# Effect of Thermal Radiation on Biomagnetic Fluid Flow and Heat Transfer over an Unsteady Stretching Sheet

### Abstract

This study examines the influence of thermal radiation on biomagnetic fluid, namely blood that passes through a two-dimensional stretching sheet in the presence of magnetic dipole. This analysis is conducted to observe the behavior of blood flow for an unsteady case, which will help in developing new solutions to treat diseases and disorders related to human body. Our model is namely biomagnetic fluid dynamics (BFD), which is consistent with two principles: ferrohydrodynamic (FHD) and magnetohydrodynamic (MHD), where blood is treated as electrically conductive. It is assumed that the implemented magnetic field is sufficiently strong to saturate the ferrofluid, and the variation of magnetization with temperature may be approximated with the aid of a function of temperature distinction. The governing partial differential equations (PDEs) converted into ordinary differential equations (ODEs) using similarity transformation and numerical results are thus obtained by using the bvp4c function technique in MATLAB software with considering applicable boundary conditions. With the help of graphs, we discuss the impact of various parameters, namely radiation parameter, unsteady parameter, permeability parameter, suction parameter, magnetic field parameter, ferromagnetic parameter, Prandtl number, velocity and thermal slip parameter on fluid (blood) flow and heat transfer in the boundary layer. The rate of heat transfer and skin friction coefficient is also computationally obtained for the requirement of this study. The fluid velocity decreases with increasing values of the magnetic parameter, ferromagnetic interaction parameter, radiation parameter whereas temperature profile increases for the unsteady parameter, Prandtl number, and permeability parameter. From the analysis, it is also observed that the skin friction coefficient decreases and the rate of heat transfer increases respectively with increasing values of the ferromagnetic interaction parameter. The most important part of the present analysis is that we neither neglect the magnetization nor electrical conductivity of the blood throughout this study. To make the results more feasible, they are compared with the data previously published in the literature and found to be in good accuracy.

### Keywords

biomagnetic fluid, magnetohydrodynamic, ferrohydrodynamic, magnetic dipole, thermal radiation, stretching sheet,### References

1. Y. Haik, J.C. Chen, V.M. Pai, Development of biomagnetic fluid dynamics, [in:] Proceedings of the IX International Symposium on Transport Properties in Thermal Fluid Engineering, Singapore, Pacific Center of Thermal Fluid Engineering, S.H. Winoto, Y.T. Chew, N.E. Wijeysundera [Eds], Hawaii, USA, June 25–28, pp. 121–126, 1996.2. P.A. Voltairas, D.I. Fotiadis, L.K. Michalis, Hydrodynamics of magnetic drug targeting, Journal of Biomechanics, 35(6): 813–821, 2002, doi: 10.1016/S0021-9290(02)00034-9.

3. E.K. Ruuge, A.N. Rusetski, Magnetic fluid as drug carriers: Targeted transport of drugs by a magnetic field, Journal of Magnetism and Magnetic Materials, 122(1–3): 335–339, 1993, doi: 10.1016/0304-8853(93)91104-F.

4. W.-L. Lin, J.-Y.Yen, Y.-Y. Chen, K.-W. Jin, M.-J. Shieh, Relationship between acoustic aperture size and tumor conditions for external ultrasound hyperthermia, Medical Physics, 26(5): 818–824, 1999, doi: 10.1118/1.598590.

5. J.C. Misra, A. Sinha, G.C. Shit, Flow of a biomagnetic viscoelastic fluid: Application to estimation of blood flow in arteries during electromagnetic hyperthermia, a therapeutic procedure for cancer treatment, Applied Mathematics and Mechanics, 31(11): 1405–1420, 2010, doi: 10.1007/s10483-010-1371-6.

6. E.E. Tzirtzilakis, A mathematical model for blood flow in magnetic field, Physics of Fluids, 17(7): 077103-1-14, 2005, doi: 10.1063/1.1978807.

7. N.G. Kafoussias, E.E. Tzirtzilakis, Biomagnetic fluid flow over a stretching sheet with nonlinear temperature dependent magnetization, The Journal of Applied Mathematics and Physics (ZAMP), 54: 551–565, 2003, doi: 10.1007/s00033-003-1100-5.

8. E.E. Tzirtzilakis, N.G. Kafoussias, Three-dimensional magnetic fluid boundary layer flow over a linearly stretching sheet, Journal of Heat Transfer, 132(1): 011702-1-8, 2010, doi: 10.1115/1.3194765.

9. M.G. Murtaza, E.E. Tzirtzilakis, M. Ferdows, Effect of electrical conductivity and magnetization on the biomagnetic fluid flow over a stretching sheet, The Journal of Applied Mathematics and Physics (ZAMP), 68: 93, 2017, doi: 10.1007/s00033-017-0839-z.

