(2) Heat and Mass Transfer on MHD Two Phase Blood Flow through a Stenosed Artery with Permeable Wall. Fig. 1 Geometry of the two-phase vertical stenosed artery Geometry of the stenosis in plasma region, which is assumed to be symmetric is defined as,. h L 2 1 cos z d 0 ; d z d L0 R( z ) 1 2 R0 L0 2 R0 1; otherwise. (2). In core region geometry of the stenosis is defined as,. h 2 1 cos R1 ( z ) 2 R0 L0 R0 ; otherwise where. L0 z d ; d z d L0 2 . (3). L 0 is the length of the stenosis as shown in Figure 1 and is the ratio of the central core radius to the normal artery. radius, R 0 is the radius of the core region of the normal artery. So, the equations of core region in terms of these dimensionless parameters can be written as. Re uc Gr Gm p 1 2 uc 1 uc 2 2 M uc c c (4) t z r r r 0 0 0 0 2 Pe K 0 c c 1 c K 0 2 (5) N c 0 s0 t r 2 r r 0. Re c t. 2 1 1 c 1 c E D S r 2 r r E c 0 c 0 . (6). Equations for plasma region in dimensionless form are as follows 2 p u p 1 u p Re 2 t z r r r. u p. 2 p p 1 p Pe 2 r r t r. 2 M u p Gr p Gm p (7) . 2 N p . p 1 p 1 p Re 2 r r t Sc r 2. (8). E p . (9). Retrieval Number G6359058719/19©BEIESP. 1225. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(3) International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-7, May 2019 where 0 is the ratio of mean radiation absorption coefficient in the plasma region to mean radiation absorption coefficient in the core region, K 0 is the ratio of thermal conductivity of plasma and core region, s0 is the specific heat ratio of plasma and core regions and E 0 is the ratio of chemical moles present in the plasma region to chemical moles in the core region. In non-dimensional form factors of thermal radiation, applied magnetic field and chemical reaction are represented by N, M and E for both core and plasma regions. Corresponding boundary conditions in non-dimensional form to solve the model for both core and plasma regions are given as. u p (u B u porous ), p 1, p 1; r R( z ) u p u B , r D a u u , , ; r R ( z ) c p c p c 1 p p c p , c p , c ; r R1 ( z ) r r r r uc 0, c 0, c 0; r 0. r r r. (10). III. MATHEMATICAL SOLUTION Since pumping action of heart results in a pulsatile blood flow, so the pressure gradient can be represented as. . p P0 ei t z. In non-dimensional form pressure gradient can be written as. . p P0 eit z. and flow variables for core and plasma regions in non-dimensional forms can be represented in terms of t as. uc (r , t ) uc0 (r )eit , u p (r , t ) u p0 (r )eit it it c (r , t ) c0 (r )e , p (r , t ) p0 (r )e it it c (r , t ) c0 (r )e , p (r , t ) p0 ( r )e. (11). Substituting expression from in to we get time independence momentum, energy and concentration equations of core region. 2 uc0 1 uc0 2 0 Re Gr c0 Gm c0 M i u P (12) c 0 r 2 r r 0 0 0 0 0 2 c0 1 c0 K 0 N 2 P K i e 0 c0 0 2 r r r 0 0 s0 c0 1 c0 r 2 r r 2. E D0 S c c0 0 iRe D0 Sc E0 . (13) (14). Same, in of plasma region substitute values from. 2 u p0 1 u p0 Gr p0 Gm p0 2 M Re i u p0 P0 2 r r r 0 0 2 p0 1 p0 2 N iPe p0 0 r 2 r r p0 1 p0 r 2 r r 2. iRe Sc ESc p0 0 . Retrieval Number G6359058719/19©BEIESP. (15) . (16) (17). 1226. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(4) Heat and Mass Transfer on MHD Two Phase Blood Flow through a Stenosed Artery with Permeable Wall To find the exact solution of heat transfer and of core and plasma regions under the given boundary conditions we apply the definition of Bessel differential equation by assuming. K N2. 1 0 0. 2 and 2 N iPe . Pe K 0 0 s0. i. So, the equations and change in form of. 2c0 r. 2. 1 c0 1c0 0 r r. . p0 2. r 2. . (18). 1 p0 2 p0 0 r r. (19). Now, the exact solution for temperature profile of core region under the given boundary conditions is calculated as. Y 2 1 c0 X 1 X Y 4 0. . 2 R1 2 R. . . . 1 R1 . X4. . 1 X 2 J 1. X. 2 J0 . . 1 (r ). . (20). where values of are expressed as. J0 X1 J 0 X 3 J1 . . J R Y R , X Y R J R Y R R Y R J R J R R Y R Y R , X X J R X 2 R1. 0. 2. 0. 2. 1. 0. 2. 1. 2. 1. 1. 2. 1. 0. 2. 0. 2. 0. 2. 0. 1. 1. 2. 1. 0. 4. 1. 1. 1. 1. 1 1. 0. 1. 2. 1. 2. 3. Solution for the temperature profile of the plasma region under the given boundary conditions is calculated as. . . . Y R 1 X 2 J1 1 R1 2 1 2 1 p0 X Y R X4 2 4 0. . J . r Y J0. 0. 2. 