J. Nig. Soc. Phys. Sci. 4 (2022) 682 Journal of the Nigerian Society of Physical Sciences Effect of Treatment Parameter on Oscillatory Flow of Blood Through an Atherosclerotic Artery with Heat Transfer R. R. Hanveya,∗, K. W. Bunonyob aDepartment of Mathematics & Statistics, Faculty of Science, SHUATS, India bDepartment of Mathematics & Statistics, Federal University, Otuoke, Nigeria Abstract This research work has been carried out to investigate the influence of treatment parameter on flow of blood in a stenosed artery in the presence of magnetic field with heat transfer. The momentum equation governing the flow field has been solved by scaling it to dimensionless structure with the aid of some dimensionless parameters. The equations have been analytically solved using modified Bessel equation and by the method of undetermined coefficients in order to obtain the temperature profile and velocity profile of the blood flow. The characteristics of the flow have been derived for a certain set of values RT , Da, θ, Gr, Re, Pr, ω, δ involved in the model analysis and are presented graphically with the help of software Mathematica. Moreover the velocity of the blood is adopting a wavy pattern as the values of the parameters vary. The study can be useful in providing a perception of the treatment caused by the superfluous consumption of fatty foods hence decreasing the risk of cancer, hypertension and many heart related diseases. DOI:10.46481/jnsps.2022.682 Keywords: Blood flow, Heat Transfer, Atherosclerosis, Treatment, Darcy number Article History : Received: 27 February 2022 Received in revised form: 01 July 2022 Accepted for publication: 01 July 2022 Published: 15 August 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: T. Latunde 1. Introduction The blood is regarded as a thick red liquid comprising of red blood cells, white blood cells and platelets which are circu- lating in through arteries and veins in a human body. It has a strong nourishing effect and serves as one of the most important substance constituting the human body. As now-a-days, heart diseases due to thickening of blood carrying artery is very com- mon which causes the accumulation of fatty substances in the lumen which in turn tends to block the side walls of the veins ∗Corresponding author tel. no: +91-9580493319 Email address: rishab.rhanvey@shiats.edu.in (R. R. Hanvey) and artery which is due to the pulsating nature of the heart. Fur- thermore, the most important function of the heart is pumping the blood throughout the circularity system and to supply oxy- gen to organs of the body. Severe stenosis may lead to critical flow conditions of blood by reducing the blood supply and resulting in serious conse- quences called carotid artery blockage which is one of the emi- nent factors contributing to fatal hemorrhages and hypertension. Regarding this, Atherosclerosis, the leading killer disease of the 21st century which is due to the hardening of the artery caused by the waxy substances, cellular wastes, calcium and choles- terol that leads to narrowing of the arterial walls resulting in brain damage or even death. Nevertheless, the significance of 1 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 2 additional factors cannot be neglected but the diet contributing to atherosclerosis is turning out to be a grave concern as the un- controlled intake of greasy and oily food enhances the risk of developing such chronic diseases. Over the decades, a lot of mathematicians and scientists have done ample of research on the blood flow with heat trans- fer in human circulatory system and have discovered a handful of useful results. Some of them supporting this research work are Bunonyo et al. [1] studied the impact of treatment param- eter on blood flow in an atherosclerotic artery. Bhatti et al. [2] examined the heat transfer analysis on peristaltic induced mo- tion of particle fluid suspension with variable viscosity: clot blood model. The influence of blood flow in large vessels on temperature distribution in hyperthermia was analyzed by La- gendijk [3]. Srivastava [4] discussed the analysis of flow char- acteristics of blood flowing through an inclined tapered porous artery with mild stenosis under the influence of inclined mag- netic field. Bunonyo and Amos [5] did the research on treat- ment and radiation effects on oscillatory blood flow through a stenosed artery. The study of slip effects on the unsteady MHD pulsating blood flow through porous