Mathematical modelling of boundary layer flow over a permeable and time-dependent shrinking sheet – A stability analysis

J. G. Tan; Yian Yian Lok; I. Pop

doi:10.15282/jmes.16.2.2022.03.0699

Authors

J. G. Tan Mathematics Section, School of Distance Education, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. Phone: +6046533931; Fax: +6046576000
Y.Y. Lok Mathematics Section, School of Distance Education, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. Phone: +6046533931; Fax: +6046576000
I. Pop Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania.

DOI:

https://doi.org/10.15282/jmes.16.2.2022.03.0699

Keywords:

Numerical solutions , Micropolar fluid, Suction, Shrinking sheet, Stability analysis

Abstract

Micropolar fluid is one type of non-Newtonian fluid which consists of non-deformable spherical particles that suspended in viscous medium. In this paper, the problem of two-dimensional boundary layer flow over a permeable shrinking sheet with time dependent velocity in strong concentration micropolar fluid is studied theoretically. The mathematical model is governed by continuity, momentum and microrotation equations. Similarity variables are introduced so that, after performing the similarity transformation on the governing equations, the resulting system of nonlinear ordinary differential equations is then numerically solved using the program bvp4c in Matlab software. The effects of the micropolar material parameter, the unsteadiness parameter, the shrinking parameter and the mass suction parameter to the skin friction coefficient, velocity profiles and microrotation profiles are investigated. It is found that triple solutions exist for some values of the parameters that were considered. Based on the stability analysis that was performed, it showed that only two branches of solutions are categorized as stable, whereas one solution branch is unstable.

References

A. C. Eringen, “Theory of micropolar fluids,” J. Math. Mech., vol. 16, no. 1, pp. 1–18, 1966.

G. Łukaszewicz, Micropolar Fluids: Theory and Applications. Boston, U.S.: Birkhäuser, 1999.

A. C. Eringen, Microcontinuum Field Theories: I. Foundations and Solids. New York, U.S.: Springer-Verlag, 1999.

T. Ariman, M. A. Turk, and N. D. Sylvester, “Microcontinuum fluid mechanics—A review,” Int. J. Eng. Sci., vol. 11, no. 8, pp. 905–930, 1973.

T. Ariman, M. A. Turk, and N. D. Sylvester, “Applications of microcontinuum fluid mechanics,” Int. J. Eng. Sci., vol. 12, no. 4, pp. 273–293, 1974.

A. J. Willson, “Boundary layers in micropolar liquids,” Math. Proc. Cambridge Philos. Soc., vol. 67, no. 2, pp. 469–476, 1970.

J. Peddieson, “Boundary layer theory for a micropolar fluid,” Virginia Polytechnic Institute and State University, U.S., 1969.

R. S. R. Gorla, “Combined forced and free convection in micropolar boundary layer flow on a vertical flat plate,” Int. J. Eng. Sci., vol. 26, no. 4, pp. 385–391, 1988.

D. A. S. Rees and A. P. Bassom, “The Blasius boundary-layer flow of a micropolar fluid,” Int. J. Eng. Sci., vol. 34, no. 1, pp. 113–124, 1996.

E. G. Fisher, E. C. Whitfield, and Plastics and Rubber Institute, Extrusion of Plastics, 3rd ed. New York, U.S.: Halsted Press, 1976.

M. Miklavčič and C. Y. Wang, “Viscous flow due to a shrinking sheet,” Quart. Appl. Math., vol. 64, no. 2, pp. 283–290, 2006.

S. Goldstein, “On backward boundary layers and flow in converging passages,” J. Fluid Mech., vol. 21, no. 1, pp. 33–45, 1965.

N. A. Yacob and A. Ishak, “Micropolar fluid flow over a shrinking sheet,” Meccanica, vol. 47, no. 2, pp. 293–299, 2012.

K. Bhattacharyya, S. Mukhopadhyay, G. C. Layek, and I. Pop, “Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet,” Int. J. Heat Mass Transf., vol. 55, no. 11, pp. 2945–2952, 2012.

H. Rosali, A. Ishak, and I. Pop, “Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction,” Int. Commun. Heat Mass Transf., vol. 39, no. 6, pp. 826–829, 2012.

M. Turkyilmazoglu, “A note on micropolar fluid flow and heat transfer over a porous shrinking sheet,” Int. J. Heat Mass Transf., vol. 72, pp. 388–391, 2014.

N. C. Roşca and I. Pop, “Boundary layer flow past a permeable shrinking sheet in a micropolar fluid with a second-order slip flow model,” Eur. J. Mech. - B/Fluids, vol. 48, pp. 115–122, 2014.

