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EDL Induced Electro-magnetized Modified Hybrid Nano-blood Circulation in an Endoscopic Fatty Charged Arterial Indented Tract

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Abstract

Purpose

The electrokinetic process for streaming fluids in magnetic environments is emerging due to its immense applications in medical and biochemical industrial domains. In this context, our proposed model seeks to inquire into the hemodynamic characteristics of electro-magnetized blood blended with trihybrid nanoparticles circulation induced by electro-osmotic forces in an endoscopic charged arterial annular indented tract. This steaming model also invokes the consequences of variable Lorentz attractive force, buoyancy force, heat source, viscous and Joule warming, arterial wall properties, and sliding phenomena for featuring more realistic problems in blood flows. Different shapes of suspended trihybrid nanoparticles, such as spheres, bricks, cylinders, and platelets, are included in the model formation. Electro-magnetized modified hybrid nano-blood is an electro-conductive solution comprising blood as base fluid and magnetized trihybrid nanoparticles (copper, gold, and alumina).

Methods

Closed-form solution in terms of Bessel’s functions is gotten for electro-osmotic potential due to the electric double layer (EDL). The homotopy perturbation methodology is implemented in order to track down the convergent series solutions of non-linear coupled flow equations being elicited. The physical attributes of distinct evolving parameters on the different dimensionless hemodynamic profiles and quantities of interest are elucidated evocatively via a sort of graphs and charts.

Results

The ancillary outcomes proved that the Debye–Hückel parameter and Helmholtz–Smoluchowski velocity have a dual impact on the ionized bloodstream. The bloodstream rapidity is alleviated/boosted for the assisting/opposing electroosmosis process. Cooling of ionized blood in the endoscopic arterial conduit is achieved with lower Hartmann numbers. Copper–gold–alumina/blood exhibits a superior heat transmission rate across the arterial wall than copper–gold–blood, copper–blood, and pure blood. Additionally, the contour topology for the bloodstream in the flow domain is briefly elaborated. The contour distribution is significantly amended due to the variant of the Debye–Hückel parameter.

Conclusion

The model’s new findings may be invaluable in electro-magneto-endoscopic operation, electro-magneto-treatment for cancer, surgical process, etc.

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Data Availability

Data will be made available on request.

Abbreviations

a :

Wave amplitude (m)

\(b_0\) :

Outer tube inlet radius (m)

\(B_0\) :

Magnetic field strength (Telsa)

Br :

Brinkman number

c :

Wave speed (m s−1)

\(c_p\) :

Specific heat at constant pressure (J kg−1 K−1)

\(\check{c}_0\) :

Viscous dumping force coefficient

\(\hat{e}\) :

Electron charge (C)

\(E_a\) :

Electric field strength (N C−1 )

\(E_1\) :

Rigidity parameter

\(E_2\) :

Stiffness parameter

\(E_3\) :

Viscous damping force parameter

g :

Gravity

Gr :

Grashof number

i :

Complex unity \((\sqrt{-1})\)

\(I_0\) :

Modified Bessel function of first kind of order zero

\(I_1\) :

Modified Bessel function of first kind of order one

\(J_0\) :

Bessel function of first kind of order zero

k :

Thermal conductivity (W m−1 K−1)

\(\hat{K}\) :

Boltzmann constant (J K−1)

l :

Constant

\(\tilde{L}\) :

Linear operator

\(\check{m}_0\) :

Mass per unit area of membrane (kg m−2)

M :

Hartmann number

n :

Shape factor

\(\hat{n}_0\) :

Bulk concentration of cations and anions

\(\hat{n}^{\pm }\) :

Number densities of cations and anions (m−3)

p :

Dimensionless blood pressure

\(\bar{p}\) :

Blood pressure (mm Hg or kg m−1 s−2)

\(p_0\) :

Pressure at membrane (kg m−1 s−2)

Pr :

Prandtl number

\(Q_0\) :

Heat source coefficient (W m−2 K−1)

\(r_0\) :

Endoscopic tube radius (m)

(r, z):

Dimensionless co-ordinates (m)

\((\bar{r}, \bar{z})\) :

Reference system co-ordinates (m)

\((r_1, r_2)\) :

Dimensionless radii of inner and outer tubes

\((\bar{r}_1, \bar{r}_2)\) :

Radii of inner and outer tubes (m))

Re :

Reynolds number (kg m−3)

\(\hat{s}\) :

Embedding parameter

S :

Joule heating parameter

t :

Dimensionless time

\(\bar{t}\) :

Time (s)

\(\hat{T}\) :

Average temperature (K)

T :

Temperature (K)

\(T_0\) :

Endoscopic wall temperature (K)

\(T_1\) :

Ambient temperature (K)

(u, w):

Dimensionless velocity components

\((\bar{u}, \bar{w})\) :

Velocity components (ms−1)

\(U_{hs}\) :

Helmholtz–Smoluchowski velocity

\(Y_0\), \(Y_1\) :

Bessel functions of second kind and zeroth order

\(Y_0\), \(Y_1\) :

Bessel functions of second kind and first order

\(\hat{z}_v\) :

Ionic valence

\(\beta\) :

Thermal expansion coefficient (K−1 )

\(\gamma\) :

Thermal slip parameter

\(\gamma ^*\) :

Thermal slip length

\(\delta\) :