medium in an artery under the effect of body acceleration was carried out by Eldesoky [6]. Makinde and Mhone [7] studied the heat transfer to MHD oscil- latory flow in a channel filled with porous medium. Unsteady heat transfer to oscillatory flow through a porous medium under slip condition was investigated by Hamza et al. [8]. Choudhury and Das [9] analyzed the heat transfer to MHD oscillatory visco elastic flow in a channel filled with porous medium. Biswas and Chakraborty [10] gave the pulsatile flow of blood in a con- stricted artery with body acceleration. Hanvey et al. [11] developed a model on heat and mass transfer in oscillatory flow of a non-Newtonian fluid between two inclined porous plates placed in a magnetic field. Plourde et al [12] looked into alterations of blood flow through arter- ies following atherectomy and the impact on pressure variation and velocity. Mathematical modeling of blood flow through vertebral artery with stenosis was studied by Ali et al. [13]. Mathur and Jain [14] discussed the mathematical model of non- Newtonian blood flow through artery in the presence of steno- sis. Ali and Asghar [15] studied the oscillatory channel flow for non-Newtonian fluid. The study of heat transfer in the flow of blood has gained much importance due to its significant role in medical sciences and in the research field as well. Many researchers such as Bunonyo, Cookey and Amos [16] studied the modeling of blood flow through stenosed artery with heat in the presence of magnetic field. Ali and Ahmad [17] gave an analytical solution of unsteady MHD blood flow and heat trans- fer through parallel plates when lower plate stretches exponen- tially. The effect of radiative heat and magnetic field on blood flow in an inclined tapered stenosed porous artery was studied by Abubakar and Adeoye [18]. Lavanya [19] made an analyti- cal study on MHD rotating flow through a porous medium with heat and mass transfer. Rawat et al. [20] considered the effect of magnetic field on oscillatory blood flow in multi stenosed artery. Varshney, Katiyar and Kumar [21] studied the effect of mag- netic field on the blood flow in artery having multiple stenosis. Elangovan and Selvaraj [22] gave the study of multiple stenosed artery with periodic body acceleration in presence of magnetic field. Transport of MHD couple stress fluid through peristal- sis in a porous medium under the influence of heat transfer and slip effects was investigated by Sankad and Nagathan [23]. Tri- pathi and Sharma [24] studied the effects of variable viscosity on MHD inclined arterial blood flow with chemical reaction. Modelling of arterial stenosis and its applications to blood flow was given by Pralhad and Schultz [25]. Kot and Elmaboud [26] studied the unsteady pulsatile fractional Maxwell viscoelastic blood flow with Cattaneo heat flux through a vertical stenosed artery with body acceleration. The study of efficient hybrid block method for numerical solution of second order partial dif- ferential problems via the method of lines was done by Olaiya, Azeez and Modeebei [27]. In the present paper, the influence of treatment parameter on the blood flow with heat transfer is examined. It has been seen that the unsteady flow of blood in stenosed artery with heat transfer placed in magnetic field has gained much importance as it can be a treatment to the fatty substances stuck at the walls of the arteries which increases the risk of having atherosclerosis, hence giving rise to such research using mathematical model. After studying the available literature, we have considered the effect of treatment parameter on the flow of blood through an atherosclerotic artery in the presence of magnetic field and heat transfer. The equations which govern the flow field are solved by using non dimensional parameters and a graphical approach has been studied which can be very useful in detection of heart related diseases which can be treated an early stage to avoid serious medical condition. 