EFFECT OF ALIGNED MAGNETIC FIELD AND INCLINED OUTER VELOCITY IN CASSON FLUID FLOW OVER A STRETCHING SHEET WITH HEAT SOURCE

Abstract

The purpose of this study was to assess the effect of the inclined outer velocity on heat and flow transportation in boundary layer Casson fluid over a stretching sheet. The flow is adopted to have non-orthogonal magnetic field with heat generation in the uniform manner on stretching surface. It has been taken that in both the directions along the x-axis, the sheet is stretched. By applying similarity transformations, the governing equations representing the heat and flow transportation are converted to ordinary differential equations. Runge-Kutta Fehlberg approach was adopted to solve numerically the moulded differential equations with the help of shooting technique. The flow is also governed by the heat source parameter, Casson fluid parameter, magnetic parameter, Prandtl number, aligned angle of magnetic field and the impinging angle parameter. The results revealed that velocity decreases with an increase in Casson fluid parameter, magnetic parameter and aligned angle of magnetic field for the case of outer velocity parameter less than one while velocity increases for the case of outer velocity parameter greater than one because of the inverted boundary layer formation for velocity profile in second case. Also, the fluid temperature increases (for the case of outer velocity parameter less than one) and temperature decreases (for the case of outer velocity parameter greater than one) with an increase in Casson fluid parameter, impinging angle parameter and aligned angle parameter. The results indicate that outer velocity and aligned magnetic field has a significant impact on fluid temperature and velocity. The behaviour of emerging fluid parameters on fluid temperature and velocity are depicted graphically and their effect on local Nusselt number (〖Nu〗_x ) and skin friction coefficient (C_f ) are represented by tables. The finding of this study may serve as to control the rate of heat transportation and fluid velocity in many manufacturing processes and industrial applications to make the desired quality of final product. Acceptance of the extant technique used in current study is correlated with the existing outcomes in the literature.

Keywords

• Casson fluid
• Aligned magnetic field
• Outer velocity
• Oblique flow
• Heat source

References

1. [1] Crane LJ. Flow past a stretching plate. Journal of Applied Mathematics and Physics (ZAMP) 1970; 21: 645–647.
2. [2] Andersson HI. MHD flow of a viscoelastic fluid past a stretching surface. ActaMechanica 1992; 95: 227–230.
3. [3] Cortell R. A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet. Applied Mathematics and Computation 2005; 168: 557-566. doi:10.1016/j.amc.2004.09.046
4. [4] Poply V, Singh P, Chaudhary KK. Analysis of laminar boundary layer flow along a stretching cylinder in the presence of thermal radiation. WSEAS Transactions on Fluid Mechanics 2013; 4: 159-164.
5. [5] Ishak A, Jafar K, Nazar R, Pop I. MHD stagnation point flow towards a stretching sheet. Physica A: Statistical Mechanics and its Applications 2009; 388: 3377–3383.
6. [6] Singh P, Tomer NS, Kumar A, Sinha D. MHD oblique stagnation-point flow towards a stretching sheet with heat transfer. International Journal of Applied Mathematics and Mechanics 2010; 6: 94–111.
7. [7] Singh P, Jangid A, Tomer NS, Sinha D. Effects of thermal radiation and magnetic field on unsteady stretching permeable sheet in presence of free stream velocity. International Journal of Physical and Mathematical Sciences 2010; 4: 160-166. doi: 10.5281/zenodo.1081565.
8. [8] Singh P, Kumar A, Tomer NS, Sinha D. Analysis of porosity effects on unsteady stretching permeable sheet. Walailak Journal of Science and Technology (WJST) 2013; 11: 611–620.