10. M. Ferdows, G. Murtaza, E.E. Tzirtzilakis, J.C. Misra, A. Alsenafi, Dual solutions in biomagnetic fluid flow and heat transfer over a nonlinear stretching/shrinking sheet: Application of Lie group transformation method, Mathematical Biosciences and Engineering, 17(5): 4852–4874, 2020, doi: 10.3934/mbe.2020264.

11. J.C. Misra, G.C. Shit, Biomagnetic viscoelastic fluid flow over a stretching sheet, Applied Mathematics and Computation, 210(2): 350–361, 2009, doi: 10.1016/j.amc.2008.12.088.

12. J.C. Misra, A. Sinha, Effect of thermal radiation on MHD flow of blood and heat transfer in a permeable capillary in stretching motion, Heat and Mass Transfer, 49: 617–628, 2013, doi: 10.1007/s00231-012-1107-6.

13. N.G. Kafoussias, E.E. Tzirtzilakis, A. Raptis, Free forced convective boundary layer flow of a biomagnetic fluid under the action of a localized magnetic field, Canadian Journal of Physics, 86(3): 447–457, 2008, doi: 10.1139/p07-166.

14. M.G. Murtaza, E.E. Tzirtzilakis, M. Ferdows, Similarity solutions of nonlinear stretched biomagnetic fluid flow and heat transfer with signum function and temperature power law geometries, International Journal of Mathematical and Computational Sciences, 12(2): 24–29, 2018, doi: 10.5281/zenodo-1315703.

15. M. Murtaza, E.E. Tzirtzilakis, M. Ferdows, Stability and convergence analysis of a biomagnetic fluid flow over a stretching sheet in the presence of a magnetic dipole, Symmetry, 12(2): 253, 2020, doi: 10.3390/sym12020253.

16. M.G. Murtaza, E.E. Tzirtzilakis, M. Ferdows, Numerical solution of three dimensional unsteady biomagnetic flow and heat transfer through stretching/shrinking sheet using temperature dependent magnetization, Archives of Mechanics, 70(2): 161–185, 2018.

17. M. Ali, F. Ahmed, S. Hussain, Analytical solution of unsteady MHD blood flow and heat transfer through parallel plates when lower plate stretches exponentially, Journal of Applied Environmental and Biological Sciences, 5(3): 1–8, 2015.

18. A. Zeeshan, A. Majed, R. Ellahi, Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation, Journal of Molecular Liquids, 215: 549–554, 2016.

19. I.H. Isaac Chen, Subrata Sana, Analysis of an intensive magnetic field on blood flow: Part 2, Journal of Bioelectricity, 4(1): 55–62, 2009, doi: 10.3109/15368378509040360.

20. S. Srinivas, P.B.A. Reddy, B.S.R.V. Prasad, Effects of chemical reaction and thermal radiation on MHD flow over an inclined permeable stretching surface with non-uniform heat source/sink: An application to the dynamics of blood flow, Journal of Mechanics in Medicine and Biology, 14(5): 1450067, 2014, doi: 10.1142/S0219519414500675.

21. A. Sinha, J.C. Misra, G.C. Shit, Effect of heat transfer on unsteady MHD flow of blood in a permeable vessel in the presence of non-uniform heat source, Alexandria Engineering Journal, 55(3): 2023–2033, 2016, doi: 10.1016/j.aej.2016.07.010.

22. C. Israel-Cookey, A. Ogulu, V.B. Omubo-Pepple, Influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction, International Journal of Heat and Mass Transfer, 46(13): 2305–2311, 2003, doi: 10.1016/S0017-9310(02)00544-6.

23. E.M.A. Elbashbeshy, D.M.Yassmin, A.A. Dalia, Heat transfer over an unsteady porous stretching surface embedded in a porous medium with variable heat flux in the presence of heat source or sink, African Journal of Mathematics and Computer Science Research, 3(5): 68–73, 2010, doi:10.5897/AJMCSR.9000071.

24. P.B.A. Reddy, N.B. Reddy, Radiation effects on MHD combined convection and mass transfer flow past a vertical porous plate embedded in a porous medium with heat generation, International Journal of Applied Mathematics and Mechanics, 6(18): 33–49, 2010.

25. S. Nadeem, S. Zaheer, T. Fang, Effects of thermal radiation on the boundary layer flow of a Jeffery fluid over an exponentially stretching surface, Numerical Algorithms, 57(2): 187–205, 2011, doi: 10.1007/s11075-010-9423-8.

26. P.B.A. Reddy, N.B. Reddy, S. Suneetha, Radiation effects on MHD flow past an exponentially accelerated isothermal vertical plate with uniform mass diffusion in the presence of heat source, Journal of Applied Fluid Mechanics, 5(3): 119–126, 2012, doi: 10.36884/jatm.5.03.19454.

27. A.A. Khan, S. Muhammad, R. Ellahi, Q.M.Z. Zia, Bionic study of variable viscosity on MHD peristaltic flow of pseudoplastic fluid in an asymmetric channel, Journal of Magnetics, 21(2): 273–280, 2016, doi: 10.4283/JMAG.2016.21.2.273.