T. G. Fang, J. Zhang, and S. S. Yao, “Viscous flow over an unsteady shrinking sheet with mass transfer,” Chinese Phys. Lett., vol. 26, no. 1, p. 14703, 2009.

J. H. Merkin and V. Kumaran, “The unsteady MHD boundary-layer flow on a shrinking sheet,” Eur. J. Mech. - B/Fluids, vol. 29, no. 5, pp. 357–363, 2010.

F. M. Ali, R. Nazar, N. M. Arifin, and I. Pop, “Unsteady shrinking sheet with mass transfer in a rotating fluid,” Int. J. Numer. Methods Fluids, vol. 66, no. 11, pp. 1465–1474, Aug. 2011.

Z. Abbas, S. Rasool, and M. M. Rashidi, “Heat transfer analysis due to an unsteady stretching/shrinking cylinder with partial slip condition and suction,” Ain Shams Eng. J., vol. 6, no. 3, pp. 939–945, 2015.

A. M. Rohni, S. Ahmad, and I. Pop, “Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids,” Int. J. Heat Mass Transf., vol. 55, no. 7, pp. 1888–1895, 2012.

N. Bachok, A. Ishak, and I. Pop, “Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet,” Int. J. Heat Mass Transf., vol. 55, no. 7, pp. 2102–2109, 2012.

M. Abd El-Aziz, “Mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation,” J. Egypt. Math. Soc., vol. 21, no. 3, pp. 385–394, 2013.

N. Sandeep and C. Sulochana, “Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink,” Eng. Sci. Technol. an Int. J., vol. 18, no. 4, pp. 738–745, 2015.

B. Mohanty, S. R. Mishra, and H. B. Pattanayak, “Numerical investigation on heat and mass transfer effect of micropolar fluid over a stretching sheet through porous media,” Alexandria Eng. J., vol. 54, no. 2, pp. 223–232, 2015.

N. C. Roy, M. A. Hossain, and I. Pop, “Analysis of dual solutions of unsteady micropolar hybrid nanofluid flow over a stretching/shrinking sheet", J. Appl. Comput. Mech., vol. 7, no. 1, pp. 19–33, 2021.

K. Naganthran, I. Hashim, and R. Nazar, “Triple solutions of Carreau thin film flow with thermocapillarity and injection on an unsteady stretching sheet,” Energies, vol. 13, no. 12. 2020.

E. H. Hafidzuddin, R. Nazar, N. M. Arifin, and I. Pop, “Stability analysis of unsteady three-dimensional viscous flow over a permeable stretching/shrinking surface,” J. Qual. Meas. Anal., vol. 11, no. 1, pp. 19–31, 2015.

I. S. Awaludin, P. D. Weidman, and A. Ishak, “Stability analysis of stagnation-point flow over a stretching/shrinking sheet,” AIP Adv., vol. 6, no. 4, p. 45308, 2016.

Y. Y. Lok, A. Ishak, and I. Pop, “Oblique stagnation slip flow of a micropolar fluid towards a stretching/shrinking surface: A stability analysis,” Chinese J. Phys., vol. 56, no. 6, pp. 3062–3072, 2018.

L. A. Lund, Z. Omar, U. Khan, I. Khan, D. Baleanu, and K. S. Nisar, “Stability analysis and dual solutions of micropolar nanofluid over the inclined stretching/shrinking surface with convective boundary condition,” Symmetry, vol. 12, no. 1. 2020.

N. C. Roşca and I. Pop, “Unsteady boundary layer flow over a permeable curved stretching/shrinking surface,” Eur. J. Mech. - B/Fluids, vol. 51, pp. 61–67, 2015.

P. D. Weidman, D. G. Kubitschek, and A. M. J. Davis, “The effect of transpiration on self-similar boundary layer flow over moving surfaces,” Int. J. Eng. Sci., vol. 44, no. 11, pp. 730–737, 2006.

J. H. Merkin, “On dual solutions occurring in mixed convection in a porous medium,” J. Eng. Math., vol. 20, no. 2, pp. 171–179, 1986.

S. D. Harris, D. B. Ingham, and I. Pop, “Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip,” Transp. Porous Media, vol. 77, no. 2, pp. 267–285, 2009.

L. F. Shampine, I. Gladwell, and S. Thompson Solving ODEs with MATLAB. Cambridge, U.K.: Cambridge University Press, 2003.

D. Gupta, L. Kumar, and B. Singh, “Finite element solution of unsteady mixed convection flow of micropolar fluid over a porous shrinking sheet,” Sci. World J., vol. 2014, p. 362351, 2014.

T. Fang and J. Zhang, “Closed-form exact solutions of MHD viscous flow over a shrinking sheet,” Commun. Nonlinear Sci. Numer. Simul., vol. 14, no. 7, pp. 2853–2857, 2009.