Hydromagnetic Pulsating Flow of Blood in a Constricted Porous Channel: A Theoretical Study G. C. Shit and M. Roy Abstract—A theoretical study of pulsatile blood flow through a constricted porous channel in the presence of an external magnetic field by considering the incompressible Newtonian fluid model is investigated. The influence of magnetic field on the flow is studied using the dimensionless magnetic parameter M and a Darcian linear impedance for low Reynolds number is taken into account in the transformed momentum equation. A perturbation method is employed to solve the governing differential equations by using a small perturbation parameter ϵ (such that 0 < ϵ << 1), which is incorporated in the time dependent transpiration velocity (suction/ injection). Using appropriate boundary conditions analytical expressions for the velocity distribution, volumetric flow rate and wall shear stress have been derived and the numerical results are presented graphically for different values of the physical parameters of interest. Index Terms—Pulsatile flow, porous channel, Time dependent suction/ injection I. I NTRODUCTION M ATHEMATICAL modeling of blood flow through a constricted porous channel/ vessel is of great con- cerned for clinical scientists and has therefore drawn serious attention of researchers. It is known that stenosis (narrowing of artery) is a dangerous disease and is caused due to the deposition of cholesterol and other various substances in an arterial wall form a plaque which grow inward and restrict the flow of blood through the lumen of the artery. If this disease takes a severe form, it may lead to morbidity, fatality and serious circulatory disorders. As a result of such undesirable formation at the endothelium of the vessel wall, reduction of regular blood flow is likely to take place in the constricted region of the channel/vessel. To understand the effects of stenosis in the lumen of an artery, many researchers ([8], [4] and [11]) have investigated the flow of blood through arteries by considering blood as a Newtonian fluid. However, most of the studies ([13], [26] and [12] ) show that, in the vicinity of a stenosis, the shear rate of blood is low and the blood behaves like a non-Newtonian fluid. It is also worth while to mention here that although blood is non-Newtonian suspension of cells in plasma, [8] remarked that for vessels of radius greater than 0.025 cm, blood may be considered as a homogeneous Newtonian fluid. Several studies ([21], [27] and [5]) of physiological fluid dynamics through stenosed arteries have been carried out to evaluate the flow pattern under steady and pulsatile conditions by treating blood as a Newtonian fluid. It has been observed that blood flow in the human circulatory Manuscript received March 22, 2012; accepted April 09, 2012. One of the authors G. C. Shit is thankful to the UGC, New Delhi for the financial support during this investigation through UGC Minor Research Project Scheme. G. C. Shit and M. Roy are with the Department of Mathematics, Jadavpur University, Kolkata-700032, India, E-mail: gcs@math.jdvu.ac.in. system depends upon the pumping action of the heart, which in turn produces a pulsatile pressure gradient throughout the system. [26] theoretically analyzed the pulsatile flow of blood in a stenosed artery, where the non-Newtonian behavior of blood was taken to be of Herschel-Bulkley type. Some excellent studies on pulsatile blood flow have made by [29]. [30] studied the fluid dynamics of pulsatile flow past a single cylinder for a non-Newtonian Casson fluid. [10] carried out the pulsatile flow of blood in an artery by considering the effects of body acceleration. [23] modeled the blood flow through arterial stenosis by treating blood as a couple stress fluid. [22] proposed a mathematical model for pulsatile blood flow in a constricted tube using the Power-law fluid. The effect of externally imposed body acceleration and magnetic field on peristaltic flow of blood through an arterial segment having stenosis has been investigated by [25]. Their studies pertains to a situation in which blood obeying micropolar fluid model, where the effect of heat transfer phenomena has been taken into account. In the recent past, engineers and scientists became inter- ested in the influence of magnetic field on blood flows with a view to utilizing MHD (magnetohydrodynamic) in controlling blood flow during surgery and also establishing the effects of magnetic field on blood flows in astronauts, citizens living in the vicinity of electromagnetic towers etc. Since blood consists of a suspension of red blood cells containing hemoglobin, which contains iron oxide, it is quite apparent that blood is electrically conducting and exhibits magnetohydrodynamic flow characteristics. Bhargava et al [1] numerically studied the pulsatile flow and mass transfer of an electrically conducting Newtonian biofluid via a channel with porous medium. The flow of blood through arteries in the presence of magnetic field under different physiological conditions were reported in ([16], [20]). Steady laminar flow of blood through a porous medium in an arterial segment having double stenoses under the influence of externally ap- plied magnetic field have been carried out by [17], [18] using numerically as well as analytically by means of Frobenius Method. The potential use of such MHD principles in various arteries have explored by [14], [15], who showed that for unsteady flow of blood in an artery of circular cross-section, a uniform magnetic fields alters the flow rate of blood. [24] have investigated using a vorticity formulation of the MHD oscillatory flows in variable cross-sectional channels, reporting a distinct reduction in velocity with a strong applied magnetic field. Many biological tissues such as bones and vascular tissues, the renal system as well as the blood vessels containing fatty deposits are assumed to be porous by nature. [7] have presented a detailed review on heat and fluid flow in a porous media having physiological applications. Pulsatile flow of blood through a stenosed porous medium has been studied by [3] under the influence of body acceleration. Proceedings of the World Congress on Engineering 2012 Vol I WCE 2012, July 4 - 6, 2012, London, U.K. ISBN: 978-988-19251-3-8 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCE 2012