Mathematical modeling of Williamson's model for blood flow inside permeable multiple stenosed arteries with electro-osmosis

Document Type : Article


Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan


An artery afflicted with multiple stenosis is the focus of this study, which emphasises the electro-osmotic effects. The artery's walls are porous and slip boundary effects are present. Blood flow problems are better modelled with a slip and porous border. It is examined extensively due to the wide range of applications in the medical field, especially in the diagnosis of drug delivery and handling of cellular irregularities. In this paper, we have visualised the non-Newtonian behaviour of blood by using viscoelastic fluids as a Williamson fluid model. A mathematical model for an incompressible fluid is being created, and the mathematical issue is then transformed into its dimensionless form by applying limitations in the case of mild multiple stenosis. As soon as the problem is put into a dimensionless form, the partial differential equations for the velocity profile and temperature profile can be found. Analytical solutions of the resulting system are calculated with the help of the Homotopy perturbation method (HPM). The visual representation of analytically obtained solutions is investigated for both symmetric and non-symmetric geometries of stenosis. For varied values of flow rate Q and electro-osmotic parameter m, the streamlines are examined in detail.


Main Subjects

1. Huang, X., Gordon, M.J., and Zare, R.N. "Currentmonitoring method for measuring the electroosmotic flow rate in capillary zone electrophoresis", Analytical Chemistry, 60(17), pp. 1837-1838 (1988).
2. Minerick, A.R., Ostafin, A.E., and Chang, H.C.  Electrokinetic transport of red blood cells in microcapillaries", Electrophoresis, 23(14), pp. 2165-2173 (2002).
3. Dolnk, V., Liu, S., and Jovanovich, S. "Capillary electrophoresis on microchip", Electrophoresis: An International Journal, 21(1), pp. 41-54 (2000).
4. Wu, R.C. and Papadopoulos, K.D. "Electroosmotic flow through porous media: Cylindrical and annular models", Colloids and Surfaces A: Physicochemical and Engineering Aspects, 161(3), pp. 469-476 (2000).
5. Yang, R.J., Fu, L.M., and Lin, Y.C. "Electroosmotic flow in microchannels", Journal of Colloid and Interface Science, 239(1), pp. 98-105 (2001).
6. Zhao, C. and Yang, C. "An exact solution for electroosmosis of non-newtonian fluids in microchannels", Journal of Non-Newtonian Fluid Mechanics, 166(17- 18), pp. 1076-1079 (2011).
7. Tang, G.H., Li, X.F., He, Y.L., et al. "Electroosmotic flow of non-Newtonian fluid in microchannels", Journal of Non-Newtonian Fluid Mechanics, 157(1-2), pp. 133-137 (2009).
8. Liu, Q., Jian, Y., and Yang, L. "Alternating current electroosmotic flow of the Jeffreys fluids through a slit microchannel", Physics of Fluids, 23(10), 102001 (2011).
9. Nadeem, S., Kiani, M.N., Saleem, A., et al. "Microvascular blood flow with heat transfer in a wavy channel having electroosmotic effects", Electrophoresis, 41(13- 14), pp. 1198-1205 (2020).
10. Narla, V.K. and Tripathi, D. "Electro-osmosis modulated transient blood flow in curved microvessels: A study of a mathematical model", Microvascular Research, 123, pp. 25-34 (2019).
11. Tripathi, D., Yadav, A., Beg, O.A., et al. "Study of microvascular non-Newtonian blood  flow modulated by electro-osmosis", Microvascular Research, 117, pp. 28-36 (2018).
12. Akram, J., Akbar, N.S., and Maraj, E.N. "A comparative study on the role of nanoparticle dispersion in electro-osmosis-regulated peristaltic  flow of water", Alexandria Engineering Journal, 59(2), pp. 943-956 (2020).
13. Saleem, S., Akhtar, S., Nadeem, S., et al. "Mathematical study of electroosmotically driven peristaltic flow of Casson fluid inside a tube having systematically contracting and relaxing sinusoidal heated walls", Chinese Journal of Physics, 71, pp. 300-311 (2021).