0. Y R 2 R. 0. 2. Y0 2 r Y 0. . r R 2. (21). 2. Now, the final expressions for temperature profile considering unsteady flow for core and plasma regions respectively are as follows. R X J R X J (r ) e X R Y R X J R J R J r Y r e X X Y R Y R . Y 2 1 c X 1 X Y 4 0. p . 2. 1. 2. 2. 1. 1. 2. 1. 1. it. 2. 1. 0. 1. 4. 2. 1. 2 1. 1 1. 0. 0. 4 0. 1. 2. 0. 4. 2. 2. 0. 2. 2. it. . r e Y R Y0 0. 2. (22) it. (23). 2. where J n ( x ) is simply the Bessel function of first kind and Yn ( x ) represents the Bessel function of second kind for integer value of n . To solve subject to the boundary conditions we assume,. . 1 iRe D0 Sc . E D0 Sc and 2 iRe Sc ESc E0 . So the Equations become. 2 c0 r 2 p0 2. r 2. 1 c0 1 c0 0 r r 1 p0 2 p0 0 r r. . Retrieval Number G6359058719/19©BEIESP. (24). (25). 1227. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(5) International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-7, May 2019 Now, apply the definition of Bessel differential equation to calculate the value of concentration profiles under the given boundary conditions for both core as well as plasma regions. So, Solution for concentration profile for core region is calculated as. . . . Y R 1 X 6 J1 1 R1 2 1 2 1 c0 X 5 X Y X8 2 R 8 0 where the values of X 5 , X 6 and X 8 are assumed as. . 6 J0 . . 1 (r ). . (26). J R Y R , X Y R J R Y R R Y R J R J R R Y R Y R , X X J R X . J0 X5 J 0 X 7 J1 . . . X. 2 R1. 0. 2. 0. 2. 1. 0. 2. 1. 6. 1. 2. 1. 0. 2. 0. 2. 0. 2. 0. 1. 1. 1. 2. 1. 0. 8. 1. 1. 1. 1. 5 1. 0. 1. 2. 1. 2. 7. Solution for concentration profile of plasma region under the given boundary conditions is calculated as. . . . Y R 1 X 6 J1 1 R1 2 1 2 1 p0 X Y R X8 2 8 0. . J . r Y J0. 0. 2. 0. Y R 2 R. 0. 2. Y0 2 r Y 0. . . R 2 r. (27). 2. So, the final solutions of concentration profile for unsteady in core and plasma regions respectively are as follows. Y 2 1 c X 5 X Y 8 0 Y 2 1 p X Y 8 0. . . 2 R1 2 R. . . . 1 X 6 J1. . 1 R1. X8. . . X. 6 J0 . Y . 1 (r ) eit . . . Y R. (28). . . Y 2 r eit 0 r eit (29) 0 2 X8 2 R Y0 2 R 0 2 To find the solution for velocity profile in core region, we put the values of c and c from and apply the method. 2 R1. . 1 X 6 J1. 1 R1 J 0 . . 2 r . 2 R. J0. . 0. 0. variation of parameters for the given non-homogenous differential equations by assuming. . 1 M 2 . 0 Re i (30) 0 . 2 uc0 1 uc0 r 2 r r . Gr c0 Gm c0 1uc0 P0 0 0 . 0 . (31). So, the momentum equation convert in the form of which can be treated as an ordinary differential equation. In variation of parameters method first, we calculate the complementary solution for homogenous differential equation by using the definition the Bessel differential equation as. uc0 C1 J 0 C. where we have. The Wronskian of these two functions is. W1 . J0. . 1 J1. . . 1 r. . . 1 r. . 1 r. . 1 r C2Y0. . 1 r , uc Y0. 1 r. . uc0 J 0 1. . . . 1 r. 02. Y0. . . 1 Y1. . 1 r. . . 2 r. Now, for finding the complete solution of the non-homogeneous we find. Retrieval Number G6359058719/19©BEIESP. 1228. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(6) Heat and Mass Transfer on MHD Two Phase Blood Flow through a Stenosed Artery with Permeable Wall. Gr c0 Gm c0 Gr c0 Gm c0 Y0 1 r P0 J 0 1 r P0 0 0 0 0 0 0 F1 dr and G1 dr W1 W1 The complete solution of the velocity profile for the core region is of the form of. uc0 C1 J 0. . . . 1 r C2Y0. . . 1 r F1 J 0. . 1 r G1Y0. . 1 r. . (32). Now for of plasma region, we assume that. 2 M 2 Re i . (33). So the convert in terms 2 as. 2 u p0 1 u p0 r 2 r r . Gr p0 Gm p0 2 u p0 P0 0 0 . (34). For given complementary solution for homogeneous differential equation can be calculated as. . u p0 C3 J 0 C. So, we have. u p0 J 0 1. The Wronskian of these two functions is. W2 . J0. . . . 2 r , u p Y0. 2r. 2 J1. . . . 2 r. . . 2r. . 2 r C4Y0. . 02. Y0. 2r. . . 2 r. 2 Y1. . . 2 r. . . 2 r. For solution of the non-homogeneous equation further, we calculate. Gr c0 Gm c0 Y0 2 r P0 0 0 F2 W2. Gr c0 Gm c0 J 0 2 r P0 0 0 0 dr and G 2 W2. 0 dr. The complete solution of the velocity profile for the plasma region is of the form of. u p0 C3 J 0. . . 1 r C4Y0. . . 1 r F2 J 0. . . 1 r G2Y0. . 1 r. . (35). Now, we calculate the values of unknowns C1 , C2 , C3 and C 4 by using boundary condition First applying the boundary condition Now, becomes. uc0 C1 J 0. . . 