Wave number

\(\epsilon _0\) :

Dielectric permittivity (F m−1 )

\(\varepsilon\) :

Dimensionless wave amplitude (m)

\(\theta\) :

Dimensionless blood temperature

\(\kappa\) :

Debye–Hückel parameter

\(\lambda\) :

Wave length (m)

\(\lambda _D\) :

Debye length (m)

\(\Lambda\) :

Operator (m)

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\nu\) :

Arterial shape factor

\(\check{\xi }_0\) :

Elastic tension per unit width (Pa)

\(\rho\) :

Density (C m−3)

\(\rho _e\) :

Net charge density (C m−3)

\(\sigma\) :

Electrical conductivity (S m−1)

\(\tau\) :

Velocity slip parameter

\(\tau ^*\) :

Velocity slip length

\(\Phi\) :

Dimensionless electro-osmotic potential

\(\bar{\Phi }\) :

Electro-osmotic potential (V)

\(\phi _1\) :

Solid volume fraction of Cu NPs

\(\phi _2\) :

Solid volume fraction of Au NPs

\(\phi _3\) :

Solid volume fraction of Al2O3 NPs

\(\Phi _0\) :

Zeta potential (kg m−2 A−1 s−3)

\(\chi\) :

Heat source parameter

\(\psi\) :

Dimensionless stream function

\(\Omega\) :

Dimensionless magnetic field strength

\(s_1\) :

Copper nanoparticles (solid)

\(s_2\) :

Gold nanoparticles (solid)

\(s_3\) :

Alumina nanoparticles (solid)

b:

Pure blood

nb:

Nano-blood

hnb:

Hybridized nano-blood

mhnb:

Modified hybridized nano-blood

NPs:

Nanoparticles

EOF:

Electro-osmotic flow

EDL:

Electric double layer

AWSS:

Endoscopic wall share stress

RTT:

Rate of thermal transport

PB:

Pure blood

NB:

Nano-blood

HNB:

Hybrid nano-blood

MHNB:

Modified hybrid nano-blood

References

  1. Choi, S.U.S. Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, D.A., Wang, H.P. (eds.) Developments and Applications of Non-Newtonian Flows ASME, New York, 1995.

  2. Skirtach, A. G., C. Dejugnat, D. Braun, A. L. Rogach, W. J. Parak, H. Möhwald, and G. B. Sukhorukov. The role of metal nanoparticles in remote release of encapsulated materials. Nano Lett. 5(7):1371–1377, 2005. https://doi.org/10.1021/nl050693n.

    Article  CAS  PubMed  Google Scholar 

  3. Ghosh, P., G. Han, M. De, C. K. Kim, and V. M. Rotello. Gold nanoparticles in delivery applications. Adv. Drug Deliv. Rev. 60:1307–1315, 2008. https://doi.org/10.1016/j.addr.2008.03.016.

    Article  CAS  PubMed  Google Scholar 

  4. Abdelsalam, S. I., and M. M. Bhatti. New insight into Au NP applications in tumour treatment and cosmetics through wavy annuli at the nanoscale. Sci. Rep. 9(1):1–14, 2019. https://doi.org/10.1038/s41598-018-36459-0.

    Article  CAS  Google Scholar 

  5. Elnaqeeb, T., N. A. Shah, and K. S. Mekheimer. Hemodynamic characteristics of gold nanoparticle blood flow through a tapered stenosed vessel with variable nano fluid viscosity. Bionanosci. 9(2):245–255, 2019. https://doi.org/10.1007/s12668-018-0593-5.

    Article  Google Scholar 

  6. Mekheimer, K. S., A. Z. Zaher, and W. M. Hasona. Entropy of AC electro-kinetics for blood mediated gold or copper nanoparticles as a drug agent for thermotherapy of oncology. Chin. J. Phys. 65:123–138, 2020. https://doi.org/10.1016/j.cjph.2020.02.020.

    Article  CAS  Google Scholar 

  7. Umadevi, C., M. Dhange, B. Haritha, and T. Sudha. Flow of blood mixed with copper nanoparticles in an inclined overlapping stenosed artery with magnetic field. Case Stud. Therm. Eng.25:100947, 2021. https://doi.org/10.1016/j.csite.2021.100947.

    Article  Google Scholar 

  8. Sarwar, L., and A. Hussain. Flow characteristics of Au-blood nano fluid in stenotic artery. Int. Commun. Heat Mass Transf.127:105486, 2021. https://doi.org/10.1016/j.icheatmasstransfer.2021.105486.

    Article  CAS  Google Scholar 

  9. Sadaf, H., and S. I. Abdelsalam. Adverse effects of a hybrid nanofluid in a wavy nonuniform annulus with convective boundary conditions. RSC Adv. 10(26):15035–15043, 2020. https://doi.org/10.1039/D0RA01134G.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  10. Gandhi, R., B.K. Sharma, C. Kumawat, and O.A. Bég. Modeling and analysis of magnetic hybrid nanoparticle Au-Al\(_2\)O\(_3\)/blood based drug delivery through a bell-shaped occluded artery with Joule heating, viscous dissipation and variable viscosity effects. Proc. Inst. Mech. Eng. E: J. Proc. Mech. Eng. 235 (5), 2022. https://doi.org/10.1177/095440892210802.

