Journal of Applied and Computational Mechanics
The Method of Lines Analysis of Heat Transfer of Ostwald-de Waele Fluid Generated by a Non-uniform Rotating Disk with a Variable Thickness
Document Type : Research Paper
Author
Department of Mathematics, Benha Faculty of Engineering, Benha University, Egypt
Abstract
In this article, it is aimed to address one of Ostwald-de Waele fluid problems that either, has not been addressed or very little focused on. Considering the impacts of heat involving in the Non-Newtonian flow, a variant thickness of the disk is additionally considered which is governed by the relation z = a (r/RO+1)-m. The rotating Non-Newtonian flow dynamics are represented by the system of highly nonlinear coupled partial differential equations. To seek a formidable solution of this nonlinear phenomenon, the application of the method of lines using von Kármán’s transformation is implemented to reduce the given PDEs into a system of nonlinear coupled ordinary differential equations. A numerical solution is considered as the ultimate option, for such nonlinear flow problems, both closed-form solution and an analytical solution are hard to come by. The method of lines scheme is preferred to obtain the desired solution which is found to be more reliable and in accordance with the required physical expectation. Eventually, some new marvels are found. Results indicate that, unlike the flat rotating disk, the local radial skin friction coefficients and tangential decrease with the fluid physical power-law exponent increases, the peak in the radial velocity rises which is significantly distinct from the results of a power-law fluid over a flat rotating disk. The local radial skin friction coefficient increases as the disk thickness index increases, while local tangential skin friction coefficient decreases, the local Nusselt number decrease, both the thickness of the velocity and temperature boundary layer increase.
Keywords
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8. Hayat, T., Rashid, M., Imtiaz, M., Alsaedi, A., Magnetohydrodynamic (MHD) flow of Cu-water nanofluid due to a rotating disk with partial slip, AIP Adv., 5, 2015, 067169.
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14. Mahanthesh, B., Gireesha, B.J., Shashikumar, N.S., Shehzad, S.A., Marangoni convective MHD flow of SWCNT and MWCNT nanoliquids due to a disk with solar radiation and irregular heat source, Physica E: Low-dimensional Systems and Nanostructures, 94, 2017, 25-30.
15. Mustafa, I., Javed, T., Ghaffari, A., Heat transfer in MHD stagnation point flow of a ferrofluid over a stretchable rotating disk, J. Mole. Liquids, 219, 2016, 526-532.
16. Rashidi, M.M., Hayat, T., Erfani, E., Hendi, A.A., Simultaneous effects of partial and thermal-diffusion and diffusion-Thermo on steady MHD convective flow due to a rotating disk, Commun. Nonlinear Sci., 16, 2011, 4303-4317.
17. Asgher, S., Jalil, M., Hussan, M., Turkyimazoglu, M., Lie group analysis of flow and heat transfer over a stretching rotating disk, Int. J. Heat Mass Transfer, 69, 2014, 140-146.
18. Hayat, T., Rashid, M., Imtiaz, M. A. Alsaedi, Nanofluid flow due to rotating disk with variable thickness and homogeneous-heterogeneous reactions, Int. J. Heat Mass Transfer, 113, 2017, 96-105.
19. Mitschka, P., Ulbrecht, J., Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids, Appl. Sci. Res., 15, 1966, 345–358.
20. Andersson, H.I., de Korte, E., Meland, R., Flow of a power-law fluid over a rotating disk revisited, Fluid Dyn. Res., 28, 2001, 75–88.
21. Attia, H.A., Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection, Commun. Nonlinear Sci., 13, 2008, 1571-1580.
22. Griffiths, P.T., Flow of a generalized Newtonian fluid due to a rotating disk, J. Non-Newtonian Fluid Mech., 221, 2015, 9–17.
23. Ming, C.Y., Zheng, L.C., Zhang, X.X., Steady flow and heat transfer of the power-law fluid over a rotating disk, Int. Commun. Heat Mass, 38, 2011, 280–284.
24. Fang, T.G., Zhang, J., Zhong, Y.F., Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput., 218, 2012, 7241–7252.
25. Rashidi, M.M., Reddy, S., Naikoti, K., MHD flow and heat transfer characteristics of Williamson nanofluid over a stretchable sheet with variable thickness and variable thermal conductivity, A. R. Mathematical Inst., 171, 2017, 195-211.