2. Material and Methods Figure 1. Geometry of the problem Let us consider blood to be Newtonian, unsteady, electri- cally conducting, incompressible and viscous flowing past an atherosclerotic artery that is presumed to be a cylindrical polar channel w′r′, x′), where r′ and x ′ are the direction of the flow. This type of flow arises due to the pumping of the blood in the region of the heart (especially when the contraction of the heart takes place). Furthermore, the pressure gradient is in the hori- zontal direction and the magnetic field is taken perpendicularly to the direction of the flow of blood. The flow field is governed by the following equations which were given by Bunonyo and 2 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 3 Amos [1] are: ρ ∂w′ ∂t′ = − ∂p′ ∂x′ + µ r′ ∂ ∂r′ ( r′ ∂w′ ∂r′ ) − µ k′ w′ −σB20w ′ + ρβT (T ′ − T8)g (1) ∂T′ ∂t′ = kT ρC p ( ∂2T′ ∂r′2 + 1 r′ ∂T′ ∂r′ ) − q′r ρC p (T′ − T′8) (2) The atherosclerosis region is supposed to be: r′ = R0 − δ′ 2 ( 1 + cos2 πx′ λ ) (3) where x′ = d0 + L0 2 (4) The boundary conditions are: w′ = 0 , T′ = Tw at r ′ = 0 w′ = 0 , T′ = T8 at r ′ = R (5) In order to write the governing equations in dimensionless form, some non-dimension variables are being introduced: w = w′ U0 , t = t′U0 R0 , Da = k′ R20 , Re = U0R0 v , M = B0R0 √ σ µ , p = R20 p ′ ρvλU0 , x = x′ λ , δ′ = 2δR0 RT , r = r′ R0 , Gr = gβT R20 vU0 T8, Rd = q′r R 2 0 µC p , Pr = µC p kT (6) The equations (1) to (4) are solved using dimensionless pa- rameter in (6), hence the following equations are obtained after dropping the primes: Re ∂w ∂t = − ∂P ∂x + 1 r ∂ ∂r ( r ∂w ∂r ) − 1 Da w − M2w + Grθ (7) Pr ∂θ ∂t = ( ∂2θ ∂r2 + 1 r ∂θ ∂r ) − RdPrθ (8) r = 1 − δ RT (1 + cos2πx ) (9) where x = 1 λ ( d0 + L0 2 ) The boundary conditions in dimensionless form are: w = 0, θ = 0 at r = 0 w = 0, θ = 1 at r = h (10) where Re, Da, M, Rd, Gr, Pr are Reynolds number, Darcy’s number, Hartmann number, Radiation parameter, Grashof num- ber and Prandtl number respectively. 3. Method of Solution The flow of blood through the arteries and veins is generally governed by the pumping action of the heart which in turn gives rise to oscillatory flow of blood, therefore the solution can be assumed to be of the form: w = w0e iωt (11) θ = θ0e iωt (12) The pressure gradient can be represented by: ∂P ∂x = −P0e iωt (13) where P0 is the pressure constant and ω represents the angular frequency of the oscillations. Now putting equations (10-12) into equations (7-9), we get: 1 r ∂ ∂r ( r ∂w0 ∂r ) − ( 1 Da + M2 + Reiω ) w0 = −P0 − Grθ0(14) ( ∂2θ0 ∂r2 + 1 r ∂θ0 ∂r ) −η2θ0 = 0 (15) where η2 = (Rd + iω)Pr The corresponding boundary conditions can be given as: w0 = 0 , θ0 = 0 at r = 0 w0 = 0, θ0 = e −iωt at r = h (16) where β2 = h2 (Rd + iω) Pr Applying the transformation ξ = r/h in the equation (15) and (16), we have: ∂2θ0 ∂ξ2 + 1 ξ ∂θ0 ∂ξ −β2θ0 = 0 (17) w0 = 0, θ0 = 0 at ξ = 0 w0 = 0, θ0 = e −iωt at ξ = 1 (18) Hence equation (17) can be re-written as: ξ2 ∂2θ0 ∂ξ2 + ξ ∂θ0 ∂ξ −β2ξ2θ0 = 0 (19) The above equation represents a modified Bessel differen- tial equation therefore its solution can be given as: θ0 (ξ) = A I0 (βξ) + B K0 (βξ) (20) where I0 and K0 are modified Bessel function of zero order. Here B = 0, as the solution is bounded at r = 0, hence otherwise θ0 (ξ) will not be finite. Therefore equation (20) will be reduced to: θ0 (ξ) = A I0 (βξ) (21) 3 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 4 Solving (21) subject to conditions given in the equations (18), we have: θ0 (ξ) = I0 (βξ) I0(β) e−iωt (22) Hence substituting equation (22) into equation (12), we have: θ (ξ) = I0 (βξ) I0(β) (23) Again, applying the transformation ξ = r/h in the equation (14) and using (23), we have: ∂2w0 ∂ξ2 + 1 ξ ∂w0 ∂ξ −ϕ w0 = χ (24) where ϕ = h2 ( 1 Da + M 2 + Reiω ) and χ = h2 ( −P0 − Gr I0 (βξ) I0 (β) e−iωt ) Hence the homogeneous solution of equation (24) is: ∂2w0 ∂ξ2 + 1 ξ ∂w0 ∂ξ −ϕ w0 = 0 (25) Equation (25) can be re-written as: ξ2 ∂2w0 ∂ξ2 + ξ ∂w0 ∂ξ −ϕ ξ2w0 = 0 (26) Equation (26) represents a modified Bessel differential equa- tion therefore its solution can be given as: w0h (ξ) = C1 I0 (√ ϕ ξ ) + C2 K0 ( √ ϕ ξ) (27) Here C2 = 0, as the solution is bounded at ξ = 0, therefore equation (27) will be reduced to: w0h (ξ) = C1 I0 (√ ϕ ξ ) . (28) Now solving equation (24) to the particular solution by us- ing the method of undetermined coefficients, we obtain: w0p (ξ) = h2 P0 ϕ + Gr h2 I0 (βξ) ϕI0(β) e−iωt (29) Further the general solution is: w0g (ξ) = w0h (ξ) + w0p (ξ) (30) Putting the values of equations (28) and (29) into (30), we get: w0g (ξ) = C1 I0 (√ ϕ ξ ) + h2 P0 ϕ + Gr h2 I0 (βξ) ϕI0(β) e−iωt (31) Substituting from (18) into (31) to calculate the value of C1, we have: C1 = − P0h2 ϕ − Grh2e−iωt ϕI0(β) (32) Keeping the value of C1 in equation (31), we have: w0 = P0h2 ϕ {( 1 − I0 (√ ϕ ξ )) + Grh2e−iωt ϕI0(β) ( I0 (βξ) − I0 (√ ϕ ξ ))} (33) The final form of general equation of the velocity profile by using equation (33) in (11), we obtain: wξ =[ P0h2 ϕ {( 1 − I0 (√ ϕ ξ )) + Grh2e−iωt ϕI0(β) ( I0 (βξ) − I0 (√ ϕ ξ ))}] eiωt (34) The expressions representing the profile for temperature and ve- locity are given by the equations (23) and (34) respectively. 