  11. Sadaf, H., N. Iftikhar, and N. S. Akbar. Physiological fluid flow analysis by means of contraction and expansion with addition of hybrid nanoparticles. Eur. Phys. J. Plus. 134(5):1–12, 2019. https://doi.org/10.1140/epjp/i2019-12598-9.

    Article  Google Scholar 

  12. Das, S., T. K. Pal, and R. N. Jana. Outlining impact of hybrid composition of nanoparticles suspended in blood flowing in an inclined stenosed artery under magnetic field orientation. Bionanosci. 11:99–115, 2021. https://doi.org/10.1007/s12668-020-00809-y.

    Article  Google Scholar 

  13. Sharma, B. K., R. Gandhi, and M. M. Bhatti. Entropy analysis of thermally radiating MHD slip flow of hybrid nanoparticles Au-Al\(_2\)O\(_3\)/blood) through a tapered multi-stenosed artery. Chem. Phys. Lett.790:139348, 2022. https://doi.org/10.1016/j.cplett.2022.139348.

    Article  CAS  Google Scholar 

  14. Abbasi, A., K. Al-Khaled, M. I. Khan, S. U. Khan, A. M. El-Refaey, W. Farooq, M. Jameel, and S. Qayyum. Optimized analysis and enhanced thermal efficiency of modified hybrid nanofluid (Al\(_2\)O\(_3\), CuO, Cu) with nonlinear thermal radiation and shape features. Case Stud. Therm. Eng.28:101425, 2021. https://doi.org/10.1016/j.csite.2021.101425.

    Article  Google Scholar 

  15. Guedri, K., T. Bashir, A. Abbasi, W. Farooq, S. U. Khan, M. I. Khan, M. Jameel, and A. M. Galal. Hall effects and entropy generation applications for peristaltic flow of modified hybrid nanofluid with electroosmosis phenomenon. J. Indian Chem. Soc.99(9):100614, 2022. https://doi.org/10.1016/j.jics.2022.100614.

    Article  CAS  Google Scholar 

  16. Abbasi, M., M. M. Heyhat, and A. Rajabpour. Study of the effects of particle shape and base fluid type on density of nanofluids using ternary mixture formula: a molecular dynamics simulation. J. Mol. Liq.305:112831, 2020. https://doi.org/10.1016/j.molliq.2020.112831.

    Article  CAS  Google Scholar 

  17. Das, S., A. Ali, R. N. Jana, and O. D. Makinde. EDL impact on mixed magneto-convection in a vertical channel using ternary hybrid nanofluid. J. Adv. Chem. Eng.12:100412, 2022. https://doi.org/10.1016/j.ceja.2022.100412.

    Article  CAS  Google Scholar 

  18. Dolui, S., B. Bhaumik, and S. De. Combined effect of induced magnetic field and thermal radiation on ternary hybrid nanofluid flow through an inclined catheterized artery with multiple stenosis. Chem. Phys. Lett.811:140209, 2023. https://doi.org/10.1016/j.cplett.2022.140209.

    Article  CAS  Google Scholar 

  19. Ali, A., S. Das, and T. Muhammad. Dynamics of blood conveying copper, gold, and titania nanoparticles through the diverging/converging ciliary micro-vessel: Further analysis of ternary-hybrid nanofluid. J. Mol. Liq.390:122959, 2023. https://doi.org/10.1016/j.molliq.2023.122959.

    Article  CAS  Google Scholar 

  20. Changdar, S., and S. De. Analytical investigation of nanoparticle as a drug carrier suspended in a MHD blood flowing through an irregular shape stenosed artery. Iran. J. Sci. Technol. Trans. A. 43(3):1259–1272, 2019. https://doi.org/10.1007/s40995-018-0601-1.

    Article  Google Scholar 

  21. Sharma, B. K., C. Kumawat, and K. Vafai. Computational biomedical simulations of hybrid nanoparticles (Au-Al\(_2\)O\(_3\)/blood-mediated) transport in a stenosed and aneurysmal curved artery with heat and mass transfer: hematocrit dependent viscosity approach. Chem. Phys. Lett.790:139666, 2022. https://doi.org/10.1016/j.cplett.2022.139666.

    Article  CAS  Google Scholar 

  22. Mekheimer, K. S., R. E. Abo-Elkhair, S. I. Abdelsalam, K. K. Ali, and A. M. A. Moawad. Biomedical simulations of nanoparticles drug delivery to blood hemodynamics in diseased organs: synovitis problem. Int. Commun. Heat Mass Transf.130:105756, 2022. https://doi.org/10.1016/j.icheatmasstransfer.2021.105756.

    Article  CAS  Google Scholar 

  23. Latham, T.W. Fluid motions in a peristaltic pump (Doctoral dissertation, Massachusetts Institute of Technology). MA, 1966.

  24. Tripathi, D., R. Jhorar, O. A. Bég, and A. Kadir. Electro-magneto-hydrodynamic peristaltic pumping of couple stress biofluids through a complex wavy micro-channel. J. Mol. Liq. 236:358–367, 2017. https://doi.org/10.1016/j.molliq.2017.04.037.

    Article  CAS  Google Scholar 

  25. Ranjit, N. K., G. C. Shit, and D. Tripathi. Joule heating and zeta potential effects on peristaltic blood flow through porous micro vessels altered by electrohydrodynamic. Microvasc. Res. 117:74–89, 2018. https://doi.org/10.1016/j.mvr.2017.12.004.