26. Imtiaz, M., Hayat, T., Asgher, S., Alsaedi, A., Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics, Ruselts in Physics, 7, 2017, 503-509.
27. Waqas, H., Ullah Khan, S., Imran, M., Bhatti, M.M., Thermally developed Falkner–Skan bioconvection flow of a magnetized nanofluid in the presence of a motile gyrotactic microorganism: Buongiorno’s nanofluid model, Physica Scripta, 94(11), 2019, 115304.
28. Tlili, I., Bhatti, M.M., Mustafa Hamad, S., Barzinjy, A.A., Sheikholeslami, M., Shafee, A., Macroscopic modeling for convection of Hybrid nanofluid with magnetic effects, Physica A: Statistical Mechanics and its Applications, 534, 2019, 122136.
29. Abdelsalam, S.I., Bhatt, M.M., Zeeshan, A., Riaz, A., Anwar Bég, O., Metachronal propulsion of a magnetized particle-fluid suspension in a ciliated channel with heat and mass transfer, Physica Scripta, 94(11), 2019, 115301.
30. Riaz, A., Ellahi, R., Mubashir Bhatti, M., Marin, M., Study of heat and mass transfer in the Eyring–Powell model of fluid propagating peristaltically through a rectangular compliant channel, Heat Transfer Research, 50(16), 2019, in press.
31. Marin, M., Vlase, S., Ellahi, R., Bhatti, M.M., On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure, Symmetry, 11(7), 2019, 863.
32. Waqas, H., Ullah Khan, S., Hassan, M., Bhatti, M.M., Imran, M., Analysis on the bioconvection flow of modified second-grade nanofluid containing gyrotactic microorganisms and nanoparticles, Journal of Molecular Liquids, 291, 2019, 111231.
33. Ali, M.R., Baleanu, D., Haar wavelets scheme for solving the unsteady gas flow in four-dimensional, Thermal Science, 2019, https://doi.org/10.2298/TSCI190101292A.
34. Ali, M.R., Hadhood, A.R., Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet scheme for solving the 2D Bratu problem, Results in Physics, 13, 2019, 12-21.
35. Ming, C.Y., Zheng, L.C., Zhang, X.X., Steady flow and heat transfer of the power-law fluid over a rotating disk, Int. Commun. Heat Mass, 38, 2011, 280–284.
36. Zheng, L.C., Xun, S., Zhang, X.X., Flow and heat transfer of Ostwald-de-Waele fluid over a variable thickness rotating disk with index decreasing, Int. Commun. Heat Mass, 103, 2016, 1214–1224.
37. Hall, G., Watt, J.M., Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1976.
38. Loeb, A.M., Schiesser, W.E., Stiffness and accuracy in the method of lines integration of partial differential equations, in Proc. of the 1973 Summer Computer Simulation Conf., 2, 1973, 25–39.
39. Schiesser, W.E., The Numerical Method of Lines, Academic Press, New York, 1991.
40. Hamdi, S., Schiesser, W.E., Griffiths, G.W., Method of lines, from Scholarpedia. http://www.scholarpedia.org/article/Method_of_Lines.
41. Hamdi, S., Enright, W.H., Schiesser, W.E., Gottlieb, J.J., Exact solutions and conservation laws for coupled generalized Korteweg deVries and quantic regularized long wave equations, Nonlinear Anal., 63, 2005, 1425–1434.
42. Schiesser, W.E., Method of lines solution of the Korteweg–de Vries equation, Comput. Math. Appl., 28, 1994, 147–154.
43. Dehghan, M., Finite-difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simulation, 71, 2006, 16–30.
44. Hossain, Md. A., Subba, R. and Gorla, R., Natural convection flow of non-Newtonian power-law fluid from a slotted vertical isothermal surface, International Journal of Numerical Methods for Heat and Fluid Flow, 19(7), 2009, 835-846.
2. Ram, P., Kumar, V., Heat transfer in FHD boundary layer flow with temperature-dependent viscosity over a rotating disk, Fluid Dyn. Mater. Process., 10, 2014, 179–196.
3. Rashidi, M., Kavyani, N., Abelman, S., Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties, Int. J. Heat Mass Transfer, 70, 2014, 892–917.