1 r F1 J 0. . uc 0 at r 0 we get C2 0 r. . 1 r G1Y0. . 1 r. . (36). After applying all the boundary conditions in to the we get the linear system of C1 , C3 and C 4 in the form of. . . J 0 1 R1 0 1 J1 1 R1 . . where S1. J0. . . 2 R1. . S1. 2 J1. J 0 ( 2 R) . . Da. . 2 R1. Y0. . . . 2 R1 C D 1 1 S2. 2 Y1. . C2 D2 2 R1 C3 D3 . . J1 ( 2 R) 2 . S2 Y0 ( 2 R) . Da. . (37). Y1 ( 2 R) 2 .. Then D1 , D2 and D3 are expressed as. Retrieval Number G6359058719/19©BEIESP. 1229. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(7) International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-7, May 2019. D1 A1 R1 J 0. . 1 R1 B1 R1 Y0. D2 A2 ( R ) J 0 . . 2 R J1 ( 2 R 2. D3 . A1 R1 . A2 R1 r. r J0. . J0. . . . 1 R1 A2 R1 J 0. . . . Da B2 ( R ) Y0 . 1 R1 A1 R1 1 J1. . 2 R1 . B2 R1 r. Y0. . . . . 1 R1 . . . 2 R1 B2 R1 Y0. . 2 R1. . 2 R Y1 ( 2 R 2. B1 R1 r. 2 R1 A2 R1 2 J1. . . Y0. . . Da u porous . . 1 R1 B1 R1 1 Y1. . 2 R1 B2 R1 2 Y1. . . 1 R1. 2 R1. . . It can be clearly seen from equation (31) that it is simply the linear system of order 3 X 3 with C1 , C3 and C 4 unknowns and which has the unique solution. Final exact solutions for velocity profiles uc0 and u p0 for core plasma regions obtain by putting the values of C1 , C3 and C 4 in and respectively. The velocity profile of the two phase blood flow in the stenosed artery is (38) U u p uc The total volumetric flow rate of the blood flow in the artery is calculated as R. Q 2 R02u0 urdr. Fig. 3 Effects of Peclet number (Pe) on profile for two phase model. (39). 0. The shear stress at the interface wall of core and the plasma region is obtained as. . U . r r R . (40). IV. RESULT AND DISCUSSION For having adequate insight into the two-phase flow behavior of blood flow through a vertical stenosed arterial segment flow resistive impedance, volumetric flow rate, wall shear stress and velocity profile have been estimated assuming pulsatile, unsteady and Newtonian nature of the blood flow for both core and plasma regions. A computational study has been carried out to graphically show the effects of RBC-depleted plasma layer on blood flow with the variation of different quantities of interest.. Fig. 4 Effect of chemical reaction parameter (E) onconcentration profile for two phase model. Fig. 5 Effect of Schmidt number (Sc) on concentration profile for two phase model. Fig. 2 Variation of temperature profile for two phasetemperature model of blood flow with (K0). Retrieval Number G6359058719/19©BEIESP. 1230. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(8) Heat and Mass Transfer on MHD Two Phase Blood Flow through a Stenosed Artery with Permeable Wall on velocity profiles as continuous lines in the figure show the magnetic field effect on velocity profile of core region and dotted lines display the effect of varying magneticfield on velocity profile of plasma region. It is clear from Figure 7 that as the value of uniform applied magnetic field increases velocity profiles for both core and plasma regions increase respectively. This happens because the mature red blood cells contain high concentration hemoglobin molecules in its content which are oxides of iron. So, when MHD blood flows under the influence of uniform applied magnetic field erythrocytes orient with their own disk plane parallel to the direction of applied magnetic field. This action forms red blood cellsand magnetic particles more suspended in blood plasma, so increased concentration of magnetic particles causes an increase in the internal blood viscosity.. Fig. 6 Variation of velocity profile for two phase model with the ratio of viscosity in plasma and core regions. V. CONCLUSION. Fig. 7 Effect of magnetic field parameter (M) on velocity profile for two phase model for plasma and core regions Figure 2 and Figure 3 illustrate the effects on the temperature profile of two phase blood flow with increase in the values of (K0) and (Pe) parameters respectively. Continuous lines in the plot show the variation of the temperature profile in the core region and dotted lines display the variations of the temperature profile in the plasma region. As it could be seen from these figures that as the ratio of the thermal conductivity of plasma