    Article  CAS  PubMed  Google Scholar 

  26. Elaqeeb, T., N. A. Shah, and K. S. Mekheimer. Hemodynamic characteristics of gold nanoparticle blood flow through a tapered stenosed vessel with variable nanofluid viscosity. Bionanosci. 9:245–255, 2019. https://doi.org/10.1007/s12668-018-0593-5.

    Article  Google Scholar 

  27. Elmaboud, Y. A., K. S. Mekheimer, and T. G. Emam. Numerical examination of gold nanoparticles as a drug carrier on peristaltic blood flow through physiological vessels: Cancer therapy treatment. Bionanosci. 9:952–965, 2019. https://doi.org/10.1007/s12668-019-00639-7.

    Article  Google Scholar 

  28. Bhatti, M. M., and S. I. Abdelsalam. Bio-inspired peristaltic propulsion of hybrid nanofluid flow with Tantalum (Ta) and Gold (Au) nanoparticles under magnetic effects. Waves Random Complex Media. 2021. https://doi.org/10.1080/17455030.2021.1998728.

    Article  Google Scholar 

  29. Sridhar, V., K. Ramesh, M. G. Reddy, M. N. Azese, and S. I. Abdelsalam. On the entropy optimization of hemodynamic peristaltic pumping of a nanofluid with geometry effects. Waves Random Complex Media. 2022. https://doi.org/10.1080/17455030.2022.2061747.

    Article  Google Scholar 

  30. Ali, A., F. Mebarek-Oudina, A. Barman, S. Das, and A. I. Ismail. Peristaltic transportation of hybrid nano-blood through a ciliated micro-vessel subject to heat source and Lorentz force. J. Therm. Anal. Calorim. 148:7059–7083, 2023. https://doi.org/10.1007/s10973-023-12217-x.

    Article  CAS  Google Scholar 

  31. Mekheimer, K. S., M. H. Haroun, and M. A. Elkot. Effects of magnetic field, porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics. Appl. Math. Mech. Engl. Ed. 32(8):1047–1064, 2011. https://doi.org/10.1007/s10483-011-1480-7.

    Article  Google Scholar 

  32. Sadaf, H., and R. Malik. Nano fluid flow analysis in the presence of slip effects and wall properties by means of contraction and expansion. Commun. Theor. Phys. 70(3):337–343, 2018. https://doi.org/10.1088/0253-6102/70/3/337.

    Article  CAS  Google Scholar 

  33. Akram, J., and N. S. Akbar. Biological analysis of Carreau nanofluid in an endoscope with variable viscosity. Phys. Scr.95(5):055201, 2020. https://doi.org/10.1088/1402-4896/ab74d7.

    Article  CAS  Google Scholar 

  34. Iftikhar, N., A. Rehman, and H. Sadaf. Inspection of physiological flow in the presence of nanoparticles with MHD and slip effects. J. Therm. Anal. Calorim. 147:987–997, 2022. https://doi.org/10.1007/s10973-020-10337-2.

    Article  CAS  Google Scholar 

  35. Dolui, S., B. Bhaumik, S. De, and S. Changdar. Effect of a variable magnetic field on peristaltic slip flow of blood based hybrid nanofluid through a non-uniform annular channel. J. Mech. Med. Biol. 23(01):2250070, 2023. https://doi.org/10.1142/S0219519422500701.

    Article  Google Scholar 

  36. Ijaz, S., and S. Nadeem. Transportation of nanoparticles investigation as a drug agent to attenuate the atherosclerotic lesion under the wall properties impact. Chaos Soliton. Fract. 112:52–65, 2018. https://doi.org/10.1016/j.molliq.2018.08.122.

    Article  CAS  Google Scholar 

  37. Ijaz, S., and S. Nadeem. Shape factor and sphericity features examination of Cu and Cu-Al\(_2\)O\(_3\) /blood through atherosclerotic artery under the impact of wall characteristic. J. Mol. Liq. 271:361–372, 2018. https://doi.org/10.1016/j.molliq.2018.08.122.

    Article  CAS  Google Scholar 

  38. Abdelsalam, S. I., and M. M. Bhatti. The impact of impinging TiO\(_2\) nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow. Multidiscip. Model. Mater. Struct. 14(3):530–548, 2018. https://doi.org/10.1108/MMMS-08-2017-0094.

    Article  Google Scholar 

  39. Das, S., B. Barman, R. N. Jana, and O. D. Makinde. Hall and ion slip currents’ impact on electromagnetic blood flow conveying hybrid nanoparticles through an endoscope with peristaltic waves. Bionanosci. 11(3):770–792, 2021. https://doi.org/10.1007/s12668-021-00873-y.

    Article  Google Scholar 

  40. Das, S., T. K. Pal, R. N. Jana, and B. Giri. Ascendancy of electromagnetic force and Hall currents on blood flow carrying Cu-Au NPs in a non-uniform endoscopic annulus having wall slip. Microvasc. Res.138:104191, 2021. https://doi.org/10.1016/j.mvr.2021.104191.