4. Sheikholeslami, M., Hatami, M., Ganji, D.D, Numerical investigation of nanofluid spraying on an inclined rotating disk for cooling process, J. Mol. Liq., 211, 2015, 577–583.
5. Bachok, N., Ishakb, A., Pop, I., Flow and heat transfer over a rotating porous disk in a nanofluid, Phys. B, 406, 2001, 1767–1772.
6. Kendoush, A.A., Similarity solution for heat convection from a porous rotating disk in a flow field, J. Heat Trans. T. ASME, 135, 2013, 1885–1886.
7. Turkyilmazoglu, M., Nanofluid flow and heat transfer due to a rotating disk, Comput. Fluids, 94, 2014, 139–146.
8. Hayat, T., Rashid, M., Imtiaz, M., Alsaedi, A., Magnetohydrodynamic (MHD) flow of Cu-water nanofluid due to a rotating disk with partial slip, AIP Adv., 5, 2015, 067169.
9. Hayat, T., Shehzad S. A., Muhammad, T., Alsaedi, A., On MHD flow of nanofluid due to rotating disk with slip effect, Comput. Methods Appl. Mech. Engrg., 315, 2017, 467-477.
10. Mahanthesh, B., Gireesha, B.J., Animasaun, I.L., Muhammad, T., Shashikumar, N.S., MHD flow of SWCNT and MWCNT nanoliquids past a rotating stretchable disk with thermal and exponential space dependent heat source, Physica Scripta, 94(8), 2019, 4.
11. Mahanthesh, B., Gireesha, B.J., Shashikumar, N.S., Hayat, T., Alsaedi, A., Marangoni convection in Casson liquid flow due to an infinite disk with exponential space dependent heat source and cross-diffusion effects, Results in Physics, 9, 2018, 78-85.
12. Mahanthesh, B., Gireesha, B.J., Shehzad, S.A., Rauf, A., Sampath Kumar, P.B., Nonlinear radiated MHD flow of nanoliquids due to a rotating disk with irregular heat source and heat flux condition, Physica B: Condensed Matter, 537, 2018, 98-104.
13. Raju, C.S.K., Hoque, M.M., Priyadharshini, P. Mahanthesh, M., Gireesha, B.J., Cross diffusion effects on magnetohydrodynamic slip flow of Carreau liquid over a slender sheet with non-uniform heat source/sink, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40, 2018, 222-242.
14. Mahanthesh, B., Gireesha, B.J., Shashikumar, N.S., Shehzad, S.A., Marangoni convective MHD flow of SWCNT and MWCNT nanoliquids due to a disk with solar radiation and irregular heat source, Physica E: Low-dimensional Systems and Nanostructures, 94, 2017, 25-30.
15. Mustafa, I., Javed, T., Ghaffari, A., Heat transfer in MHD stagnation point flow of a ferrofluid over a stretchable rotating disk, J. Mole. Liquids, 219, 2016, 526-532.
16. Rashidi, M.M., Hayat, T., Erfani, E., Hendi, A.A., Simultaneous effects of partial and thermal-diffusion and diffusion-Thermo on steady MHD convective flow due to a rotating disk, Commun. Nonlinear Sci., 16, 2011, 4303-4317.
17. Asgher, S., Jalil, M., Hussan, M., Turkyimazoglu, M., Lie group analysis of flow and heat transfer over a stretching rotating disk, Int. J. Heat Mass Transfer, 69, 2014, 140-146.
18. Hayat, T., Rashid, M., Imtiaz, M. A. Alsaedi, Nanofluid flow due to rotating disk with variable thickness and homogeneous-heterogeneous reactions, Int. J. Heat Mass Transfer, 113, 2017, 96-105.
19. Mitschka, P., Ulbrecht, J., Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids, Appl. Sci. Res., 15, 1966, 345–358.
20. Andersson, H.I., de Korte, E., Meland, R., Flow of a power-law fluid over a rotating disk revisited, Fluid Dyn. Res., 28, 2001, 75–88.
21. Attia, H.A., Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection, Commun. Nonlinear Sci., 13, 2008, 1571-1580.
22. Griffiths, P.T., Flow of a generalized Newtonian fluid due to a rotating disk, J. Non-Newtonian Fluid Mech., 221, 2015, 9–17.
23. Ming, C.Y., Zheng, L.C., Zhang, X.X., Steady flow and heat transfer of the power-law fluid over a rotating disk, Int. Commun. Heat Mass, 38, 2011, 280–284.
