Sains Malaysiana 49(5)(2020): 1175-1190

http://dx.doi.org/10.17576/jsm-2020-4905-23

Hall Current and Joule Heating Effects on Peristaltic Flow of a Sisko Fluid with Mild Stenosis through a Porous Medium in a Tapered Artery with Slip and Convective Boundary Conditions

(Arus Hall dan Kesan Pemanasan Joule pada Aliran Peristalsis Bendalir Sisko dengan Stenosis Lembap melalui Medium Berliang dalam Arteri Tirus dengan Syarat Gelinciran dan Syarat Sempadan Olakan)

 NABIL T.M. EL-DABE & DOAA R. MOSTAPHA*

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Diserahkan: 11 Julai 2019/Diterima: 15 Januari 2020

ABSTRACT

This work is arranged to investigate the Hall current and Joule heating effects on peristaltic flow of a Sisko fluid through a porous medium. The streaming is through tapered artery with mild stenosis. The influences of radiative heat transfer and chemical reactions are taken in consideration. Convective conditions are considered for heat and mass transfer. However, the slip conditions are adopted for the velocity distribution. The combined effects of viscous dissipation and radiation in energy expression are presented. Soret and Dufour features produce the coupled differential systems. The presumptions of the long wavelength and low Reynolds number are adopted to examine the governing equations of motion. The analytical solutions of these equations are given by two methods. The first one uses regular perturbation technique, which based on small wave number for small artery. The second one is based on utilizing the Homotopy perturbation technique. The approximate analytical solutions of the pressure rise and friction force are predestined along a numerical integration. The influences of various physical parameters of the problem are debated and depicted graphically through a set of figures. It is found that the axial velocity increases with the increase of Hall current parameter and with the decrease of Hartmann number. Also, it can be observed that the Brickmann number and Dufour number give rise to the fluid temperature. Meanwhile, reverse effect is observed towards concentration for both Schmidt number and chemical reaction number. Furthermore, the stream lines are graphically shown.

Keywords: Hall current; Joule heating; peristaltic flow; porous medium; Sisko model

ABSTRAK

Kajian ini mengkaji arus Hall dan kesan pemanasan Joule terhadap aliran peristalsis bendalir Sisko melalui medium berliang. Aliran adalah melalui arteri tirus dengan stenosis lembut. Pengaruh pemindahan haba sinaran dan tindak balas kimia dipertimbangkan. Syarat perolakan dipertimbangkan bagi pemindahan haba dan jisim. Walau bagaimanapun, syarat gelinciran digunakan dalam taburan halaju. Kesan gabungan lesapan likat dan radiasi dalam ungkapan tenaga dipersembahkan. Ciri-ciri Soret dan Dufour menghasilkan sistem persamaan pembezaan terganding. Andaian panjang gelombang dan nombor Reynolds yang rendah digunakan untuk mengkaji persamaan yang mengawal gerakan. Penyelesaian beranalisis persamaan ini diberikan oleh dua kaedah. Yang pertama menggunakan teknik usikan biasa, yang berdasarkan nombor gelombang kecil untuk arteri kecil. Yang kedua adalah berdasarkan teknik usikan homotopi. Penyelesaian analisis anggaran kenaikan tekanan dan daya geseran ditentukan menggunakan kamiran berangka. Pengaruh pelbagai parameter fizikal bagi masalah ini dibahas dan digambarkan secara grafik. Didapati bahawa halaju bertambah dengan peningkatan parameter arus Hall dan dengan penurunan nombor Hartmann. Juga, boleh diperhatikan bahawa nombor Brickmann dan nombor Dufour meningkatkan suhu bendalir. Sementara itu, kesan bertentangan diperhatikan terhadap kepekatan untuk kedua-dua nombor Schmidt dan nombor tindak balas kimia. Juga, garis arus ditunjukkan secara grafik.

Kata kunci: Aliran peristalsis; arus Hall; pemanasan Joule; medium berliang; model Sisko

RUJUKAN

Abdelsalam, S.I. & Bhatti, M.M. 2019. New insight into AuNP applications in tumour treatment and cosmetics through wavy annuli at the nanoscale. Scientific Reports 9(260): 1-14.

Abdelsalam, S.I. & Bhatti, M.M. 2018. The study of non-Newtonian nanofluid with hall and ion slip effects on peristaltically induced motion in a non-uniform channel. RSC Advances 8: 7904-7915.

Ang, K.C. & Mazumdar, J.N. 1997. Mathematical modeling of three dimensional flow through an asymmetric arterial stenosis. Mathematical and Computer Modelling 25(1): 19-29.

Arora, C.P. 1997. Heat and Mass Transfer. 2nd edition. Delhi: Khanna Publishers.

