Mathematical Modelling of MHD Blood Flow with Gold Nanoparticles in Slip Small Arteries

Document Type : Research Paper


Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia


Nanofluid is an innovative technology that is essential in biomedical applications. A nanofluid study of human blood flow mathematically is more favorable since it provides a hypothesis for complex systems faster and is cost-saving. Academic researchers have expressed interest in investigating the characteristics of Casson nanofluid flow within a cylindrical structure, which serves as a representative model for the flow of blood in constricted human arteries. However, slip velocity boundary conditions were considered by only a certain number of researchers. The goal of this study is to develop mathematical modelling of Casson fluid flow with gold nanoparticles in the slip cylinder. The impacts of convective heat transfer, magnetohydrodynamics (MHD), and porous medium are also investigated. The Tiwari-Das nanofluid model is utilized in the governing equations. Then, the governing equations with the related boundary conditions are transformed into dimensionless form. The analytical solutions were obtained through the use of the Laplace transform and the finite Hankel transform in combination. The results of nanofluid velocity, temperature, skin friction, and Nusselt number are analyzed through the use of graphs and tables containing relevant parameters. Slip velocity causes an increment in blood velocity and a decrement in skin friction. Blood velocity and temperature are enhanced as the nanoparticles' volume fraction is increased. It is significant in cancer treatment to increase the heat transfer rate at targeted cancerous cells.


Main Subjects

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[1] Yáñez, C., DeMas-Giménez, G., Royo, S., Overview of Biofluids and Flow Sensing Techniques Applied in Clinical Practice, Sensors, 22, 2022, 6836.
[2] Liepsch, D., Biofluid mechanics, Biomed. Tech., 43, 4, 1998, 94–99.
[3] Chhabra, R. P., Non-Newtonian Fluids: An Introduction, SERC School-cum-Symposium on Rheology of complex fluids, 2010.
[4] Baieth, H. A. E., Physical parameters of blood as a non-Newtonian fluid, Int. J. Biomed. Sci., 4, 2008, 323–329.
[5] Windberger, U., Sparer, A., Elsayad, K., The role of plasma in the yield stress of blood, Clin. Hemorheol. Microcirc., 84, 2023, 369-383.
[6] Wajihah, S. A., Sankar, D. S., A review on non-Newtonian fluid models for multi-layered blood rheology in constricted arteries, Arch. Appl. Mech., 93, 2023, 1771–1796.
[7] Venkatesan, J., Sankar, D. S., Hemalatha, K., Yatim, Y., Mathematical analysis of Casson fluid model for blood rheology in stenosed narrow arteries, J. Appl. Math., 2013, 2013, 583809.
[8] Sochi, T., Non-Newtonian flow in porous media, Polymer, 51, 2010, 5007–5023.
[9] Misra, J. C., Adhikary, S. D., Shit, G. G., Mathematical analysis of blood flow through an arterial segment with time-dependent stenosis, Math. Model. Anal., 13, 2008, 401-412.
[10] Siddiqui, S. U., Verma, N. K., Mishra, S., Gupta, R. S., Mathematical modelling of pulsatile flow of Casson’s fluid in arterial stenosis, Appl. Math. Comput., 210, 2009, 1–10.
[11] Kauffman, R. B., Thermoregulatory Physiology, Doomsday Preppers and Surviving the Unexpected Emergency, 2015, 1-4.
[12] Zolfaghari, A., Maerefat, M., Bioheat Transfer, Developments in Heat Transfer, 2011, 153–170.
[13] Luchakov, Y. I., Nozdrachev, A. D., Mechanism of heat transfer in different regions of human body, Biol. Bull., 36, 2009, 53–57.
[14] Pop, I., Sheremet, M., Free convection in a square cavity filled with a Casson fluid under the effects of thermal radiation and viscous dissipation, Int. J. Numer. Methods Heat Fluid Flow, 27, 2017, 2318–2332.
[15] Aghighi, M. S., Metivier, C., Masoumi, H., Natural convection of Casson fluid in a square enclosure, Multidiscip. Model. Mater. Struct., 16, 2020, 1245–1259.
[16] Sheikholeslami, M., Ganji, D. D., Magnetohydrodynamic and ferrohydrodynamic, External Magnetic Field Effects on Hydrothermal Treatment of Nanofluid, William Andrew Publishing, Norwich, New York, USA, 2016.
[17] Scanlon V. C., Sanders, T., Essentials of Anatomy and Physiology, FA Davis, 2018.
