1. LITERATURE SURVEY:
It is well experienced that the most of the
fluids are not satisfies the common assumption of a linear relationship between
the rate of strain and shear stress, therefore these types of fluids called as
non-Newtonian fluids. The fluid which holds the Newton’s law of viscosity is
called Newtonian fluid. The Newton’s law of viscosity is where the kinematic
viscosity of the fluid is
and the shear stress is
. All the
fluids are not following the stress-strain relation. Those are not obeys the
Newton law of viscosity is known as non-Newtonian fluids. The kinematic
viscosity of the non-Newtonian fluids is a function of strain rate. These types
of fluid flows have presented exciting the challenges the mathematicians,
numerical engineering simulators and physicists. The non-Newtonian fluids
equations are more difficult than the second order Navier-Stoke’s equations.
Such types of fluids have been modelled by constitutive equations. The
resultant governing non-linear equations are not possible to solve
analytically, due to this reason recently many researchers are trying to
solving the non-Newtonian fluid models by using numerical techniques. It also
has various science and engineering, industrial applications such as thermal
insulation of buildings, cooling of electronic devices, food processing system,
enhanced oil recovery system, drying of solid systems, geophysical systems and
geothermal energy systems.
The laminar boundary layer flow driven by a continuously moving laminar boundary layer flow driven by a continuously moving solid surface has received considerable attention since the pioneering investigations by Sakiadis1-2. Magnetohydrodynamic Carreau fluid slip flow over a porous stretching sheet with variable thermal conductivity and viscous dissipation is discussed by Shah et al.3. Falkner Skan flow of a magnetic Carreau fluid past a wedge in the presence of cross diffusion studied by Raju et al.4. Durga Prasad et al.5. Examined the Blasius and Sakiadis flow of MHD Jeffrey fluid with non-uniform heat source/sink. Vinod et al. 6reported the heat transfer analysis on Blasius and Sakiadis flow of magnetohydrodynamic Maxwell fluid with Cattaneo-Christov heat flux model. Later on, Hashim et al.7 studied a revised model to analyse the heat and mass transfer mechanisms in the flow of Carreau nanofluids. Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation is talked about by Khan et al.8. Amongst these the Carreau rheological model9 is a subclass of generalized Newtonian fluids. Recently, Hayat et al.10 proposed a model stagnation point flow with Catteneo-Christov heat flux and homogeneous-heterogeneous reactions.There are also several experimental and numerical studies on the natural and forced convection using non-Newtonian nanofluids related with different heat enclose, and we mention here those by Madhu et al.11. Kishan et al.12, Tibullo et al.13, Sakiadis et al.14, Chang et al. 15, Tiwari and Das 16, Oztop and Abu-Nada 17, and Muthtamilselvan et al. 18. The book by Das et al. 19 and the recent review paper by Kakaç and Pramunjaroenkij20. Present excellent collection of up to now published papers on magneto hydrodynamic. Few useful articles are cited for reader's interest see ref. 21-26.
Motivated by the above mentioned studies and challenges, but no studies have been described yet up to the author’s knowledge on heat transfer analysis of Blasius and Sakiadis flow of Magnetohydrodynamic Radiated Carreau fluid with Cattaneo-Christov heat flux. The transformed ordinary differential equations are solved numerically by using shooting technique. Also, calculated the friction factor coefficient and heat transfer rate for the various non-dimensional governing parameters.
2. BASIC EQUATIONS:
3. RESULT AND DISCUSSION
4. CONCLUSIONS:
In this study, we have theoretically investigated the impacts of leading parameters i.e., magnetic field parameter, radiation parameter, thermal radiation, porosity parameter and power law index and thermal relaxation parameters are analyzed on heat transfer analysis of Blasius and Sakiadis flow of Magnetohydrodynamic Radiated Carreau fluid with Cattaneo-Christov heat flux. Numerical solutions are carried out by using Runge-Kutta based Shooting technique. The effects of various governing parameters on the flow quantities are demonstrated graphically. The local Nusselt numbers for the Blasius and Sakiadis fluid flows. The following are the brief conclusions drawn from the present study:
1. The thermal relaxation and thermal radiation parameters are help to improve the rate of heat transfer.
2. The thickness of the momentum boundary layer and thermal boundary layers enhanced by rise in power index parameter.
3. The effect of porosity parameter was to reduce the velocity of the fluid.
4. The wall temperature accelerated by the small values of thermal relaxation parameter, but decelerates the velocity of the fluid.