Bhatti, M.M., Zeeshan, A. & Ellah, R. 2016. Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism. Microvascular Research 110: 32-42.

Burns, J.C. & Parkes, T. 1967. Peristaltic motion. Journal of Fluid Mechanics 29(4): 731-743.

Chakravarty, S. & Sen, S. 2005. Dynamic response of heat and mass transfer in blood flow through stenosed bifurcated arteries. Korea-Australia Rheology Journal 17(2): 47-62.

Chakravarty, S., Datta, A. & Mandal, P.K. 1995. Analysis of nonlinear blood flow in a stenosed flexible artery. International Journal of Engineering Science 33(12): 1821-1837. 

Di Federico, V., Longo, S., King, S.E., Chiapponi, L., Petrolo, D. & Ciriello, V. 2017. Gravity driven flow of Herschel-Bulkley fluid in a fracture and in a 2D porous medium. Journal of Fluid Mechanics 821: 59-84.

El-dabe, N.T. & Abou-zeid, M. 2014. MHD peristaltic flow with heat and mass transfer of micropolar biviscosity fluid through a porous medium between two co-axial tubes. Arabian Journal for Science and Engineering 39(6): 5045-5062.

El-dabe, N.T., Moatimid, G.M., Hassan, M.A. & Mostapha, D.R. 2016. Effect of partial slip on peristaltic flow of a Sisko fluid with mild stenosis through a porous medium. Applied Mathematics & Information Sciences 10(2): 1-15.

El-dabe, N.T., Hassan, M.A. & Abou-zeid, M. 2015. Wall properties effect on the peristaltic motion of a coupled stress fluid with heat and mass transfer through a porous medium. Journal of Engineering Mechanics 142(3): 04015102.

El-dabe, N.T., Kamel, K.A., Galila Abd-Allah, M. & Ramadan, S.F. 2013. Heat absorption and chemical reaction effects on peristaltic motion of micropolar fluid through a porous medium in the presence of magnetic field. African Journal of Mathematics and Computer Science Research 6(5): 94-101.

Eldabe, N.T., El-Sayed, M.F., Ghaly, A.Y. & Sayed, H.M. 2008. Mixed convective heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity. Archive of Applied Mechanics 78: 599-624.

Eldabe, N.T., Ghaly, A.Y. & Sayed, H.M. 2007. MHD Peristaltic flow of non-Newtonian fluid through a porous medium in circular cylindrical tube. Bulletin of the Calcutta Mathematical Society 99: 123-136.

Fung, Y.C. & Yin, F. 1969. Peristaltic waves in circular cylindrical tubes. Journal of Applied Mechanics 36(3): 579-587.

Hayat, T., Javed, S., Khan, M., Imran Khan, M. & Alsaedi, A. 2019. Physical aspects of irreversibility in radiative flow of viscous material with cubic autocatalysis chemical reaction. European Physical Journal Plus 134(4): 1-24.

Hayat, T., Aslam, N., Khan, M., Imran Khan, M. & Alsaedi, A. 2018a. Physical significance of heat generation/absorption and Soret effects on peristalsis flow of pseudoplastic fluid in an inclined channel. Journal of Molecular Liquids 275: 599-615.

Hayat, T., Qayyum, S., Khan, M. & Alsaedi, A. 2018b. Entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and Joule heating. Physics of Fluids 30(1): 017101.

Hayat, T., Zahir, H., Alsaedi, A. & Ahmad, B. 2017. Hall current and Joule heating effects on peristaltic flow of viscous fluid in a rotating channel with convective boundary conditions. Results in Physics 7: 2831-2836.

Hayat, T., Nawaz, S., Alsaedi, A. & Rafiq, M. 2016a. Mixed convective peristaltic flow of water based nanofluids with Joule heating and convective boundary conditions. PLoS ONE 11(4): 1-28.

Hayat, T., Rafiq, M. & Ahmad, B. 2016b. Soret and Dufour effects on MHD peristaltic flow of Jeffrey fluid in a rotating system with porous medium. PLoS ONE 11(1): 1-18.

Hayat, T., Iqbal, R., Tanveer, A. & Alsaedi, A. 2016. Influence of convective conditions in radiative peristaltic flow of pseudoplastic nanofluid in a tapered asymmetric channel. Journal of Magnetism and Magnetic Materials 408: 168-176.

Hayat, T., Abbasi, F.M., Ahmad, B. & Alsaedi, A. 2014. MHD mixed convection peristaltic flow with variable viscosity and thermal conductivity. Sains Malaysiana 43(10): 1583-1590.