14. Ponalagusamy, R. "Blood flow through stenosed tube", Ph.D Thesis, Indian Institute of Technology Bombay (1986).
15. Ponalagusamy, R. "Blood flow through an artery with mild stenosis: A two-layered model, different shapes of stenoses and slip velocity at the wall", Journal of Applied Sciences, 7(7), pp. 1071-1077 (2007).
16. Varshney, G., Katiyar, V., and Kumar, S. "Effect of magnetic field on the blood  flow in an artery having multiple stenoses: A numerical study", International Journal of Engineering, Science, and Technology, 2(2), pp. 967-82 (2010).
17. Nadeem, S. and Ijaz, S. "Single wall carbon nanotube (SWCNT) examination on blood flow through a multiple stenosed artery with variable nanofluid viscosity", AIP Advances, 5(10), 107217 (2015).
18. Haider, J.A. and Ahmad, S. "Dynamics of the Rabinowitsch fluid in a reduced form of elliptic duct using finite volume method", International Journal of Modern Physics, B, 36(30), 2250217 (2022).
19. Akbar, N.S., Nadeem, S., and Ali, M. "Jeffrey fluid model for blood flow through a tapered artery with a stenosis", Journal of Mechanics in Medicine and Biology, 11(03), pp. 529-545 (2011).
20. Akbar, N.S. and Nadeem, S. "Simulation of heat and chemical reactions on Reiner Rivlin fluid model for blood flow through a tapered artery with stenosis", Heat and Mass Transfer, 46(5), pp. 531-539 (2010).
21. Haider, J.A., Asghar, S., and Nadeem, S. "Travelling wave solutions of the third-order KdV equation using Jacobi elliptic function method", International Journal of Modern Physics, B, 37(12), 2350117 (2023).
22. Toghraie, D., Esfahani, N.N., Zarringhalam, M., et al. "Blood flow analysis inside different arteries using non- Newtonian Sisko model for application in biomedical engineering", Computer Methods and Programs in Biomedicine, 190, 105338 (2020).
23. Yan, S.R., Zarringhalam, M., Toghraie, D., et al. "Numerical investigation of non-Newtonian blood flow within an artery with cone shape of stenosis in various stenosis angles", Computer Methods and Programs in Biomedicine, 192, 105434 (2020).
24. Riaz, A., Bobescu, E., Ramesh, K., et al. "Entropy analysis for cilia-generated motion of Cu-blood flow of nanofluid in an annulus", Symmetry, 13(12), 2358 (2021).
25. Tripathi, D., Prakash, J., Tiwari, A.K., et al. "Thermal, microrotation, electromagnetic field, and nanoparticle shape effects on Cu-CuO/blood flow in microvascular vessels", Microvascular Research, 132, 104065 (2020).
26. Shehzad, N., Zeeshan, A., and Ellahi, R. "Electroosmotic flow of MHD power law Al2O3-PVC nanofluid in a horizontal channel: Couette-Poiseuille flow model", Communications in Theoretical Physics, 69(6), p. 655 (2018).
27. Bhatti, M.M., Arain, M.B., Zeeshan, A., et al. "Swimming of gyrotactic microorganism in MHD Williamson nanofluid flow between rotating circular plates embedded in a porous medium: Application of thermal energy storage", Journal of Energy Storage, 45, 103511 (2022).
28. Ouellette, J., An-Ti-Ci-Pa-Tion: The Physics of Dripping Honey, Scientific America (2013).
29. Williamson, R.V. "The flow of pseudoplastic materials", Industrial & Engineering Chemistry, 21(11), pp. 1108-1111 (1929).
30. Khan, M., Malik, M.Y., Salahuddin, T., et al. "Heat and mass transfer of Williamson nano fluid flow yield by an inclined Lorentz force over a nonlinear stretching sheet", Results in Physics, 8, pp. 862-868 (2018).
31. Khan, M., Salahuddin, T., Malik, M.Y., et al. "Change in viscosity of Williamson nanofluid flow due to thermal and solutal stratification", International Journal of Heat and Mass Transfer, 126, pp. 941-948 (2018).