[18] Abdalla, S., Al-ameer, S. S., Al-Magaishi, S. H., Electrical properties with relaxation through human blood, Biomicrofluidics, 4, 2010, 1–16.
[19] Uchikawa, Y., Kotani, M., Measurement of Magnetic Field Produced from the Human Body, IEEE Transl. J. Magn. Japan, 7, 1992, 600–607.
[20] Kivrak, E., Yurt, K., Kaplan, A., Alkan, I., Altun, G., Effects of electromagnetic fields exposure on the antioxidant defense system, J. Microsc. Ultrastruct., 5, 2017, 167.
[21] Keltner, J. R., Roos, M. S., Brakeman, P. R., Budinger, T. F., Magnetohydrodynamics of blood flow, Magn. Reson. Med., 16, 1990, 139–149.
[22] Rashidi, S., Esfahani, J. A., Maskaniyan, M., Applications of magnetohydrodynamics in biological systems-a review on the numerical studies, J. Magn. Magn. Mater., 439, 2017, 358–372.
[23] Ali, F., Imtiaz, A., Khan, I., Sheikh, N. A., Hemodynamic flow in a vertical cylinder with heat transfer: Two-phase caputo fabrizio fractional model, J. Magn., 23, 2018, 179–191.
[24] Ali, F., Imtiaz, A., Khan, I., Sheikh, N.A., Flow of magnetic particles in blood with isothermal heating: A fractional model for two-phase flow, J. Magn. Magn. Mater., 456, 2018, 413–422.
[25] Ali, F., Khan, N., Imtiaz, A., Khan, I., Sheikh, N. A., The impact of magnetohydrodynamics and heat transfer on the unsteady flow of Casson fluid in an oscillating cylinder via integral transform: A Caputo–Fabrizio fractional model, Pramana - J. Phys., 93, 2019, 1–12.
[26] Kumar, G., Rizvi, S. M., Casson fluid flow past on vertical cylinder in the presence of chemical reaction and magnetic field, Appl. Appl. Math. An Int. J., 16, 2021, 524–537.
[27] Mehmood, O. U., Mustapha, N., Shafie, S., Unsteady Two-Dimensional Blood Flow in Porous Artery with Multi-Irregular Stenoses, Transp. Porous Media, 92, 2012, 259–275.
[28] Khaled, A. R. A., Vafai, K., The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Transf., 46, 2003, 4989–5003.
[29] Omamoke, E., Amos, E., Jatari, E., Impact of Thermal Radiation and Heat Source on MHD Blood Flow with an Inclined Magnetic Field in Treating Tumor and Low Blood, Asian Res. J. Math., 2020, 77–87.
[30] Dash, R. K., Mehta, K. N., Jayaraman, G., Casson Fluid Flow in a Pipe Filled with a Homogeneous Porous Medium, Int. J. Engng Sci., 34, 1996, 1145–1156.
[31] Anurag, Singh, A. K., Role of heat source / sink in transient free convective flow through a vertical cylinder filled with a permeable medium: An analytical approach, Heat Transf., 50, 2021, 3154-3175.
[32] Anurag, Maurya, J. P., Singh, A. K., Significance of time-dependent magnetohydrodynamic transient free convective flow in vertical annuli: An analytical approach with the finite Hankel transform, Heat Transf., 50, 2021, 6719–6736.
[33] Raje, A., Koyani, F., Bhise, A. A., Ramesh, K., Heat transfer and entropy optimization for unsteady MHD Casson fluid flow through a porous cylinder: Applications in nuclear reactors, Int. J. Mod. Phys. B, 7, 2023, 2350293.
[34] Mahian, O., Kianifar, A., Kleinstreuer, C., Al-Nimr, M. A., Pop, I., Sahin, A. Z., Wongwises, S., A review of entropy generation in nanofluid flow, Int. J. Heat Mass Transf., 65, 2013, 514–532.
[35] Mahian, O., Kianifar, A., Kalogirou, S. A., Pop, I., Wongwises, S., A review of the applications of nanofluids in solar energy, Int. J. Heat Mass Transf., 57, 2013, 582–594.
[36] Buongiorno, J., et al., A benchmark study on the thermal conductivity of nanofluids, J. Appl. Phys., 106, 2009, 094312.
[37] Choi, S. U. S., Enhancing thermal conductivity of fluids with nanoparticles, Am. Soc. Mech. Eng. Fluids Eng. Div. FED, 231, 1995, 99–105.