4. Results and Discussion This study is carried out in view of combined effects of var- ious parameters on blood flow flowing past an atherosclerotic artery. Authors have derived the numerical results for velocity profile and temperature profile. The results calculated above are obtained for different values of parameters like Radiation pa- rameter, Prandtl number, Treatment parameter, Reynolds num- ber, Darcy number, Grashoff number, height of the stenosis and Hartmann number. In this paper, the velocity profile is calculated for Rd, RT , Pr, M, t, Gr, Da, δ,ω and Re as it is shown in Figures 1 to 10. It is clear from Figure 1 that the impact of Radiation parameter dominates the flow of blood and as the value of Rd increases the velocity takes a decreasing pattern. In addition, the velocity increases and attains a maximum position for different values of Radiation parameter, viz, Rd = 0.2, 0.4, 0.6, 0.8, 1.0 and then it starts to decrease in a converging manner and comes down to zero as r approaches 1. Figure 2 is obtained for various values of Prandtl number. On increasing the Prandtl number (Pr = 3, 5, 7, 9, 11) an in- crease in the blood velocity is observed. Furthermore, the am- plitude of the velocity profile keeps on increasing and reaches a maximum point and then it starts to decrease and converges at a same point. In Figure 3, the treatment parameter highly impacts the flow of blood as at first the velocity shows pulsating char- acter for RT = 0.1 and then starts decreasing as the parameter RT increases in the arterial channel. The treatment parameter controls the plague that is build up in the arterial section due to the consumption of fatty and oily food. The results obtained shows an obvious reading according to the physical behavior of human body and the velocity of the blood depends on the doses of treatment parameter values. Figure 4 expresses the curve for various values of Reynolds number (Re = 5, 10, 15, 20, 25). The velocity of the blood takes a curvilinear pattern and the velocity decreases in a regular trend as the Reynolds number increases. The graph attains a maximum position at r = 0 and gradually starts to drop and further converges to a point r = 1. Figure 5 illustrates that the growth of plague matter can cause many cardiovascular diseases which is quite clear from 4 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 5 Figure 2. Effect of Radiation Rd on Blood velocity w(r, t) with other parameters values Pr = 21, Re = 2, RT = 0.5, Gr = 15 and variation of r Figure 3. Effect of Prandtl number Pr on Blood velocity w(r, t) with other parameters values Rd = 2, Re = 2, RT = 0.5, Gr = 15, t = 2 and variation of r Figure 4. Effect of Treatment parameter RT on Blood velocity w(r, t) with other parameters values Rd = 2, Re = 2, M = 2, Gr = 15, t = 2 and variation of r the graph as the height of the stenosis increases the blood veloc- ity declines. This retardation of blood in arteries and veins gives Figure 5. Effect of Treatment parameter RT on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, M = 2, Gr = 15, t = 2 and variation of r Figure 6. Effect of Height of Steonsis δ on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, M = 2, Gr = 15, t = 2 and variation of r Figure 7. Effect of Darcy Number Da on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, M = 2, Gr = 15, t = 2 and variation of r a crystal clear proof that the glands, organs and tissues of the human body gets famished from oxygen rich blood which ul- timately results in hypertension and other cardiovascular prob- 5 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 6 Figure 8. Effect of Hartmann Number M on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, Pr = 21, Gr = 15, t = 2 and variation of r Figure 9. Effect of Oscillatory frequency ω on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, Pr = 21, Gr = 15, t = 2 and variation of r lems. The result comes out be very obvious from the fact that velocity becomes maximum at the midpoint and minimal at both the ends. Figure 6 explains the effect of Darcy’s num- ber of the flow of blood as the presence of porosity directly effects the flow. As the value of the parameter increases (Da = 0.05, 0.06, 0.07, 0.08, 0.09) the velocity shows