Exact solution for heat transport of Newtonian fluid with quadratic order thermal slip in a porous medium

Öz

In this communication, an analytical solution for the thermal transfer of Newtonian fluid flow with quadratic order thermal and velocity slips is presented for the first time. The flow of a Newtonian fluid over a stretching sheet which is embedded in a porous medium is considered. Karniadakis and Beskok’s quadratic order slip boundary conditions are taking into account. A closed form of analytical solution of momentum equation is used to derive the analytical solution of heat transfer equation in terms of confluent hyper-geometric function with quadratic order thermal slip boundary condition. Accuracy of present results is assured with the numerical solution obtained by Iterative Power Series method with shooting technique. The impacts of porous medium parameter, tangential momentum accommodation coefficient, energy accommodation coefficient on velocity and temperature profiles, skin friction coefficient and reduced Nusselt number are discussed. The Nusselt number increases with the higher estimations of tangential momentum and energy accommodation coefficients.

Anahtar Kelimeler

• Analytical solution,Hyper-geometric functions,Newtonian fluid,Quadratic order thermal slip,Porous medium

Kaynakça

1. [1]. Sakiadis, B. C., (1961a). ‘‘Boundary-Layer Behaviour on Continuous Solid Surfaces: I. Boundary- Layer Equations for Two-dimensional and Axisymmetric Flow,’’ AIChE J., 7, 26.
2. [2]. Sakiadis, B. C., (1961b). ‘‘Boundary-Layer Behaviour on Continuous Solid Surfaces: II. The Boundary Layer on a Continuous Flat Surface,’’ AIChE J., 7, 221.
3. [3]. Crane, L. J., (1970). ‘‘Flow Past a Stretching Plate,’’ Z. Angew. Math. Phys., 21, 645.
4. [4]. Banks, W.H.H., 1983. Similarity solutions of the boundary-layer equations for a stretching wall. Journal de Mécanique théorique et appliquée, 2, pp.375-392.
5. [5]. Magyari, E. and Keller, B., 2000. Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls. European Journal of Mechanics-B/Fluids, 19(1), pp.109-122.
6. [6]. Wang, C.Y., 2002. Flow due to a stretching boundary with partial slip—an exact solution of the Navier–Stokes equations. Chemical Engineering Science, 57(17), pp.3745-3747.
7. [7]. Wang, C.Y., 2009. Analysis of viscous flow due to a stretching sheet with surface slip and suction. Nonlinear Analysis: Real World Applications, 10(1), pp.375-380.
8. [8]. Sajid, M., Ali, N., Abbas, Z. and Javed, T., 2010. Stretching flows with general slip boundary condition. International Journal of Modern Physics B, 24(30), pp.5939-5947

