani P, Kaloni PN (1976) Two-fluid Poiseuille flow model for blood flow through arteries of small diameter and arterioles. Biorheology 13:243–250

- 11.
Chaturani P, Ponalagusamy R (1982) A two-fluid model for blood flow through stenosed arteries. In: Proceedings of 11th national conference on fluid mechanics and fluid power, B.H.E.L. (R & D), Hyderabad, India, pp 16–22

- 12.
Chaturani P, Ponalagusamy R (1983) Blood flow through stenosed arteries. In: Proceedings of first international conference on physiological fluid dynamics, vol 1, pp 63–67

- 13.
Chaturani P, Ponalagusamy R (1984) Analysis of pulsatile blood flow through stenosed arteries and its applications to cardiovascular diseases. In: Proceedings of 13th national conference on fluid mechanics and fluid power (FMFP-1984), pp 463–468

- 14.
Cogley ACL, Vincenti WG, Gilles ES (1968) Differential approximation for radiative heat transfer in a nonlinear equations-grey gas near equilibrium. Am Inst Aeronaut Astronaut J 6:551–553

- 15.
Craciunescu OI, Clegg ST (2001) Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels. J Biomech Eng 123:500–505

- 16.
Dintenfass L (1977) Viscosity factors in hypertensive and cardiovascular diseases. Cardiovasc Med 2:337–354

- 17.
El-Shehawey EF, Mekheimer K, El Kot MA (2008) The micropolar fluid model for blood flow through a stenotic arteries. Proc Math Phys Soc Egypt 86:215–234

- 18.
Forrester JH, Young DF (1970) Flow through a converging—diverging tube and its implications in occlusive vascular disease-I and II. J Biomech 3:297–316

- 19.
Fry DL (1968) Acute vascular endothelial changes associated with increased blood velocity gradients. Circ Res 22:165–197

- 20.
Fry DL (1973), Responses of the arterial wall to certain physical factors. In: Atherogenesis: initiating factors, Ciba foundation symposium, Elsevier, Experta Medica, North Holland, Amsterdam, vol 12, pp 93–125

- 21.
Haldar K (1985) Effects of the shape of stenosis on the resistance to blood flow through an artery. Bull Math Biol 47:545–550

- 22.
Haldar K, Ghosh SN (1994) Effect of a magnetic field on blood flow through an indented tube in the presence of erythrocytes. Indian J Pure Appl Math 25(1994):345–352

- 23.
Hershey D, Byrnes RE, Deddens RL, Rao AM (1964) Blood rheology: temperature dependence of the power law model. In AIChE Meeting, Boston

- 24.
Ikbal M, Chakravarty AS, Mandal PK (2009) Two-fluid micropolar fluid flow through stenosed artery: effect of peripheral layer thickness. Comput Math Appl 58:1328–1339

- 25.
Imaeda K, Goodman FO (1980) Analysis of non-linear pulsatile blood flow in arteries. J Biomech 13:1007–1021

- 26.
Israel-Cookey C, Ogulu A, Omubo-pepple VB (2003) Influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction. Int J Heat Mass Transf 46:2305–2311

- 27.
Kumar S, Kumar S, Kumar D (2009) Oscillatory MHD flow of blood through an artery with mild stenosis (Research Note). Int J Eng Trans A Basics 22:125–130

- 28.
Mac Donald DA (1979) On steady flow through modeled vascular stenoses. J Biomech 12:13–20

- 29.
Mauro Greppi (1978) Numerical solution of a pulsatile flow problem. Meccanica 13:230–237

- 30.
Midya C, Layek GC, Gupta AS, Mahapatra TR (2003) Mageneto-hydrodynamic viscous flow separation in a channel with constrictions. Trans ASME J Fluids Eng 125:952–962

- 31.
Motta M, Haik Y, Gandhari A, Chen CJ (1998) High magnetic field effects on human deoxygenated hemoglobin light absorption. Bioelectrochem Bioenerg 47:297–300

- 32.
Ogulu A, Bestman AR (1993) Deep heat muscle treatment a mathematical model—I & II. Acta Phys Hung 73:3–16, 17–27

- 33.
Ogulu A, Abbey TM (2005) Simulation of heat transfer on an oscillatory blood flow in an indented porous artery. Int Commun Heat Mass Transf 32:983–989

- 34.
Philip D, Chandra P (1996) Flow of Eringen fluid (simple micro fluid) through an artery with mild stenosis. Int J Eng Sci 34:87–99