an increasing pat- tern which reaches to an amplitude and then starts to converge at r = 1. The application of applied magnetic field holds ut- most importance in the study of blood flow as it can be a cure to many heart related diseases. It is examined in Figure 7 that as the intensity of the magnetic is increased the blood velocity is decreased. It is because when the magnetic field is applied on the blood flow which is electrically conducting in nature, gener- ates Lorentz force which makes the velocity of blood in arteries and veins to retard. Figure 8 represents the effects of oscillatory frequency pa- Figure 10. Effect of Grashoff Number Gr on Blood velocity w(r, t) with other parameters values Rd = 2, RT = 0.5, Pr = 21, Da = 0.05, t = 2 and variation of r Figure 11. Effect of Time parameter t on Blood velocity w(r, t) with other pa- rameters values Rd = 2, RT = 0.5, Pr = 21, Da = 0.05, Gr = 15 and variation of r Figure 12. Effect of Prandlt Number Pr on Temperature Profile θ(r, t) with other parameters values Rd = 2, RT = 0.5, t = 2, δ = 0.2 and variation of r rameter of the velocity of blood. As the frequency parameter increases (ω = 0.5, 0.6, 0.7, 0.8, 0.9) the velocity also adopts an increasing pattern and every time the amplitude keeps on esca- lating for higher values of frequency parameter. In Figure 9, the influence of Grashoff number is observed and its impact on 6 Hanvey & Bunonyo / J. Nig. Soc. Phys. Sci. 4 (2022) 682 7 Figure 13. Effect of Treatment Parameter RT on Temperature Profile θ(r, t) with other parameters values Rd = 2, , t = 2,ω = 2 δ = 0.2 and variation of r Figure 14. Effect of Radiation Parameter Rd on Temperature Profile θ(r, t) with other parameters values RT = 0.5, δ = 2, t = 2,ω = 2, and variation of r Figure 15. Effect of Height of Stenosis δ on Temperature Profile θ(r, t) with other parameters values RT = 0.5, Rd = 2, t = 2,ω = 2, and variation of r the velocity of blood is perceived. As the Grashoff number in- creases (Gr = 5, 10, 15, 20, 25), the velocity takes an increasing trend and the amplitude uniformly increases. Figure 10 explains the effect of time ‘t′ on the flow of blood as the value of t is in- creased the value of velocity also increases which further takes a maximum value and then converges to a point at r = 1. The influence of other parameters on the temperature pro- file can be seen from Figure 11 to Figure 16. The parameters Figure 16. Effect of Oscillatory Frequency ω on Temperature Profile θ(r, t) with other parameters values RT = 0.5, Rd = 2, t = 2 and variation of r affect the temperature profile by increasing and decreasing it at different levels. 5. Conclusion In the present study, we have investigated the influence of treatment parameter on the blood flowing past an atheroscle- rotic artery with heat transfer. On solving the mathematical model and carrying out the graphical study using the software Mathematica and furthermore the results have been discussed. The main conclusions of this work are as follows: It is observed that the velocity takes a oscillatory pattern with increasing values of radiation parameter Rd, however the effect of other parameters can’t be ignored. The effect of Prandlt number causes the velocity of blood to increase as Pr increases. The treartment parameter causes the flow of blood to retard which is due to the magnetic field that is applied on the surface portion however the contribution of other parameters cannot be neglected. The height of the stenosis is responsible for the decreases in the blood flow through arteries which inturns lowers the oxygen level in the human body leading to plethora of cardiovascular issues. The velocity of the blood decreases as the Reynolds number increases while other parameters are held. The magnetic field intensity decelerates the blood flow in the human system which can be very useful in the detection of formation of plagues in arteries and veins, detection of tumors through MRI scan and detects the diseases in liver and bile duct. The porosity factor holds great importance in circulation of blood flow through arteries as the porosity factor increases the velocity of blood increases and healthy oxygenated blood can reach different organs and tissues. Therefore the level of poros- ity factor can be maintained by implication of treatment param- eter. The velocity of the blood increases as frequency ω, Grashof number Gr, and time t increases. 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