9. [9] Poply V, Singh P, Yadav AK. A study of temperature-dependent fluid properties on mhd free stream flow and heat transfer over a non-linearly stretching sheet. Procedia Engineering 2015; 127: 391–397. doi:10.1016/j.proeng.2015.11.386.
10. [10] Poply V, Singh P, Yadav AK. Stability analysis of MHD outer velocity flow on a stretching cylinder. Alexandria Engineering Journal 2017; 57: 2077-2083. doi:10.1016/j.aej.2017.05.025.
11. [11] Hayat T, Qasim M, Shehzad SA, Alsaedi A. Unsteady stagnation point flow of second grade fluid with variable free stream. Alexandria Engineering Journal 2014; 53: 455–461. doi: 10.1016/j.aej.2014.02.004
12. [12] Siddheshwar PG, Meenakshi, N. Effects of suction and freestream velocity on a hydromagnetic stagnation-point flow and heat transport in a Newtonian fluid toward a stretching sheet. Journal of Heat Transfer 2016; 138: 1-4. doi.10.1115/1.4033460.
13. [13] Mukhopadhyay S, Layek GC. Effects of variable fluid viscosity on flow past a heated stretchingsheet embedded in a porous medium in presence of heat source/sink.Meccanica 2012; 47: 863–876. doi: 10.1007/s11012-011-9457-6.
14. [14] Singh P, Tomer NS, Sandeep K, Deepa S. Effect of radiation and porosity parameter on magnetohydrodynamic flow due to stretching sheet in porous media. Thermal Science 2011; 15: 517–526. doi:10.2298/TSCI1102517S.
15. [15] Seddeek MA. Heat and mass transfer on a stretching sheet with a magnetic field in a viscoelastic fluid flow through a porous medium with heat source or sink. Computational Materials Science 2007; 38: 781–787. doi:10.1016/j.commatsci.2006.05.015
16. [16] Ahmad K, Halim SA, Hanouf Z. Variable viscosity of casson fluid flow over a stretching sheet in porous media with newtonian heating. Journal of Informatics and Mathematical Sciences 2018; 10: 359–370.
17. [17] Mustafa M, Hayat T, Pop I, Hendi A. Stagnation-point flow and heat transfer of a casson fluid towards a stretching sheet. Zeitschrift für Naturforschung A 2012; 67: 70-76.doi: 10.5560/ZNA.2011-0057.
18. [18] Kameswaran PK, Shaw S, Sibanda P. Dual solutions of Casson fluid flow over a stretching or shrinking sheet, Sadhana 2014; 39: 1573–1583.
19. [19] Bhattacharyya K, Hayat T, Alsaedi A. Exact solution for boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet. ZAMM - J. Appl. Math. Mech. Z. FürAngew. Math. Mech 2014; 94: 522–528. doi: 10.1002/zamm.201200031
20. [20] Megahed AM. Effect of slip velocity on Casson thin film flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and viscous dissipation. Appl. Math. Mech. 2015; 36: 1273–1284.
21. [21] Bhattacharyya K. MHD Stagnation-Point Flow of Casson Fluid and Heat Transfer over a Stretching Sheet with Thermal Radiation.Journal of Thermodynamics2013; 2013: 1–9. doi.10.1155/2013/169674.
22. [22] Yaragani HK, Reddy GVR, Makinde OD. Chemical reaction effect on mhd flow of casson fluid with porous stretching sheet. Defect and Diffusion Forum 2018; 389: 100–109.
23. [23] Mabood F, Das K. Outlining the impact of melting on MHD Casson fluid flow past a stretching sheet in a porous medium with radiation. Heliyon 2019; 5. doi.10.1016/j.heliyon.2019.e01216.
24. [24] Raza J. Thermal radiation and slip effects on magneto hydrodynamic (MHD) stagnation point flow of Casson fluid over a convective stretching sheet. Propulsion Power Research 2019; 8: 138-146.
25. [25] [25] Prabhakar B, Bandari S, Kumar CK. Effects of inclined magnetic field and chemical reaction on flow of a casson nanofluid with second order velocity slip and thermal slip over an exponentially stretching sheet. International Journal of Applied and Computational Mathematics 2017; 3: 2967–2985.