9. [9]. Turkyilmazoglu, M., 2011. Analytic heat and mass transfer of the mixed hydrodynamic/thermal slip MHD viscous flow over a stretching sheet. International Journal of Mechanical Sciences, 53(10), pp.886-896.
10. [10]. Das, K., 2015. Nanofluid flow over a non-linear permeable stretching sheet with partial slip. Journal of the Egyptian Mathematical Society, 23(2), pp.451-456.
11. [11]. Soomro, F.A., Haq, R.U., Al-Mdallal, Q.M. and Zhang, Q., 2018. Heat generation/absorption and nonlinear radiation effects on stagnation point flow of nanofluid along a moving surface. Results in physics, 8, pp.404-414.
12. [12]. Besthapu, P., Haq, R.U., Bandari, S. and Al-Mdallal, Q.M., 2019. Thermal radiation and slip effects on MHD stagnation point flow of non-Newtonian nanofluid over a convective stretching surface. Neural Computing and Applications, 31(1), pp.207-217
13. [13]. Karniadakis, G.E., Beskok, A. and Gad-el-Hak, M., 2002. Micro flows: fundamentals and simulation. Appl. Mech. Rev., 55(4), pp.B76-B76.
14. [14]. Xiao, N., Elsnab, J. and Ameel, T., 2009. Microtube gas flows with second-order slip flow and temperature jump boundary conditions. International Journal of Thermal Sciences, 48(2), pp.243-251.
15. [15]. Hamdan M. A., M. A. Al- Nimr, V. A. Hammoudeh, 2010. Effect of second order velocity slip/ temperature -jump on the basic gaseous fluctuating micro- flows, J. Fluids Eng., 132 (2010) 074503.
16. [16]. Fang, T., Yao, S., Zhang, J. and Aziz, A., 2010. Viscous flow over a shrinking sheet with a second order slip flow model. Communications in Nonlinear Science and Numerical Simulation, 15(7), pp.1831-1842.
17. [17]. Wu, L., 2008. A slip model for rarefied gas flows at arbitrary Knudsen number. Applied Physics Letters, 93(25), p.253103.
18. [18]. Nandeppanavar, M.M., Vajravelu, K., Abel, M.S. and Siddalingappa, M.N., 2012. Second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition. International Journal of Thermal Sciences, 58, pp.143-150.
19. [19]. Turkyilmazoglu, M., 2013. Heat and mass transfer of MHD second order slip flow. Computers & Fluids, 71, pp.426-434.
20. [20]. Hakeem, A.A., Ganesh, N.V. and Ganga, B., 2015. Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect. Journal of Magnetism and Magnetic Materials, 381, pp.243-257.
21. [21]. Rashidi, M.M., Hakeem, A.A., Ganesh, N.V., Ganga, B., Sheikholeslami, M. and Momoniat, E., 2016. Analytical and numerical studies on heat transfer of a nanofluid over a stretching/shrinking sheet with second-order slip flow model. International Journal of Mechanical and Materials Engineering, 11(1), p.1.
22. [22]. Roşca, N.C. and Pop, I., 2014. Boundary layer flow past a permeable shrinking sheet in a micropolar fluid with a second order slip flow model. European Journal of Mechanics-B/Fluids, 48, pp.115-122.
23. [23]. Sharma, R. and Ishak, A., 2014. Second order slip flow of cu-water nanofluid over a stretching sheet with heat transfer. WSEAS Trans. Fluid Mech, 9, pp.26-33.
24. [24]. Hayat, T., Imtiaz, M. and Alsaedi, A., 2015. Impact of magnetohydrodynamics in bidirectional flow of nanofluid subject to second order slips velocity and homogeneous–heterogeneous reactions. Journal of magnetism and magnetic materials, 395, pp.294-302.
25. [25]. Liu, Y. and Guo, B., 2017. Effects of second-order slip on the flow of a fractional Maxwell MHD fluid. Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, pp.232-241.
26. [26]. Mabood, F., Shafiq, A., Hayat, T. and Abelman, S., 2017. Radiation effects on stagnation point flow with melting heat transfer and second order slip. Results in Physics, 7, pp.31-42.
27. [27]. Ibrahim, W., 2017. MHD boundary layer flow and heat transfer of micropolar fluid past a stretching sheet with second order slip. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39(3), pp.791-799.
28. [28]. Aman, S., Al-Mdallal, Q. and Khan, I., 2018. Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium. Journal of King Saud University-Science.
29. [29]. Uddin, M.J., Khan, W.A. and Ismail, A.M., 2018. Melting and second order slip effect on convective flow of nanofluid past a radiating stretching/shrinking sheet. Propulsion and Power Research, 7(1), pp.60-71.
30. [30]. Aman, S. and Al-Mdallal, Q., 2019, July. Flow of ferrofluids under second order slip effect. In AIP Conference Proceedings (Vol. 2116, No. 1, p. 030012). AIP Publishing LLC.
31. [31]. Kumar, P.S., Gireesha, B.J., Mahanthesh, B. and Chamkha, A.J., 2019. Thermal analysis of nanofluid flow containing gyrotactic microorganisms in bioconvection and second-order slip with convective condition. Journal of Thermal Analysis and Calorimetry, 136(5), pp.1947-1957.
32. [32]. Waqas, H., Khan, S.U., Bhatti, M.M. and Imran, M., 2020. Significance of bioconvection in chemical reactive flow of magnetized Carreau–Yasuda nanofluid with thermal radiation and second-order slip. Journal of Thermal Analysis and Calorimetry, pp.1-14.
33. [33]. Arikoglu, A., Komurgoz, G., Gunes, A.Y. and Ozkol, I., 2014. Effect of second-order velocity slip and temperature jump conditions on rotating disk flow in the case of blowing and suction with entropy generation. Heat Transfer Research, 45(2).
34. [34]. Ganesh, N.V., Al-Mdallal, Q.M. and Chamkha, A.J., 2019. A numerical investigation of Newtonian fluid flow with buoyancy, thermal slip of order two and entropy generation. Case Studies in Thermal Engineering, 13, p.100376.
35. [35]. Al Khawaja, U. and Al-Mdallal, Q.M., 2018. Convergent power Series of and solutions to nonlinear differential equations. International Journal of Differential Equations, 2018.