5. REFERENCES:
1. B.C.Sakiadis, Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J. 1961, 7, 26–28.
2. H. Blasius, Grenzschichten in Flüssigkeiten mitkleiner Reibung. Z. Math. Phys.1908, 56, 1–37.
3. R.A. Shah, T.Abbas, M. Idrees and M. Ullah, MHD Carreau fluid slip flow over a porous stretching sheet with viscous dissipation and variable thermal conductivity, Boundary Value Problems2017, 94, DOI 10.1186/s13661-017-0827-4.
4. C.S. K. Raju, N. Sandeep, Falkner Skan flow of a magnetic Carreau fluid past a wedge in the presence of cross diffusion, European Physical Journal Plus, 2016, 131, 267.
5. P. Durga Prasad, C.S.K. Raju, S.V.K. Varma, S. U. Mamatha, Blasius and Sakiadis flow of magnetohydrodynamic Jeffrey fluid with non-uniform heat source or sink, International Journal of Research in Science and Engineering, Special Issue- NCRAPAM. 2017, 39-49
6. G. Vinod, C. S. K. Raju, R. Fatima, S. V. K. Varma, heat transfer analysis of Blasius and Sakiadis flow of MHD radiated Maxwell fluid with Cattaneo-Christov heat flux model, International Journal of Research In Science and Engineering, , Special Issue-NCRAPAM. 2017, 198-206.
7. Hashim, M. Khan, A revised model to analyse the heat and mass transfer mechanisms in the flow of Carreau nanofluids, International Journal of Heat and Mass Transfer, 2016, 103, 291-297.
8. M.Khan, Hashim, M. Hussain, M.Azam, Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation, Journal of Magnetism and Magnetic Materials,2016, 412, 63-68.
9. P.J.Carreau, Rheological equations from molecular network theories, Transactions of the Society Rheology, 1972, 16 (1), 99-127.
10. T.Hayat, M.I.Khan, M. Farooq, A.Aslaedi, T.Yasmen, A.Asaedi, Stagnation point flow with Catteneo-Christov heat flux and homogeneous-heterogeneous reactions, Journal of Molecular Liquids, 2016, 220, 49-55.
11. Macha Madhu, Naikoti Kishan and A. Chamkha, Boundary layer flow and heat transfer of a non-Newtonian nanofluid over a non-linearly stretching sheet IJNM for Heat &Fluid Flow, 2015, 26(7), 2198-2217.
12. Naikoti Kishan, P.Kavitha, MHD Non-Newtonian Power Law Fluid Flow and Heat Transfer Past a Non-Linear Stretching Surface with Thermal Radiation and Viscous Dissipation, 2014, 17(3), 267-274.
13. V. Tibullo and V. Zampoli, “A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids, ”Mech. Res. Commun.2011, 38, 77–79.
14. B.C. Sakiadis, Boundary-layer behaviour on continuous solid surfaces. The boundary-layer on a continuous flat plate, AIChE J. 1961, 7, 221–225.
15. C.W. Chang et al.TheLie-group shooting method for boundary layer equations in fluid mechanics, J. Hydrodynam. 2006, 18 (3), 103–108.
16. R.K. Tiwari, M.K. Das. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 2007,50, 2002–2018.
17. H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow, 2008,29, 1326–1336.
18. M. Muthtamilselvan, P. Kandaswamy, J. Lee. Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure. Commun. Nonlinear. Sci. Numer. Simul. 2010, 15, 1501–1510.
19. S.K. Das, Choi, S.U.S., Yu, W., T. Pradet, Nanofluids: Science and Technology. Wiley, New Jersey 2007.
20. S Kakaç, A Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 2009,52, 3187–3196.
21. S. Ahmad, A.M. Rohni, and I. Pop, Blasius and Sakiadis problems in nanofluids. Acta Mechanica, 2011, 218, 195–204.
22. A. Aziz, A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Communications in Nonlinear Science and Numerical Simulation, 2009, 14, 1064–1068.
23. R.C. Bataller, Radiation effects for the Blasius and Sakiadis flows with a convective surfaceboundary condition. Applied Mathematics and Computation, 2008,206, 832–840.
24. I. Pop, Gorla, R.S.R., and M. Rashidi, The effect of variable viscosity on the flow and heat transfer to a continuous moving flat plate. Int. J. Eng. Sci. 1992, 30, 1–6.
25. A. Pantokratoras, Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate. Int. J. Eng. Sci. 2004, 42, 1891–1896.
26. H.I. Andersson, J.B. Aarseth, Sakiadis flow with variable fluid properties revisited. Int. J. Eng. Sci. 2007, 45, 554–561.
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Received on 27.10.2017 Accepted on 10.12.2017 ©A&V Publications all right reserved Research J. Engineering and Tech. 2018;9(1): 14-20 DOI: 10.5958/2321-581X.2018.00003.X |
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