[38] Sheikholeslami, M., Ganji, D. D., Nanofluid convective heat transfer using semi analytical and numerical approaches: A review, J. Taiwan Inst. Chem. Eng., 65, 2016, 43–77.
[39] Guo, Z., A review on heat transfer enhancement with nanofluids, J. Enhanc. Heat Transf., 27, 2020, 1–70.
[40] Mahian, O., Pop, I., Sahin, A. Z., Oztop, H. F., Wongwises, S., Irreversibility analysis of a vertical annulus using TiO2/water nanofluid with MHD flow effects, Int. J. Heat Mass Transf., 64, 2013, 671–679.
[41] Mabrouk, M., Das, D. B., Salem, Z. A., Beherei, H. H., Nanomaterials for biomedical applications: Production, characterisations, recent trends and difficulties, Molecules, 26, 2021, 1–27.
[42] Wong, K. V., De Leon, O., Applications of nanofluids: Current and future, Adv. Mech. Eng., 2010, 1–11.
[43] Malik, M. Y., Naseer, M., Nadeem, S., Rehman, A., The boundary layer flow of Casson nanofluid over a vertical exponentially stretching cylinder, Appl. Nanosci., 4, 2014, 869–873.
[44] Alebraheem, J., Ramzan, M., Flow of nanofluid with Cattaneo – Christov heat flux model, Appl. Nanosci., 10, 2020, 2989-99.
[45] Khan, A., Shah, Z., Alzahrani, E., Islam, S., Entropy generation and thermal analysis for rotary motion of hydromagnetic Casson nanofluid past a rotating cylinder with Joule heating effect, Int. Commun. Heat Mass Transf., 119, 2020, 104979.
[46] Walelign, T., Haile, E., Kebede, T., Walelgn, A., Analytical study of heat and mass transfer in MHD flow of chemically reactive and thermally radiative Casson nanofluid over an inclined stretching cylinder, J. Phys. Commun., 4, 2020, 1–20.
[47] Farooq, U., Waqas, H., E.Alhazmi, S., Alhushaybari, A., Imran, M., Sadat, R., Muhammad, T., Ali, M. R., Numerical treatment of Casson nanofluid Bioconvectional flow with heat transfer due to stretching cylinder/plate: Variable physical properties, Arab. J. Chem., 16, 2023, 1-15.
[48] Alharbi, K. A. M., Shahmir, N., Ramzan, M., Almusawa, M. Y., Kadry, S., Bioconvective radiative unsteady Casson nanofluid flow across two concentric stretching cylinders with variable viscosity and variable thermal conductivity, Numer. Heat Transf. Part A-Applications, 2023.
[49] Mahdi, R. A., Mohammed, H. A., Munisamy, K. M., Saeid, N. H., Review of convection heat transfer and fluid flow in porous media with nanofluid, Renew. Sustain. Energy Rev., 41, 2015, 715–734.
[50] Kasaeian, A., Daneshazarian, R., Mahian, O., Kolsi, L., Chamkha, A. J., Wongwises, S., Pop, I., Nanofluid flow and heat transfer in porous media: A review of the latest developments, Int. J. Heat Mass Transf., 107, 2017, 778–791.
[51] Ghadimi, A., Saidur, R., Metselaar, H. S. C., A review of nanofluid stability properties and characterization in stationary conditions, Int. J. Heat Mass Transf., 54, 2011, 4051–4068.
[52] Merkin, J. H., Pop, I., Lok, Y. Y., Grosan, T., Basic equations and mathematical methods, Similarity Solutions for the Boundary Layer Flow and Heat Transfer of Viscous Fluids, Nanofluids, Porous Media, and Micropolar Fluids, Academic Press, 2022.
[53] Yan, J. F., Liu, J., Nanocryosurgery and its mechanisms for enhancing freezing efficiency of tumor tissues, Nanomedicine Nanotechnology, Biol. Med., 4, 2008, 79–87.
[54] Yu, Z., Gao, L., Chen, K., Zhang, W., Zhang, Q., Li, Q., Hu, K., Nanoparticles: A New Approach to Upgrade Cancer Diagnosis and Treatment, Nanoscale Res. Lett., 16, 2021, 88.
[55] Hamad, E. M., Khaffaf, A., Yasin, O., El-Rub, Z. A., Al-Gharabli, S., Al-Kouz, W., Chamkha, A. J., Review of Nanofluids and Their Biomedical Applications, J. Nanofluids, 10, 2021, 463–477.