and core regions (K0)and value of the Peclet number (Pe) increase, the temperature profile of the two-phase blood flow decrease in both core and plasma regions respectively. Figure 4 and Figure 5 reveal that under the purview of the present computational study, concentration profiles for both core and plasma regions in two phase blood flow decrease as the values of the chemical reaction parameter and Schmidt number increase respectively. Now, the following figures are plotted to analyze the effects of various dimensionless parameters on velocity, temperature and concentration profiles. Figure 6 displays the variation of velocity profile for different values of (µ0) which is the ratio of the viscosity in plasma region to the core region. In figure continuous and dotted lines show the velocity variations in core and plasma regions respectively, in which plasma region varies from 0.8 to 1 radius of the artery.From Figure 6 it is clear that as the value of viscosity ratio increases, the value of velocity profiles also increase for both core and plasma regions. Figure 7 shows the effect of varying applied magnetic field. Retrieval Number G6359058719/19©BEIESP. It is analyzed through our results that as the value of both thermal Radiation and chemical reaction parameter increase as the temperature and concentration profiles in both core and plasma regions decrease respectively. To show the effectiveness of the two-phase model we carried out the comparative study between the single-phase and two-phase model of the blood flow and found that the two-phase model fits with the experimental data more accurately than the single phase model of blood flow. To get the proper understanding of the effects of two-phase blood flow on stenosed artery of the graphs of the variations of wall shear stress, resistive impedance, volumetric flow rate and velocity profile have been plotted which show their sinusoidal behavior with time. The model predicts increase in wall shear stress with peripheral layer viscosity. Predicted trends are found to exist in artery and hence validate the model. More experimental results are required for further development from clinical point of view. REFERENCES 1. Cogley,A.C., Gilles,S.E. and Vincenti,W.G. (1968). Differential approximation for radiative transfer in a nongrey gas near equilibrium, AIAA Journal, 6(3), 551–553. 2. Bugliarello,G. and Sevilla,J. (1970). Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes, Biorheology, 7, 85–107. 3. Bhavya Tripathi. and Bhupendra Kumar Sharma. (2017). Flow Characteristics of MHD Oscillatory Two-Phase blood flow through a stenosed artery with Heat and Mass Transfer, Physics.flu-dyn. 4. Ponalagusamy,R. (2016). Two-fluid model for blood flow through a tapered arterial stenosis: Effect of non-zero couple stress boundary condition at the interface, International Journal of Applied and Computational Mathematics, 1–18. 5. Sankar,D.S. and Hemalatha,K. (2007). Pulsatile flow of Herschelbulkley fluid through catheterized arteries- a mathematical model, Applied Mathematical Modelling, 31, 1497-1517. 6. Sankar,D.S. and Lee,U. (2007). Two-phase non-linear model for the flow through stenosed blood vessels, Journal of Mechanical Science and Technology, 21(4), 678–689. 7. Biswas,D. and Chakraborty. (2009). Steady flow of blood through a catheterized tapered artery with stenosis, A Theoritical model, Assam University Journal of Science and Technology, 4(2), 7-16.. 1231. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

(9) International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue-7, May 2019 8. Chaturani,P. and Ponalagusamy,R. (1982). A two-layered model for blood flow through stenosed arteries, Proceedings of the 11th National Conference on Fluid Mechanics and Fluid Power, B.H.E.L(R and D),Hyderabad,India, 16–22. 9. Hazarika,G.C. and Barnali Sharma. (2014). Two layered Model of Blood flow through composite stenosed artery in porous medium under the effect of Applied Magnetic field, International Journal of Mathematics Trends and Technology, 15, 11-20.. Retrieval Number G6359058719/19©BEIESP. 1232. Published By: Blue Eyes Intelligence Engineering & Sciences Publication.

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