32. Khan, N.A., Khan, S., and Riaz, F. "Boundary layer flow of Williamson fluid with chemically reactive species using scaling transformation and homotopy analysis method", Mathematical Sciences Letters, 3(3), p. 199 (2014).
33. Das, S., Pal, T.K., Jana, R.N., et al. "Significance of Hall currents on hybrid nano-blood flow through an inclined artery having mild stenosis: Homotopy perturbation approach", Microvascular Research, 137, 104192 (2021).
34. Ahmad, I. and Ilyas, H. "Homotopy perturbation method for the nonlinear MHD Jeffery-Hamel blood flows problem", Applied Numerical Mathematics, 141, pp. 124-132 (2019).
35. Hamrelaine, S., Mebarek-Oudina, F., and Sari, M.R. "Analysis of MHD Jeffery Hamel flow with suction/ injection by homotopy analysis method", Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 58(2), pp. 173-186 (2019).
36. Sudha, T., Umadevi, C., Dhange, M., et al. "Effects of stenosis and dilatation on flow of blood mixed with suspended nanoparticles: A study using homotopy technique", International Journal of Applied Mechanics and Engineering, 26(1), pp. 251-265 (2021).
37. Tripathi, B. and Sharma, B.K. "Effect of variable viscosity on MHD inclined arterial blood flow with chemical reaction", International Journal of Applied Mechanics and Engineering, 23(3), pp. 767-785 (2018).
38. Ullah, I., Hayat, T., Alsaedi, A., et al. "Modeling for radiated Marangoni convection flow of magneto-nano liquid subject to activation energy and chemical reaction", Scientia Iranica, 27(6), pp. 3390-3398 (2020).
39. Makinde, O.D. and Gnaneswara Reddy, M. "MHD peristaltic slip  flow of Casson fluid and heat transfer in channel filled with a porous medium", Scientia Iranica, 26(4), pp. 2342-2355 (2019).
40. Kaveh, A. and Kooshkebaghi, M. "Artificial coronary circulation system: A new bio-inspired metaheuristic algorithm", Scientia Iranica, 26(5), pp. 2731-2747 (2019).
41. Akhtari, H., Mirzaee, I., and Pourmahmoud, N. "Lattice Boltzmann simulation of blood flow properties and vessel geometry in open and closed vessels: A numerical study", Scientia Iranica, 26(6), pp. 3283- 3292 (2019).
42. Basar, A., Kabak, O., and Topcu, Y.I. "A tabu search algorithm for a multi-period bank branch location problem: A case study in a Turkish bank", Scientia Iranica, 26(6), pp. 3728-3746 (2019).
43. Tang, L.N., Ma, Y.Z., Wang, J.J., et al. "Robust parameter design of supply chain inventory policy considering the uncertainty of demand and lead time", Scientia Iranica, 26(5), pp. 2971-2987 (2019).
44. Sohrabi, M., Zandieh, M., and Afshar-Nadjafi, B. "An equity-oriented multi-objective inventory management model for blood banks considering the patient condition: a real-life case", Scientia Iranica, pp. 1-39 (2021).
45. Fallahi, H., Shirani, E., and Zohravi, E. "Hemodynamic analysis of coronary artery bypass grafting with elastic walls and different stenoses", Scientia Iranica, 28(2), pp. 773-784 (2021).
46. Gitinavard, H., Ghodsypour, S.H., and Akbarpour Shirazi, M. "A bi-objective multi-echelon supply chain model with Pareto optimal points evaluation for perishable products under uncertainty", Scientia Iranica, 26(5), pp. 2952-2970 (2019).
47. Hamdi-Asl, A., Amoozad-Khalili, H., Tavakkoli- Moghaddam, R., et al. "Toward sustainability in designing agricultural supply chain network: A case study on palm date", Scientia Iranica, pp. 1-27 (2021).
48. Liu, X., Chen, X., Zhang, Y., et al. "The thermal behavior of blood flow in the arteries with various radii and various stenosis angles using non-Newtonian Sisko model", Alexandria Engineering Journal, 61(9), pp. 7195-7201 (2022).