- 35.
Ponalagusamy R (1986) Blood flow through stenosed tube. Ph.D. Thesis, IIT, Bombay, India

- 36.
Ponalagusamy R, Kawahava M (1989) A finite element analysis of laminar unsteady flows of viscoelastic fluids through channels with non-uniform cross-sections. Int J Numer Methods Fluids 9:1487–1501

- 37.
Ponalagusamy R (2007) Blood flow through an artery with mild stenosis: a two-layered model, different shapes of stenoses and slip velocity at the wall. J Appl Sci 7:1071–1077

- 38.
Ponalagusamy R, Tamil Selvi R (2011) A study on two-fluid model (Casson–Newtonian) for blood flow through an arterial stenosis: axially variable slip velocity at the wall. J Franklin Inst 348:2308–2321

- 39.
Ponalagusamy R, Tamil Selvi R (2013) Blood flow in stenosed arteries with radially variable viscosity, peripheral plasma layer thickness and magnetic field. Meccanica 48:2427–2438

- 40.
Ponalagusamy R (2012) Mathematical analysis on effect of non-Newtonian behavior of blood on optimal geometry of microvascular bifurcation system. J Franklin Inst 349:2861–2874

- 41.
Ponalagusamy R, Tamil Selvi R, Banerjee AK (2012) Mathematical model of pulsatile flow of non-Newtonian fluid in tubes of varying cross-sections and its implications to blood flow. J Franklin Inst 349:1681–1698

- 42.
Ponalagusamy R (2013) Pulsatile flow of Hershel–Bulkley fluid in tapered blood vessels. In: Proceedings of the 2013 international conference on scientific computing (CSC 2013), WORLDCOMP’13, held in Las Vegas Nevada, USA, pp 67–73

- 43.
Rao AR, Deshikachar KS (1988) Physiological type flow in a circular pipe in the presence of a transverse magnetic field. J Indian Inst Sci 68:247–260

- 44.
Sankar DS, Lee U (2010) Two-fluid Casson model for pulsatile blood flow through stenosed arteries: a theoretical model. Commun Nonlinear Sci Numer Simul 15:2086–2097

- 45.
Shahmohamadi H (2011) Reliable treatment of a new analytical method for solving MHD boundary-layer equations. Meccanica 46:921–933

- 46.
Shukla JB, Parihar RS, Rao BRP (1980) Effects of stenosis on non-Newtonian flow of the blood in an artery. Bull Math Biol 42:283–294

- 47.
Shukla JB, Parihar RS, Rao BRP (1980) Effect of peripheral layer viscosity on blood flow through the artery with mild stenosis. Bull Math Biol 42:797–805

- 48.
Shukla JB, Parihar RS, Rao BRP (1980) Biorheological aspects of blood flow through artery with mild stenosis: effects of peripheral layer. Biorheology 17:403–410

- 49.
Srivastava VP (1996) Two-phase model of blood flow through stenosed tubes in the presence of a peripheral layer: applications. J Biomech 29:1377–1382

- 50.
Srivastava VP, Srivastava R (2009) Particulate suspension blood flow through a narrow catheterized artery. Comput Math Appl 58:227–238

- 51.
Vardanian VA (1973) Effect of magnetic field on blood flow. Biofizika 18:491–496

- 52.
Victor SA, Shah VL (1975) Heat transfer to blood flowing in a tube. Biorheology 12:361–368

- 53.
Voltairas PA, Fotiadis DI, Michalis LK (2002) Hydrodynamics of magnetic drug targeting. J Biomech 35:813–821

- 54.
Whitmore RL (1968) The rheology of the circulation. Pergamon Press, New York

- 55.
Young DF (1968) Effects of a time-dependent stenosis on flow through a tube. J Eng Ind 90:248–254

- 56.
Young DF (1979) Fluid mechanics of arterial stenoses. J Biomech Eng Trans ASME 101:157–175

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Ponalagusamy, R., Tamil Selvi, R. Influence of magnetic field and heat transfer on two-phase fluid model for oscillatory blood flow in an arterial stenosis.
*Meccanica* **50, **927–943 (2015). https://doi.org/10.1007/s11012-014-9990-1

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### Keywords

- Stenosed artery
- Blood flow
- Heat transfer
- Peripheral plasma layer
- Magnetic field
- Phase lag
- Oscillatory flow