    Article  CAS  PubMed  Google Scholar 

  41. Das, S., T. K. Pal, and R. N. Jana. Electromagnetic hybrid nano-blood pumping via peristalsis through an endoscope having blood clotting in presence of Hall and ion slip currents. Bionanosci. 11(3):848–870, 2021. https://doi.org/10.1007/s12668-021-00853-2.

    Article  Google Scholar 

  42. Ayub, S., H. Zahir, and A. Tanveer. Mixed convection and non-linear thermal radiative analysis for Carreau–Yasuda nanofluid in an endoscope. Int. Commun. Heat Mass Tranf.138:106371, 2022. https://doi.org/10.1016/j.icheatmasstransfer.2022.106371.

    Article  CAS  Google Scholar 

  43. Das, S., and T. K. Pal. Hall impact on peristaltic stream of Prandtl hybrid nano-blood via a tapered endoscopic annular vessel with blood clotting. Waves in Random and Complex Media. 2023. https://doi.org/10.1080/17455030.2023.2252922.

    Article  Google Scholar 

  44. Karmakar, P., A. Barman, and S. Das. EDL transport of blood-infusing tetra-hybrid nano-additives through a cilia-layered endoscopic arterial path. Mater. Today Commun. 36:06772, 2023. https://doi.org/10.1016/j.mtcomm.2023.106772.

    Article  CAS  Google Scholar 

  45. Reuss, F. F. Charge-induced flow. Proc. Imp. Soc. Nat. Moscow. 3:327–344, 1809.

    Google Scholar 

  46. Wiedemann, G. First quantitative study of electrical endosmose. Pogg. Ann. 87:321–323, 1852.

    Google Scholar 

  47. Abdelsalam, S. I., K. S. Mekheimer, and A. Z. Zaher. Alterations in blood stream by electroosmotic forces of hybrid nanofluid through diseased artery: aneurysmal/stenosed segment. Chin. J. Phys. 67:314–329, 2020. https://doi.org/10.1016/j.cjph.2020.07.011.

    Article  CAS  Google Scholar 

  48. Das, S., B. N. Barman, and R. N. Jana. Hall and ion-slip currents’ role in transportation dynamics of ionic Casson hybrid nano-liquid in a microchannel via electroosmosis and peristalsis. Korea Aust. Rheol. J. 33(4):367–391, 2021. https://doi.org/10.1007/s13367-021-0029-6.

    Article  Google Scholar 

  49. Saleem, S., S. Akhtar, S. Nadeem, A. Saleem, M. Ghalambaz, and A. Issakhov. Mathematical study of electroosmotically driven peristaltic flow of Casson fluid inside a tube having systematically contracting and relaxing sinusoidal heated walls. Chin. J. Phys. 71:300–311, 2021. https://doi.org/10.1016/j.cjph.2021.02.015.

    Article  Google Scholar 

  50. Noreen, S., S. Waheed, D. C. Lu, and D. Tripathi. Heat stream in electroosmotic bio-fluid flow in straight microchannel via peristalsis. Int. Commun. Heat Mass Transf.123:105180, 2021. https://doi.org/10.1016/j.icheatmasstransfer.2021.105180.

    Article  Google Scholar 

  51. Akhtar, S., L. B. McCash, S. Nadeem, S. Saleem, and A. Issakhov. Mechanics of non-Newtonian blood flow in an artery having multiple stenosis and electroosmotic effects. Sci. Prog. 104(3):1–9, 2021. https://doi.org/10.1177/00368504211031693.

    Article  Google Scholar 

  52. Das, S., and B. Barman. Ramification of hall and ion-slip currents on electro-osmosis of ionic hybrid nanofluid in a peristaltic microchannel. Bionanosci. 12(3):957–78, 2022. https://doi.org/10.1007/s12668-022-01002-z.

    Article  Google Scholar 

  53. Asghar, Z., M. Waqas, M. A. Gondal, and W. A. Khan. Electro-osmotically driven generalized Newtonian blood flow in a divergent micro-channel. Alex. Eng. J. 61:4519–4528, 2022. https://doi.org/10.1016/j.aej.2021.10.012.

    Article  Google Scholar 

  54. Das, S., P. Karmakar, and A. Ali. Electrothermal blood streaming conveying hybridized nanoparticles in a non-uniform endoscopic conduit. Med. Biol. Eng. Comput. 60:3125–3151, 2022. https://doi.org/10.1007/s11517-022-02650-9.

    Article  CAS  PubMed  Google Scholar 

  55. Mehmood, O. U., S. Bibi, A. Zeeshan, M. M. Maskeen, and F. Alzahrani. Electroosmotic impacts on hybrid antimicrobial blood stream through catheterized stenotic aneurysmal artery. Eur. Phys. J. Plus. 137:585, 2022. https://doi.org/10.1140/epjp/s13360-022-02783-8.

    Article  Google Scholar 

  56. Karmakar, A., A. Ali, and S. Das. Circulation of blood loaded with trihybrid nanoparticles via electro-osmotic pumping in an eccentric endoscopic arterial canal. Int. Commun. Heat Mass Transf. 141:106593, 2023. https://doi.org/10.1016/j.icheatmasstransfer.2022.106593.

    Article  CAS  Google Scholar 

  57. Manchi, R., and R. Ponalagusamy. Pulsatile flow of EMHD micropolar hybrid nanofluid in a porous bifurcated artery with an overlapping stenosis in the presence of body acceleration and Joule heating. Braz. J. Phys. 52:52, 2022. https://doi.org/10.1007/s13538-022-01061-3.

