Document Type : Review Paper
Authors
Department of Mathematics, S.V. University, Tirupati-517502, A.P, India.
Abstract
This paper aimed to model and analyze the unsteady hydromagnetic boundary layer stagnation point nanofluid flow over a non-linear stretching surface through porous medium with variable wall thickness. The effects of radiation, dissipation, and slip velocity are taken into account. The formulation of the problem is made through Buongiorno’s model, which involves the aspects of thermophoresis and Brownian motion. The set of governing non-linear Ordinary Differential Equations (ODE’s) are solved numerically by using boundary value problem default solver in MATLAB bvp4c package. The impact of different flow quantities on fluid velocity, temperature, and nanoparticle concentration are analyzed and examined through graphs. The physical parameters like friction factor coefficient , rates of heat transfer , and nanoparticle friction are derived and presented through tables. It is found that the wall thickness parameter depreciates the nanofluid velocity for and accelerates when . Also, the unsteadiness parameter shows a significant effect on the stagnation point flow.
Keywords
Main Subjects
[1] Hiemenz, K. (1911). Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden kreiszylinder. Gottingen dissertation. Dingler's polytech. J, 326, 311.
[2] Crane, L. J. (1970). Flow past a stretching plate. Zeitschrift für angewandte mathematik und physik zamp, 21(4), 645-647.
[3] Wang, C. Y. (1984). The three‐dimensional flow due to a stretching flat surface. The physics of fluids, 27(8), 1915-1917.
[4] Suali, M., Nik Long, N. M. A., & Ariffin, N. M. (2012). Unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet with suction or injection. Journal of applied mathematics, 2012.
[5] Zhong, Y., & Fang, T. (2011). Unsteady stagnation-point flow over a plate moving along the direction of flow impingement. International journal of heat and mass transfer, 54(15-16), 3103-3108.
[6] Ishak, A., Jafar, K., Nazar, R., & Pop, I. (2009). MHD stagnation point flow towards a stretching sheet. Physica A: Statistical mechanics and its applications, 388(17), 3377-3383.
[7] Hayat, T., Qayyum, S., Alsaedi, A., & Waqas, M. (2016). Simultaneous influences of mixed convection and nonlinear thermal radiation in stagnation point flow of Oldroyd-B fluid towards an unsteady convectively heated stretched surface. Journal of molecular liquids, 224, 811-817.
[8] Hayat, T., Khan, M. I., Tamoor, M., Waqas, M., & Alsaedi, A. (2017). Numerical simulation of heat transfer in MHD stagnation point flow of Cross fluid model towards a stretched surface. Results in physics, 7, 1824-1827.
[9] Choi, S. U., & Eastman, J. A. (1995). Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the ASME international mechanical engineering congress and Exposition, 66. Argonne National Lab., IL (United States).
[10] Khanafer, K., Vafai, K., & Lightstone, M. (2003). Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. International journal of heat and mass transfer, 46(19), 3639-3653.
[11] Buongiorno, J. (2006). Convective transport in nanofluids. Journal of heat transfer, 128(3), 240-250.
[12] Khalili, S., Dinarvand, S., Hosseini, R., Tamim, H., & Pop, I. (2014). Unsteady MHD flow and heat transfer near stagnation point over a stretching/shrinking sheet in porous medium filled with a nanofluid. Chinese physics B, 23(4), 048203.
[13] Hayat, T., Khan, M. I., Waqas, M., Alsaedi, A., & Farooq, M. (2017). Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid. Computer methods in applied mechanics and engineering, 315, 1011-1024.
[14] Hady, F. M., Eid, M. R., & Ahmed, M. A. (2014). Slip effects on unsteady MHD stagnation point flow of a nanofluid over stretching sheet in a porous medium with thermal radiation. Journal of pure and applied mathematics: Advances and applications, 12(2), 181-206.
[15] Salem, A. M., Ismail, G., & Fathy, R. (2015). Unsteady MHD boundary layer stagnation point flow with heat and mass transfer in nanofluid in the presence of mass fluid suction and thermal radiation. The European physical journal plus, 130(6), 113.
[16] Das, K., Duari, P. R., & Kundu, P. K. (2014). Nanofluid flow over an unsteady stretching surface in presence of thermal radiation. Alexandria engineering journal, 53(3), 737-745.