49. Cherkaoui, I., Bettaibi, S., Barkaoui, A., et al. "Magnetohydrodynamic blood flow study in stenotic coronary artery using lattice Boltzmann method", Computer Methods and Programs in Biomedicine, 106850 (2022).
50. Carvalho, V., Lopes, D., Silva, J., et al., Comparison of CFD and FSI Simulations of Blood Flow in Stenotic Coronary Arteries (2022).
51. Ellahi, R., Sait, S.M., Shehzad, N., et al. "A hybrid investigation on numerical and analytical solutions of electro-magnetohydrodynamics flow of nanofluid through porous media with entropy generation", International Journal of Numerical Methods for Heat & Fluid Flow, 30(2), pp. 834-854 (2019).
52. Bakker, L.M.M.L., Xiao, N., Van De Ven, A.A.F., et al. "Image-based blood flow estimation using a semianalytical solution to the advection-diffusion equation in cylindrical domains", Journal of Fluid Mechanics, 924, pp. 924A18-1-924A18-11 (2021).
53. "Simulation of micropolar fluid flows: Validation of  umerical results with analytical solutions" Bachelor's Thesis (2018).
54. Ramesh, K., Tripathi, D., Beg, O.A., et al. "Slip and hall current effects on Jeffrey fluid suspension flow in a peristaltic hydromagnetic blood micropump", Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 43(4), pp. 675-692 (2019).
55. Zhang, L., Bhatti, M.M., and Michaelides, E.E. "Thermally developed coupled stress particle-fluid motion with mass transfer and peristalsis", Journal of Thermal Analysis and Calorimetry, 143(3), pp. 2515-2524 (2021).
56. Raje, A., Devakar, M., and Ramgopal, N.C. "Influence of heat transfer on the flow of immiscible fluids through pipes: An analytical study", Journal of Porous Media, 24(11), pp. 85-99 (2021).
57. Bai, H.G., Jeyanthi, M.P., Hemavathy, P., et al. "Blood flow in arteries with stenoses: A threedimensional unsteady flow", Journal of Positive School Psychology, 6(2), pp. 3336-3343 (2022).
58. Saleem, A., Akhtar, S., and Nadeem, S. "Biomathematical analysis of electro-osmotically modulated hemodynamic blood 
flow inside a symmetric and nonsymmetric stenosed artery with joule heating", International Journal of Biomathematics, 15(02), 2150071 (2022).
59. Shao, S. and Sun, Q. "Evaluation of intracranial artery stenosis using time-of-flight magnetic resonance angiography: New wine in an old bottle", European Radiology, 32(6), pp. 1-2 (2022).
60. Kamangar, S. "Influence of multi stenosis on hemodynamic parameters in an idealized coronary artery model", Journal of Applied Fluid Mechanics, 15(1), pp. 15-23 (2022).
61. Jamali, M.S.A. and Ismail, Z. "Generalized power law model of blood flow in a stenosed bifurcated artery", Annals of Mathematical Modeling, 1(2), pp. 35-46 (2022).
62. Akhtar, S., McCash, L.B., Nadeem, S., et al. "Mechanics of non-Newtonian blood flow in an artery having multiple stenoses and electroosmotic effects", Science Progress, 104(3), 00368504211031693 (2021).
63. Mekheimer, K.S., Haroun, M.H., and Elkot, M.A. "Effects of magnetic field, porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood  flow characteristics", Applied Mathematics and Mechanics, 32(8), pp. 1047-1064 (2011).
64. Liechty, B.C., Webb, B.W., and Maynes, R.D. "Convective heat transfer characteristics of electroosmotically generated flow in microtubes at high wall potential", International Journal of Heat and Mass Transfer, 48(12), pp. 2360-2371 (2005).
65. Akbar, N.S. and Butt, A.W. "Entropy generation analysis in convective ferromagnetic nano blood flow through a composite stenosed arteries with permeable wall", Communications in Theoretical Physics, 67(5), p. 554 (2017).