[56] Hou, Y., Z. Sun, W. Rao, and J. Liu, Nanoparticle-mediated cryosurgery for tumor therapy, Nanomedicine Nanotechnology, Biol. Med., 14, 2018, 493–506.
[57] Sheikhpour, M., Arabi, M., Kasaeian, A., Rabei, A. R., Taherian, Z., Role of nanofluids in drug delivery and biomedical technology: Methods and applications, Nanotechnol. Sci. Appl., 13, 2020, 47–59.
[58] Liu, J., Deng, Z. S., Nano-cryosurgery: Advances and challenges, J. Nanosci. Nanotechnol., 9, 2009, 4521–4542.
[59] Khan, U., Bilal, S., Zaib, A., Makinde,O. D., Wakif, A., Numerical simulation of a nonlinear coupled differential system describing a convective flow of Casson gold–blood nanofluid through a stretched rotating rigid disk in the presence of Lorentz forces and nonlinear thermal radiation, Numer. Methods Partial Differ. Equ., 38, 2022, 308–328.
[60] Khan, U., Zaib, A., Khan, I., Nisar, K. S., Insight into the dynamics of transient blood conveying gold nanoparticles when entropy generation and Lorentz force are significant, Int. Commun. Heat Mass Transf., 127, 2021, 105415.
[61] Hussain, M., Farooq, U., Sheremet, M., Nonsimilar convective thermal transport analysis of EMHD stagnation Casson nanofluid flow subjected to particle shape factor and thermal radiations, Int. Commun. Heat Mass Transf., 137, 2022, 106230.
[62] Hussain, F., Nazeer, M., Altanji, M., Saleem, A., Ghafar, M. M., Thermal analysis of Casson rheological fluid with gold nanoparticles under the impact of gravitational and magnetic forces, Case Stud. Therm. Eng., 28, 2021, 101433.
[63] Upreti, H., Bartwal, P., Pandey, A. K., Makinde, O. D., Heat transfer assessment for Au-blood nanofluid flow in Darcy-Forchheimer porous medium using induced magnetic field and Cattaneo-Christov model, Numer. Heat Transf. Part B-Fundamentals, 84, 2023, 415–431.
[64] Imtiaz, A., Foong, O. M., Aamina, A., Khan, N., Ali, F., Khan, I., Generalized model of blood flow in a vertical tube with suspension of gold nanomaterials: Applications in the cancer therapy, Comput. Mater. Contin., 65, 2020, 171–192.
[65] Wang, R., Chai, J., Luo, B., Liu, X., Zhang, J., Wu, M., Wei, M., Ma, Z., A review on slip boundary conditions at the nanoscale: recent development and applications, Beilstein J. Nanotechnol., 12, 2021, 1237–1251.
[66] Nubar, Y., Blood Flow, Slip, and Viscometry, Biophys. J., 11, 1971, 252–264.
[67] Khan, M., Hashim, Hafeez, A., A review on slip-flow and heat transfer performance of nanofluids from a permeable shrinking surface with thermal radiation: Dual solutions, Chem. Eng. Sci., 173, 2017, 1–11.
[68] Rao, I. J., Rajagopal, K. R., Effect of the slip boundary condition on the flow of fluids in a channel, Acta Mech., 135, 1999, 113–126.
[69] Afify, A. A., The Influence of Slip Boundary Condition on Casson Nanofluid Flow over a Stretching Sheet in the Presence of Viscous Dissipation and Chemical Reaction, Math. Probl. Eng., 2017, 1-12.
[70] Gbadeyan, J. A., Titiloye, E. O., Adeosun, A. T., Effect of variable thermal conductivity and viscosity on Casson nanofluid flow with convective heating and velocity slip, Heliyon, 6, 2020, e03076.
[71] Noor, N. A. M., Shafie, S., Admon, M. A., Effects of viscous dissipation and chemical reaction on MHD squeezing flow of Casson nanofluid between parallel plates in a porous medium with slip boundary condition, Eur. Phys. J. Plus, 123, 2020, 855.
[72] Thirupathi, G., Govardhan, K., Narender, G., Radiative Magnetohydrodynamics Casson Nanofluid Flow and Heat and Mass Transfer past on Nonlinear Stretching Surface, J. Adv. Res. Numer. Heat Transf. J., 5, 2021, 1–21.
[73] Usman, M., Soomro, F. A., Ul Haq, R., Wang, W., Defterli, O., Thermal and velocity slip effects on Casson nanofluid flow over an inclined permeable stretching cylinder via collocation method, Int. J. Heat Mass Transf., 122, 2018, 1255–1263.