24. Fang, T.G., Zhang, J., Zhong, Y.F., Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput., 218, 2012, 7241–7252.
25. Rashidi, M.M., Reddy, S., Naikoti, K., MHD flow and heat transfer characteristics of Williamson nanofluid over a stretchable sheet with variable thickness and variable thermal conductivity, A. R. Mathematical Inst., 171, 2017, 195-211.
26. Imtiaz, M., Hayat, T., Asgher, S., Alsaedi, A., Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics, Ruselts in Physics, 7, 2017, 503-509.
27. Waqas, H., Ullah Khan, S., Imran, M., Bhatti, M.M., Thermally developed Falkner–Skan bioconvection flow of a magnetized nanofluid in the presence of a motile gyrotactic microorganism: Buongiorno’s nanofluid model, Physica Scripta, 94(11), 2019, 115304.
28. Tlili, I., Bhatti, M.M., Mustafa Hamad, S., Barzinjy, A.A., Sheikholeslami, M., Shafee, A., Macroscopic modeling for convection of Hybrid nanofluid with magnetic effects, Physica A: Statistical Mechanics and its Applications, 534, 2019, 122136.
29. Abdelsalam, S.I., Bhatt, M.M., Zeeshan, A., Riaz, A., Anwar Bég, O., Metachronal propulsion of a magnetized particle-fluid suspension in a ciliated channel with heat and mass transfer, Physica Scripta, 94(11), 2019, 115301.
30. Riaz, A., Ellahi, R., Mubashir Bhatti, M., Marin, M., Study of heat and mass transfer in the Eyring–Powell model of fluid propagating peristaltically through a rectangular compliant channel, Heat Transfer Research, 50(16), 2019, in press.
31. Marin, M., Vlase, S., Ellahi, R., Bhatti, M.M., On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure, Symmetry, 11(7), 2019, 863.
32. Waqas, H., Ullah Khan, S., Hassan, M., Bhatti, M.M., Imran, M., Analysis on the bioconvection flow of modified second-grade nanofluid containing gyrotactic microorganisms and nanoparticles, Journal of Molecular Liquids, 291, 2019, 111231.
33. Ali, M.R., Baleanu, D., Haar wavelets scheme for solving the unsteady gas flow in four-dimensional, Thermal Science, 2019, https://doi.org/10.2298/TSCI190101292A.
34. Ali, M.R., Hadhood, A.R., Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet scheme for solving the 2D Bratu problem, Results in Physics, 13, 2019, 12-21.
35. Ming, C.Y., Zheng, L.C., Zhang, X.X., Steady flow and heat transfer of the power-law fluid over a rotating disk, Int. Commun. Heat Mass, 38, 2011, 280–284.
36. Zheng, L.C., Xun, S., Zhang, X.X., Flow and heat transfer of Ostwald-de-Waele fluid over a variable thickness rotating disk with index decreasing, Int. Commun. Heat Mass, 103, 2016, 1214–1224.
37. Hall, G., Watt, J.M., Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1976.
38. Loeb, A.M., Schiesser, W.E., Stiffness and accuracy in the method of lines integration of partial differential equations, in Proc. of the 1973 Summer Computer Simulation Conf., 2, 1973, 25–39.
39. Schiesser, W.E., The Numerical Method of Lines, Academic Press, New York, 1991.
40. Hamdi, S., Schiesser, W.E., Griffiths, G.W., Method of lines, from Scholarpedia. http://www.scholarpedia.org/article/Method_of_Lines.
41. Hamdi, S., Enright, W.H., Schiesser, W.E., Gottlieb, J.J., Exact solutions and conservation laws for coupled generalized Korteweg deVries and quantic regularized long wave equations, Nonlinear Anal., 63, 2005, 1425–1434.
42. Schiesser, W.E., Method of lines solution of the Korteweg–de Vries equation, Comput. Math. Appl., 28, 1994, 147–154.
43. Dehghan, M., Finite-difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simulation, 71, 2006, 16–30.
44. Hossain, Md. A., Subba, R. and Gorla, R., Natural convection flow of non-Newtonian power-law fluid from a slotted vertical isothermal surface, International Journal of Numerical Methods for Heat and Fluid Flow, 19(7), 2009, 835-846.
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