[17] Haq, R. U., Nadeem, S., Khan, Z. H., & Akbar, N. S. (2015). Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet. Physica E: Low-dimensional systems and nanostructures, 65, 17-23.
[18] Akbar, N. S., Nadeem, S., Haq, R. U., & Khan, Z. H. (2013). Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition. Chinese journal of aeronautics, 26(6), 1389-1397.
[19] Nagendramma, V., Kumar, R. K., Prasad, P. D., Leelaratnam, A., & Varma, S. V. K. (2016). Multiple slips and radiation effects on Maxwell nanofluid flow over a permeable stretching surface with dissipation. Journal of nanofluids, 5(6), 817-825.
[20] Hayat, T., Qayyum, S., Waqas, M., & Alsaedi, A. (2016). Thermally radiative stagnation point flow of Maxwell nanofluid due to unsteady convectively heated stretched surface. Journal of molecular liquids, 224, 801-810.
[21] Farooq, M., Khan, M. I., Waqas, M., Hayat, T., Alsaedi, A., & Khan, M. I. (2016). MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects. Journal of molecular liquids, 221, 1097-1103.
[22] Kumar, R. K., & Varma, S. V. K. (2017). Multiple Slips and Thermal Radiation Effects on MHD Boundary Layer Flow of a Nanofluid Through Porous Medium Over a Nonlinear Permeable Sheet with Heat Source and Chemical Reaction. Journal of nanofluids, 6(1), 48-58.
[23] Das, K. (2015). Nanofluid flow over a non-linear permeable stretching sheet with partial slip. Journal of the Egyptian mathematical society, 23(2), 451-456.
[24] Lee, L. L. (1967). Boundary layer over a thin needle. The physics of fluids, 10(4), 820-822.
[25] Fang, T., Zhang, J., & Zhong, Y. (2012). Boundary layer flow over a stretching sheet with variable thickness. Applied mathematics and computation, 218(13), 7241-7252.
[26] Acharya, N., Das, K., & Kundu, P. K. (2016). Ramification of variable thickness on MHD TiO2 and Ag nanofluid flow over a slendering stretching sheet using NDM. The European physical journal plus, 131(9), 303.
[27] Prasad, K. V., Vajravelu, K., Vaidya, H., & Van Gorder, R. A. (2017). MHD flow and heat transfer in a nanofluid over a slender elastic sheet with variable thickness. Results in physics, 7, 1462-1474.
[28] KiranKumar, R. V. M. S. S., & Varma, S. V. K. (2017). Hydromagnetic Boundary Layer Slip Flow of Nanofluid Through Porous Medium Over a Slendering Stretching Sheet. Journal of nanofluids, 6, 1-10.
[29] Hayat, T., Waqas, M., Alsaedi, A., Bashir, G., & Alzahrani, F. (2017). Magnetohydrodynamic (MHD) stretched flow of tangent hyperbolic nanoliquid with variable thickness. Journal of molecular liquids, 229, 178-184.
[30] Hayat, T., Zubair, M., Waqas, M., Alsaedi, A., & Ayub, M. (2017). Application of non-Fourier heat flux theory in thermally stratified flow of second grade liquid with variable properties. Chinese journal of physics, 55(2), 230-241.
[31] Khader, M. M., & Megahed, A. M. (2013). Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity. The European physical journal plus, 128(9), 100.
[32] Devi, S. A., & Prakash, M. (2016). Thermal radiation effects on hydromagnetic flow over a slendering stretching sheet. Journal of the brazilian society of mechanical sciences and engineering, 38(2), 423-431.
[33] Ascher, U. M., Mattheij, R. M., & Russell, R. D. (1994). Numerical solution of boundary value problems for ordinary differential equations (Vol. 13). Siam.
[34] Ibrahim, W., Shankar, B., & Nandeppanavar, M. M. (2013). MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet. International journal of heat and mass transfer, 56(1-2), 1-9.
[35] Mahapatra, T. R., & Gupta, A. S. (2002). Heat transfer in stagnation-point flow towards a stretching sheet. Heat and mass transfer, 38(6), 517-521.
[36] Bhattacharyya, K. (2013). MHD stagnation-point flow of Casson fluid and heat transfer over a stretching sheet with thermal radiation. Journal of thermodynamics. http://dx.doi.org/10.1155/2013/169674
)