849 ISSN 1061-933X, Colloid Journal, 2017, Vol. 79, No. 6, pp. 849–856. © Pleiades Publishing, Ltd., 2017. Original Russian Text © Bhupesh Dutt Sharma, Pramod Kumar Yadav, Anatoly Filippov, 2017, published in Kolloidnyi Zhurnal, 2017, Vol. 79, No. 6, pp. 813–821. A Jeffrey-Fluid Model of Blood Flow in Tubes with Stenosis 1 Bhupesh Dutt Sharma a , Pramod Kumar Yadav a , and Anatoly Filippov b, * a Motilal Nehru National Institute of Technology Allahabad Allahabad, 211004 (U.P.) India b Gubkin University, Moscow, 119991 Russia *e-mail: anatoly.filippov@gmail.com Received May 21, 2017 Abstract—In this paper, we discuss the two-layered Jeffrey-fluid model with mild stenosis in narrow tubes. The blood flow in narrow arteries is treated as a two-f luid model with the suspension of erythrocytes, leuko- cytes, etc., as a Jeffrey fluid, which is a non-Newtonian fluid, in the core region and plasma, a Newtonian fluid, in the peripheral region. An analytical solution has been obtained for the velocity in the core and peripheral region, volume flow rate, resistance to flow, and wall-shear stress. The effect of Jeffrey-fluid parameters, like the height of stenosis, viscosity, etc., on volume flow rate, resistance to flow (impedance), and wall-shear stress has been discussed graphically. Through the present study, it is found that the wall-shear stress and resistance to flow increases with the increase in height of stenosis and decreases with the increase in the ratio of relaxation time. It is also found that the velocity decreases with an increase in stenosis height in both the core and the peripheral region. A previous result has been also verified. DOI: 10.1134/S1061933X1706014X 1. INTRODUCTION Blood is an important fluid of our body, like other bio-fluids. The study of blood flow in blood vessels, such as arteries and veins, under the effect of stenosis been a topic of long-standing interest for researchers. Today, we can see that arteries are choked because of modern lifestyles, high blood cholesterol, smoking, and possibly a genetic problem. Many researchers have developed a model of blood flow through arteries with atherosclerosis by assuming the nature of blood to be both Newtonian and non-Newtonian. Experien- tially, it has been proven that blood is the suspension of red blood cells (RBCs), white blood cells (WBCs), and platelets. Hence, it behaves in a Newtonian way when high shear stress is applied, i.e., when f lowing through large arteries, and in a non-Newtonian way when low shear stress, is applied, i.e., when flowing through small arteries and veins. Some researchers have been modelled blood flow as both Newtonian and non-Newtonian, i.e., using a two-fluid model considered to be Newtonian in the peripheral region and non-Newtonian in the core region. D.F. Young [1] has studied the effect of time- dependent stenosis on flow through a tube and found increased resistance to flow and flow variation for dif- ferent conditions of stenosis. K. Haldar and S.N. Ghosh [2] discussed the effect of a magnetic field on blood flow through an indented tube in the pres- ence of stenosis. In that paper, they discussed pres- sure-gradient variation with hematocrit and magnetic field. They also discussed the velocity problem in the tube and noted that the magnetic field and the hema- tocrit control the velocity profile. Two-layer models of blood flow have been discussed by Shukla et al. [3]. In their paper, they found that the flow resistance and wall-shear stress decrease with decrease in peripheral- layer viscosity. The variation of plasma-layer thickness and wall-shear stress, as well as pressure-drop varia- tion, for a two-layer model of blood flow through ste- nosed arteries has been discussed by K. Haldar and H.I. Andersson [4]. They found that the plasma-layer thickness varies along the stenosis length and the wall- shear stress increases rapidly when the plug-core radius increases. Srivastava and Saxena [5] discussed the two-fluid model assuming blood to be non-New- tonian in form of a peristaltic wave flow. A non-Newtonian-fluid flow model has been dis- cussed by D.S. Sankar and Hemalatha [6], assuming blood to be a Hershel–Bulkley f luid. They studied the effect of catheterization on wall-shear stress, pressure drop, and impedance to flow in a narrow artery. They noted that wall-shear stress and flow rate decrease with increasing of catheter radius and yield stress. They also observed that wall-shear stress and f low rate increase with an increase in pressure gradient. D.S. Sankar and Hemalatha [7, 8] also studied pulsa- tile blood flow for a Hershel–Bulkley fluid with and without a catheter. In these papers, they studied the effect of pulsatility, stenosis size, the non-Newtonian behavior of blood in small arteries, and the Womersley 1 The article was translated by the authors.