28. P. Sreenivasulu, T. Poornima, N. Bhaskar Reddy, Thermal radiation effects on MHD boundary layer slip flow past a permeable exponential stretching sheet in the presence of joule heating and viscous dissipation, Journal of Applied Fluid Mechanics, 29(1): 267–278, 2016.

29. F.T. Akyildiz, D.A. Siginer, K. Vajravelu, J.R. Cannon, R.A. Van Gorder, Similarity solutions of the boundary layer equations for a nonlinearly stretching sheet, Mathematical Methods in the Applied Sciences, 33(5): 601–606, 2010, doi: 10.1002/mma.1181.

30. K. Bhattacharyya, G.C. Layek, R.S.R. Gorla, Boundary layer slip flow and heat transfer past a stretching sheet with temperature dependent viscosity, Thermal Energy and Power Engineering, 2(1): 38–43, 2013.

31. A. Majeed, A. Zeeshan, F.M. Noori, U. Masud, Influence of rotating magnetic field on Maxwell saturated ferrofluid flow over a heated stretching sheet with heat generation/absorption, Mechanics and Industry, 20(5): 502, 9 pp., 2019, doi: 10.1051/meca/2019022.

32. P. Ram, V. Kumar, Heat transfer in FHD boundary layer flow with temperature dependent viscosity over a rotating disk, Fluid Dynamics and Materials Processing, 10(2): 179–196, 2014.

33. G. Bognár, K. Hriczó, Ferrofluid flow in magnetic field above stretching sheet with suction and injection, Mathematical Modeling and Analysis, 25(3): 461–472, 2020, doi: 10.3846/mma.2020.10837.

34. A. Gizachew, B. Shanker, MHD flow of non-Newtonian viscoelastic fluid on stretching sheet with the effects of slip velocity, International Journal of Engineer and Manufacturing Science, 8(1): 1–14, 2018.

35. A. Majeed, A. Zeeshan, R.S.R. Gorla, Convective heat transfer in a dusty ferromagnetic fluid over a stretching surface with prescribed surface temperature/heat flux including heat source/sink, Journal of the National Science Foundation of Sri Lanka, 46(3): 399–409, 2018, doi: 10.4038/jnsfsr.v46i3.8492.

36. L.S.R. Titus, A. Abraham, Heat transfer in ferrofluid flow over a stretching sheet with radiation, International Journal of Engineering Research and Technology, 3(6): 2198–2203, 2014.

37. L. Wahidunnsia, K. Subbarayudu, S. Suneetha, A novel technique for unsteady Newtonian fluid flow over a permeable plate with viscous dissipation, non-uniform heat source/sink and chemical reaction, International Journal of Research in Engineering Application and Management, 4(10): 58–67, 2019.

38. S.R.R. Reddy, P.B.A. Reddy, S. Suneetha, Magnetohydrodynamic flow of blood in a permeable inclined stretching surface with viscous dissipation, non-uniform heat source/sink and chemical reaction, Frontiers in Heat and Mass Transfer, 10(22), 10 pp., 2018, doi: 10.5098/hmt.10.22.

39. M.Q. Brewster, Thermal Radiative Transfer and Properties, John Wiley & Sons, New York, 1992.

40. E.E. Tzirtzilakis, A simple numerical methodology for BFD problems using stream function vortices formulation, Communications in Numerical Methods in Engineering, 24(8): 683–700, 2008, doi: 10.1002/cnm.981.

41. H.I. Anderson, O.A. Valens, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mechanica, 128: 39–47, 1998, doi: 10.1007/BF01463158.

42. E.E. Tzirtzilakis, M.A. Xenos, Biomagnetic fluid flow in a driven cavity, Meccanica, 48(1): 187–200, 2013, doi: 10.1007/s11012-012-9593-7.

43. V.C. Loukopoulos, E.E. Tzirtzilakis, Biomagnetic channel flow in a spatially varying magnetic field, International Journal of Engineering Sciences, 42: 571–590, 2014, doi: 10.1016/j.ijengsci.2003.07.007.

44. E. Magyari, B. Keller, Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface, Journal of Physics D: Applied Physics, 32(5): 577–585, 1999, doi: 10.1088/0022-3727/32/5/012.

45. M.A. El-Aziz, Viscous dissipation effect on mixed convection flow of a micropolar fluid over an exponentially stretching sheet, Canadian Journal of Physics, 87(4): 359–368, 2009, doi: 10.1139/P09-047.

46. B. Bidin, R. Nazar, Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation, European Journal Scientific Research, 33(4): 710–717, 2009.

47. A. Ishak, MHD boundary layer flow due to an exponentially stretching sheet with radiation effect, Sains Malaysiana, 40(4): 391–395, 2011.

**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 28, n. 2, p. 81–104, july 2021. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/327>. Date accessed: 30 nov. 2021. doi: http://dx.doi.org/10.24423/cames.327.