    Article  Google Scholar 

  58. Ramesh, K., D. Tripathi, M. M. Bhatti, K. Ghachem, S. U. Khan, and L. Kolsi. Mathematical modeling and simulation of electromagnetohydrodynamic bionanomaterial flow through physiological vessels. J. Appl. Biomater. Funct. Mater. 2022. https://doi.org/10.1177/22808000221114.

    Article  PubMed  Google Scholar 

  59. Ali, A., A. Barman, and S. Das. Electromagnetic phenomena in cilia actuated peristaltic transport of hybrid nano-blood with Jeffrey model through an artery sustaining regnant magnetic field. Waves Random Complex Media. 2022. https://doi.org/10.1080/17455030.2022.2072533.

    Article  Google Scholar 

  60. Ali, A., A. Barman, and S. Das. EDL aspect in cilia-regulated bloodstream infused with hybridized nanoparticles via a microtube under a strong field of magnetic attraction. Therm. Sci. Eng. Prog.36:101510, 2022. https://doi.org/10.1016/j.tsep.2022.101510.

    Article  CAS  Google Scholar 

  61. Paul, P., and S. Das. EDL flow designing of an ionized Rabinowitsch blood doped with gold and go nanoparticles in an oblique skewed artery with slip events. BioNanoSci. 13:2307–2336, 2023. https://doi.org/10.1007/s12668-023-01176-0.

    Article  Google Scholar 

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Acknowledgements

The authors extend their sincere thanks to the respected editor and reviewers.

Funding

The author(s) received no specific funding for this research study.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Gour Banga, Malda, 732 103, India

    Poly Karmakar & Sanatan Das

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  1. Poly Karmakar
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  2. Sanatan Das
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Correspondence to Sanatan Das.