[74] Iqbal, W., Jalil, M., Khadimallah, M. A., Hussain, M., Naeem, M. N., Al Naim, A. F., Tounsi, A., Interaction of casson nanofluid with Brownian motion:Temperature profile with shooting method, Adv. Nano Res., 10, 2021, 349–357.
[75] Azmi, W. F. W., Mohamad, A. Q., Jiann, L. Y., Shafie, S., Unsteady natural convection flow of blood Casson nanofluid (Au) in a cylinder: nano ‑ cryosurgery applications, Sci. Rep., 13, 2023, 1–15.
[76] Rogers, K., Blood Vessel, Encyclopedia Britannica, 2023. Available at:
[77] Body, V., Blood Vessels, Circulatory Anantomy, 2023. Available at:
[78] Tietjen, G. T., Saltzman, W. M., Nanomedicine gets personal, Sci. Transl. Med., 7, 2015, 1–4.
[79] Maiti, S., Shaw, S., Shit, G. C., Fractional order model for thermochemical flow of blood with Dufour and Soret effects under magnetic and vibration environment, Colloids Surfaces B Biointerfaces, 197, 2021, 111395.
[80] Raza, J., Thermal radiation and slip effects on magnetohydrodynamic (MHD) stagnation point flow of Casson fluid over a convective stretching sheet, Propuls. Power Res., 18, 2019, 138-146.
[81] Benhanifia, K., Redouane, F., Lakhdar, R., Brahim, M., Al‑Farhany, K., Jamshed, W., Eid, M. R., El Din, S. M., Raizah, Z., Investigation of mixing viscoplastic fluid with a modified anchor impeller inside a cylindrical stirred vessel using Casson–Papanastasiou model, Sci. Rep., 12, 2022, 1–19.
[82] Noranuar, W. N. N., Mohamad, A. Q., Shafie, S., Khan, I., Jiann, L. Y., Ilias, M. R., Non-coaxial rotation flow of MHD Casson nanofluid carbon nanotubes past a moving disk with porosity effect, Ain Shams Eng. J., 12, 2021, 4099-4110.
[83] Mackolil, J., Mahanthesh, B., Exact and statistical computations of radiated flow of nano and Casson fluids under heat and mass flux conditions, J. Comput. Des. Eng., 6, 2019, 593–605.
[84] Oztop, H. F., Abu-Nada, E., Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow, 29, 2008, 1326–1336.
[85] Padma, R., Selvi, R. T., Ponalagusamy, R., Effects of slip and magnetic field on the pulsatile flow of a Jeffrey fluid with magnetic nanoparticles in a stenosed artery, Eur. Phys. J. Plus, 134, 2019, 1–15.
[86] Khan, I., Shah, N. A., Tassaddiq, A., Mustapha, N., Kechil, S. A., Natural convection heat transfer in an oscillating vertical cylinder, PLoS One, 13, 2018, e0188656.
[87] Mirza, I. A., Akram, M. S., Siddique, I., Flows of a generalized second grade fluid in a cylinder due to a velocity shock, Chinese J. Phys., 60, 2019, 720–730.
[88] Maiti, S., Shaw, S., Shit, G. C., Caputo–Fabrizio fractional order model on MHD blood flow with heat and mass transfer through a porous vessel in the presence of thermal radiation, Phys. A Stat. Mech. its Appl., 540, 2020, 123149.
[89] Esfe, M. H., Bahiraei, M., Torabi, A., Valadkhani, M., A critical review on pulsating flow in conventional fluids and nanofluids: Thermo-hydraulic characteristics, Int. Commun. Heat Mass Transf., 120, 2021, 104859.
[90] Tripathi, J., Vasu, B., Gorla, R. S. R., Chamkha, A. J., Murthy, P. V. S. N., Bég, O. A., Blood flow mediated hybrid nanoparticles in human arterial system: Recent research, development and applications, J. Nanofluids, 10, 2021, 1–30.
[91] Reyaz, R., Mohamad, A. Q., Lim, Y. J., Saqib, M., Shafie, S., Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects, Fractal Fract., 6, 2022, 38.
[92] Onyiriuka, E. J., Ighodaro, O. O., Adelaja, A. O., Ewim, D. R. E., Bhattacharyya, S., A numerical investigation of the heat transfer characteristics of water-based mango bark /nanofluid flowing in a double-pipe heat exchanger, Heliyon, 5, 2019, e02416.