He, J.H. 1999. Homotopy perturbation technique. Computer Methods in Applied Mechanics & Engineering 178(3-4): 257-262.

Ijaz, N., Zeeshan, A., Bhatti, M.M. & Ellahi, R. 2008. Analytical study on liquid-solid particles interaction in the presence of heat and mass transfer through a wavy channel. Journal of Molecular Liquids 250: 80-87.

Jaffrin, M.Y. 1973. Inertia and streamline curvature effects on peristalsis pumping. International Journal of Engineering Science 11(6): 681-699.

Jamil, D.F., Roslan, R., Abdulhameed, M. & Hashim, I. 2018. Controlling the blood flow in the stenosed porous artery with magnetic field. Sains Malaysiana 47(10): 2581-2587.

Khan, M., Abbas, Q. & Duru, K. 2010. Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study. International Journal for Numerical Method in Fluids 62(10): 1169-1180.

Lauriola, I., Felisa, G., Petrolo, D., Di Federico, V. & Longo, S. 2018. Porous gravity currents: Axisymmetric propagation in horizontally graded medium and a review of similarity solutions. Advances in Water Resources 115: 136-150.

Longo, S. & Di Federico, V. 2015. Unsteady flow of shear-thinning fluids in porous media with pressure-dependent properties. Transport in Porous Media 110(3): 429-447.

Mandal, P.K. 2005. An unsteady of non-Newtonian blood flow through tapered arteries with a stenosis. International Journal of Non-Linear Mechanics 40(1): 151-164.

Manton, M.J. 1975. Long-wavelength peristaltic pumping at low Reynolds number. Journal of Fluid Mechanics 68(3): 467-476.

Mehmood, O.U., Mustapha, N. & Hayat, T. 2014. Partial slip effect on heat and mass transfer of MHD peristaltic transport in a porous medium. Sains Malaysiana 43(7): 1109-1118.

Mekheimer, K.S. & El Kot, M.A. 2012. Mathematical modelling of unsteady flow os a Sisko fluid through an anisotropically tapered elastic arteries with time variant overlapping stenosis. Applied Mathematical Modelling 36(11): 5393-5407.

Nadeem, S. & Akbar, N.S. 2010. Peristaltic flow of Sisko fluid in a uniform inclined tube. Acta Mechanica Sinica 26: 675-683.

Nadeem, S. & Akbar, N.S. 2009. Influence of heat transfer on a peristaltic transport of Herschel Bulkley fluid in a non-uniform inclined tube. Nonlinear Science & Numerical Simulation 14(12): 4100-4113.

Nichols, W.W. & Orourke, M.F. 1973. McDonald's Blood Flow in Arteries. New York: Oxford University Press Inc.

Pal, D. & Talukdar, B. 2011. Combined effects of Joul heating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation. Mathematical and Computer Modelling 54(11-12): 3016-3036.

Rashid, M., Khan, M., Hayat, T., Imran Khan, M. & Alsaedi, A. 2019. Entropy generation in flow of ferromagnetic liquid with nonlinear radiation and slip condition. Journal of Molecular Liquids 276: 441-452.

Rohsenow, W.M., Hartnett, J.P. & Cho, Y.I. 1998. Handbook of Heat Transfer. New York: McGraw-Hill.

Sadaf, H. & Nadeem, S. 2017. Analysis of combined convective and viscous dissipation effects for peristaltic flow of Rabinowitsch fluid model. Journal of Bionic Engineering 14(1): 182-190.

Scheidegger, A.E. 1963. The Physics of Flow through Porous Media. New York: McGraw-Hill.

Shapiro, A.H., Jaffrin, M.Y. & Weinberg, S.L. 1969. Peristaltic pumping with long wavelengths at low Reynolds number. Journal of Fluid Mechanics 37(4): 799-825.

Siddiqui, A.M., Azim, Q.A., Ashraf, A. & Ghori, Q.K. 2008. Homotopy perturbation solution for peristaltic flow of a third order fluid. Topological Methods in Nonlinear Analysis 31: 331-339.

Sisko, A.W. 1958. The flow of lubricating greases. Industrial & Engineering Chemistry 50(12): 1789-1792.

Sohail, M., Naz, R. & Abdelsalam, S.I. 2019. On the onset of entropy generation for a nanofluid with thermal radiation and gyrotactic microorganisms through 3D flows. Physica Scripta. https://doi.org/10.1088/1402-4896/ab3c3f.

Verma, N. & Parihar, R.S. 2010. Mathematical model of blood flow through a tapered artery with mild stenosis and hematocrit. Journal of Modern Mathematics and Statistics 4(1): 38- 43.

*Pengarang untuk surat-menyurat; email: doaaroshdy@edu.asu.edu.eg