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Appendix

Appendix

$$\begin{aligned} \zeta _1^*= & {}\, \frac{\rho _{mhnb}}{\rho _{b}},\, \zeta _2^*= \frac{\mu _{mhnb}}{\mu _{b}},\, \zeta _3^*= \frac{{(\rho \beta )}_{mhnb}}{{(\rho \beta )}_{b}},\, \zeta _4^*= \frac{\sigma _{mhnb}}{\sigma _{b}},\\ \zeta _5^*= & {}\, \frac{{(\rho c_p)}_{mhnb}}{{(\rho c_p)}_{b}},\, \zeta _6^*= \frac{k_{mhnb}}{k_{b}},\\ \zeta _1= & {}\, \zeta _2^*,\, \zeta _2 = \zeta _3^*, \, \zeta _3 = \zeta _4^*,\, \zeta _4 = \zeta _6^*,\,\\ E(z, t)= & {}\, 8 \pi ^3 \varepsilon \left[ \frac{E_3}{2 \pi } \sin {2 \pi (z-t)} - (E_1 + E_2) \cos {2 \pi (z-t)}\right] ,\\ a_1= & {}\, -\frac{\zeta _3}{\zeta _1} M^2,\, a_2 = \frac{1}{\zeta _1} \kappa ^2 U_{hs},\, a_3 =-\frac{1}{\zeta _1} \frac{\partial p}{\partial z},\, a_4 = \frac{\zeta _2}{\zeta _1} Gr,\\ b_1= & {}\, \frac{\zeta _1}{\zeta _4} Br,\, b_2 = \frac{\zeta _3}{\zeta _4} Br M^2,\, b_3 = \frac{\zeta _3 S + \chi }{\zeta _4},\\ c_1= & {}\, -\frac{1}{\log {r_2} - \log {r_1} + \frac{\tau \zeta _1}{r_2}},\, c_2 = \frac{a_4}{9 (r_1 - r_2)} (r_1^3 - r_2^3 - 3 \tau \zeta _1 r_2^2),\\ c_3= & {}\, (1 + a_1) (r_2^2 - r_1^2 + \tau \zeta _1),\\ c_4= & {}\, \frac{1}{4 (r_1 - r_2)} (r_2^2 - r_1^2 + 2 \tau \zeta _1 r_2) [(r_2 - r_1) (4 + a_1 + a_3) + a_4 r_2],\\ c_5= & {}\, \frac{a_2}{\kappa ^2 \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )- J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] } Y_0(-i r_1 \kappa ) [I_0(r_1 \kappa ) - I_0(r_2 \kappa )],\\ c_6= & {}\, \frac{a_2}{\kappa ^2 \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )- J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] } J_0(-i r_1 \kappa ) [Y_0(- i r_2 \kappa )-Y_0(-i r_1 \kappa )],\\ c_7= & {}\, \frac{a_1}{2} r_1 r_2 (\log ^2{r_1} - \log ^2{r_2}),\, c_8 = \tau \zeta _1 (r_2 - r_1 - a_1 r_1 \log {r_2}),\\ c_9= & {}\, \frac{a_2 \tau \zeta _1}{\kappa \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] } \left[ i J_0(i r_1 \kappa ) Y_1(-i r_2 \kappa ) - I_1(r_2 \kappa ) Y_0(-i r_1 \kappa )\right] ,\\ c_{10}= & {}\, -\frac{1}{36 r_2 \kappa ^3 (r_2 - r_1) [r_2 (\log {r_1} - \log {r_2}) - \tau \zeta _1] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] },\\ c_{11}= & {}\, 36 a_2 \kappa r_2 (r_1 - r_2) \left[ r_2 (\log {r_2} - \log {r_1}) + \tau \zeta _1\right] Y_0(-i r_1 \kappa ),\\ c_{12}= & {}\, 36 \kappa r_2^2 \left[ a_2 (r_1 - r_2) \left\{ Y_0(-i r_1 \kappa ) I_0(r_2 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right\} \right. \nonumber \\{} & {} + \left. r_1 r_2 \kappa ^2 (r_1 - r_2 + a_1 r_1) \left\{ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) - J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right\} \right] \log {r_1},\\ c_{13}= & {}\, 36 \kappa r_2^2 \left[ a_2 (r_2 - r_1) \left\{ I_0(r_1 \kappa ) - J_0(i r_1 \kappa )\right\} Y_0(-i r_1 \kappa ) \right. \nonumber \\{} & {} + \left. r_2 \kappa ^2 \left\{ (r_1 - r_2 - a_1 r_2) J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) + r_2 (1 + a_1) J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right\} \right] \log {r_2},\\ c_{14}= & {}\, 36 \tau \zeta _1 \kappa r_2 \left[ a_2 (r_1 - r_2) \left\{ \kappa r_2 I_1(r_2 \kappa ) + J_0(i r_1 \kappa ) - I_0(r_1 \kappa )\right\} Y_0(-i r_1 \kappa ) \right. \nonumber \\{} & {} + \left. \kappa ^2 r_2 (r_1 - r_2 + a_1 r_2) \left\{ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) - J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right\} \right] ,\\ c_{15}= & {}\, 36 a_1 \tau \zeta _1 r_1 r_2^2 \kappa ^3 \left[ (r_1 - r_2) J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - r_1 J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] \log {r_1} \log {r_2}\nonumber \\{} & {} + 36 \tau \zeta _1 r_2^2 \kappa ^2 \left[ i a_2 (r_2 - r_1) J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) + \kappa r_1^2 a_1 \left\{ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) \right. \right. \nonumber \\ {}- & {} \left. \left. \kappa r_1^2 a_1 J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right\} \right] \log {r_1},\\ c_{16}= & {}\, 27 a_1 r_2 \kappa ^3 [r_2^4 \log {r_1} + r_1^3 (\tau \zeta _1 + r_2 \log {r_2})] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] ,\\ c_{17}= & {}\, 18 a_1 r_1 r_2^3 \kappa ^3 (r_2 - r_1) (\log {r_2} - \log {r_1}) \left[ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) \right. \nonumber \\- & {} \left. J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right] \log {r_2} \log {r_1},\\ c_{18}= & {}\, 18 a_1 \tau \zeta _1 r_1 r_2^2 \kappa ^3 (r_2 - r_1) \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] \log ^2{r_1},\\ c_{19}= & {}\, 18 \tau \zeta _1 r_2^3 \kappa ^3 \left[ a_1 (r_1 + r_2) + a_3 (r_1 - r_2)\right] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) \right. \nonumber \\- & {} \left. J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] \log {r_1},\\ c_{20}= & {}\, 9 r_2^4 \kappa ^3 [r_1 (a_1 + a_3) - r_2 a_3] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] \log {r_1},\\ c_{21}= & {}\, 9 r_1^2 r_2^2 \kappa ^3 [r_2 (a_1 + a_3 + a_4) - r_1 a_3] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] \log {r_2},\\ c_{22}= & {}\, 9 \tau \zeta _1 r_1^2 r_2 \kappa ^3 [r_2 (a_1 + a_3 + a_4) - r_1 a_3] \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] ,\\ c_{23}= & {}\, 6 a_4 r_2^5 \tau \zeta _1 \kappa ^3 \left[ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) - J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right] \log {r_1},\\ c_{24}= & {}\, 5 a_4 r_2^5 \kappa ^3 \left[ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) - J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right] \log {r_1},\\ c_{25}= & {}\, 4 a_4 r_1^3 r_2 \kappa ^3 (r_2 \log {r_2} + \tau \zeta _1) \left[ J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa ) - J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa )\right] ,\\ C_1= & {}\, c_1 (c_2 + c_3 + c_4 + c_5 + c_6 + c_7 + c_8 + c_9),\\ C_2= & {}\, c_{10} (c_{11} + c_{12} + c_{13} + c_{14} + c_{15} + c_{16} \\{} & {} + c_{17} + c_{18} + c_{19} + c_{20} + c_{21} \\{} & {} + c_{22} + c_{23} + c_{24} + c_{25}), \end{aligned}$$
$$\begin{aligned} w_{1}= & {}\, C_2,\, w_2 = C_1,\, w_3 = -\frac{1}{2} a_1 r_1 r_2,\, w_4 = (1 + a_1) (r_1 + r_2),\\ w_5= & {}\, \frac{(r_2 - r_1) (4 + a_1 + a_3) + a_4 r_2}{4 (r_1 - r_2)},\, w_6 = -\frac{a_4}{9 (r_1 - r_2)},\\ w_7= & {}\, \frac{a_2}{\kappa ^2 \left[ J_0(i r_2 \kappa ) Y_0(-i r_1 \kappa ) - J_0(i r_1 \kappa ) Y_0(-i r_2 \kappa )\right] },\\ d_{1}= & {}\, -\frac{1}{\log {r_2}-\log {r_1} + \frac{\gamma \zeta _4}{r_2}},\\ d_{2}= & {}\, (r_1^2 - r_2^2) \left[ \frac{b_3}{4} - \frac{11}{9} b_2 r_1 r_2 + \frac{13}{144} b_2 (r_1^2 + r_2^2)\right] ,\\ d_{3}= & {}\, \frac{b_1}{18} (r_1 + r_2) (r_1^3 - r_2^3),\\ d_{4}= & {}\, \frac{b_2}{2} r_1^2 r_2^2 \left( \log ^2{r_1} - \log ^2{r_2}\right) ,\\ d_{5}= & {}\, \gamma \zeta _4 \left[ r_1 r_2 \left\{ b_1 \left( \frac{r_2}{3}-\frac{r_1}{2}\right) +b_2 \left( \frac{3}{2} r_1+\frac{2}{3} r_2\right) \right\} \right. \nonumber \\{} & {} - \left. r_2 \left\{ \frac{b_3}{2} + \frac{1}{12} (2 b_1 + b_2) r_2^2 + b_2 r_1^2 \log {r_2}\right\} + \frac{1}{r_1 - r_2}\right] ,\\ d_{6}= & {}\, -\frac{1}{144 (r_1 - r_2) [r_2 (\log {r_1} - \log {r_2}) - \gamma \zeta _4]},\\ d_{7}= & {}\, 36 b_3 (r_1 - r_2) (r_2^3 \log {r_1}\\{} & {} + r_1^2 r_2 \log {r_2} + r_1^2 \gamma \zeta _4),\\ d_{8}= & {}\, -36 b_1 r_1^2 r_2^3 (r_1 \log {r_1}\\{} & {} + r_2 \log {r_2} + r_1 \gamma x_4),\\ d_{9}= & {}\, 4 r_1^2 r_2^2 (7 b_1 - 19)(r_2^2 \log {r_1}\\{} & {} + r_1 r_2 \log {r_2} + r_1 \gamma \zeta _4),\\ d_{10}= & {}\, 252 b_2 r_1^2 r_2^2 (r_2^2 \log {r_2}\\{} & {} + r_1 r_2 \log {r_1} + r_2 \gamma \zeta _4),\\ d_{11}= & {}\, - 189 b_2 r_1 r_2 (r_2^4 \log {r_1}\\{} & {} + r_1^3 r_2 \log {r_2} + r_1^3 \gamma \zeta _4),\\ d_{12}= & {}\, (8 b_1 + 13 b_2) (r_2^6 \log {r_1}\\{} & {} + r_1^5 r_2 \log {r_2} + r_1^5 \gamma \zeta _4),\\ d_{13}= & {}\, 72 b_2 r_1^3 r_2^3 (r_1 - r_2) (\log {r_1} - \log {r_2}) \log {r_1} \log {r_2},\\ d_{14}= & {}\, 72 r_2^2 \gamma \zeta _4 (r_1 - r_2) (b_2 r_1^2 \log {r_1} - b_3) \log {r_1},\\ d_{15}= & {}\, -72 b_1 r_1 r_2^2 \gamma \zeta _4 (r_2^2 + r_1^2) \log {r_1},\\ d_{16}= & {}\, 144 r_2 \gamma \zeta _4 [1 - b_2 r_1^2 r_2 (r_1 - r_2) \log {r_2}]\log {r_1},\\ d_{17}= & {}\, 120 r_1^2 r_2^3 \gamma \zeta _4 (b_1 - b_2) \log {r_1},\\ d_{18}= & {}\, 12 \gamma \zeta _4 r_2^2 [r_2^3 (b_2 + 2 b_1)\\{} & {} + 9 b_2 r_1 (2 r_1^2 - r_2^2)] \log {r_1},\\ D_1= & {}\, d_1 (d_2 + d_3 + d_4 + d_5),\\ D_2= & {}\, d_6 (d_7 + d_8 + d_9 + d_{10} + d_{11}\\{} & {} + d_{12} + d_{13} + d_{14} + d_{15} + d_{16} + d_{17} + d_{18}),\\ \theta _{1}= & {}\, D_2,\, \theta _{2} = D_1, \theta _3 = -\frac{1}{2} b_2 r_1^2 r_2^2,\, \theta _4=2 b_2 r_1 r_2 (r_1 + r_2),\\ \theta _5= & {}\, -\left[ \frac{1}{4} (b_1 + b_2) (r_1^2 + r_2^2)\right. \\{} & {} \left. + \frac{1}{2} r_1 r_2 (b_1 + 2 b_2) + \frac{b_3}{4}\right] ,\\ \theta _6= & {}\, \frac{2}{9} (2 b_1 + b_2) (r_1 + r_2),\, \theta _7 = -\frac{1}{16} (4 b_1 + b_2). \end{aligned}$$

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Karmakar, P., Das, S. EDL Induced Electro-magnetized Modified Hybrid Nano-blood Circulation in an Endoscopic Fatty Charged Arterial Indented Tract. Cardiovasc Eng Tech (2023). https://doi.org/10.1007/s13239-023-00705-y

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