26. [26] Jusoh R, Nazar R. Effect of heat generation on mixed convection of micropolar Casson fluid over a stretching/shrinking sheet with suction. Journal of Physics: Conference Series 2019; 1212: 1-6. doi:10.1088/1742-6596/1212/1/012024.
27. [27] Megahed AM. MHD viscous Casson fluid flow and heat transfer with second-order slip velocity and thermal slip over a permeable stretching sheet in the presence of internal heat generation/absorption and thermal radiation. The European Physical Journal Plus 2015; 130: 1-17. doi: 10.1140/epjp/i2015-15081-9.
28. [28] Mahapatra TR, Dholey S, Gupta AS. Heat transfer in oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface. Heat and Mass Transfer 2007; 43: 767–773. doi: 10.1007/s00231-006-0116-8.
29. [29] Lok YY, Amin N, Pop I. Non-orthogonal stagnation point flow towards a stretching sheet. International Journal of Non-Linear Mechanics 2006; 41: 622–627. doi:10.1016/j.ijnonlinmec.2006.03.002
30. [30] Labropulu F, Li D, Pop I. Non-orthogonal stagnation-point flow towards a stretching surface in a non-Newtonian fluid with heat transfer. International Journal of Thermal Sciences 2010; 49: 1042–1050. doi: 10.1016/j.ijthermalsci.2009.12.005.
31. [31] Singh P, Sinha D, Tomer NS. Oblique stagnation-point darcy flow towards a stretching sheet. Journal of Applied Fluid Mechanics 2012; 5: 29-37.
32. [32] Lok YY, Pop I, Ingham DB, Amin N. Mixed convection flow of a micropolar fluid near a non‐orthogonal stagnation‐point on a stretching vertical sheet. International Journal of Numerical Methods for Heat & Fluid Flow2009; 19: 459–483. doi:10.1108/09615530910938380.
33. [33] Lok YY, Merkin JH, Pop I. MHD oblique stagnation-point flow towards a stretching/shrinking surface. Meccanica 2015; 50: 2949–2961. doi: 10.1007/s11012-015-0188-y.
34. [34] Sheikholeslami M. Influence of magnetic field on Al2O3-H2O nanofluid forcedconvection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM. Journal of Molecular Liquids 2018; 263: 472–488. doi:10.1016/j.molliq.2018.04.111.
35. [35] Sheikholeslami M. Solidification of NEPCM under the effect of magnetic field in aporous thermal energy storage enclosure using CuO nanoparticles. Journal of Molecular Liquids 2018; 263: 303–315. doi: 10.1016/j.molliq.2018.04.144.
36. [36] Sheikholeslami M, Zeeshan A. Analysis of flow and heat transfer in water basednanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM. Computer Methods in Applied Mechanics and Engineering 2017; 320: 68–81.
37. [37] Asirvatham LG. Nanofluid heat transfer and applications. Journal of Thermal Engineering 2015; 1: 113-115. doi: 10.18186/jte.93344.
38. [38] Sulochana C, Sandeep N, Sugunamma V, Kumar BR. Aligned magnetic field and cross-diffusion effects of a nanofluid over an exponentially stretching surface in porous medium. Applied Nanoscience 2016; 6: 737–746.
39. [39] Devi R, Poply V, Manimala. Impact of inclined outer velocity in MHD Casson fluid over stretching sheet. International Journal of Advanced Trends in Computer Applications 2019; 1: 32-38.
40. [40] Vinita V, Poply V. Impact of outer velocity MHD slip flow and heat transfer of nanofluid past a stretching cylinder. Materials Today Proceedings 2019; 26: 3429-3435. doi: 10.1016/j.matpr.2019.11.304.
41. [41] Mahabaleshwar US, Vinay Kumar PN, Sheremet M. Magneto hydrodynamics flow of a nanofluid driven by a stretching/shrinking sheet with suction. SpringerPlus 2016; 5: 1-9. doi: 10.1186/s40064-016-3588-0.
42. [42] Gireesha BJ, Ramesh GK, Bagewadi CS. Heat transfer in MHD flow of a dusty fluid over a stretching sheet with viscous dissipation. Advances in Applied Science Research 2012; 3: 2392-2401.
43. [43] Abel MS, Mahesha N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Applied Mathematical Modelling 2008; 32: 1965–1983. doi:10.1016/j.apm.2007.06.038.