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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 2

Issues

Numerical Exploration of Heat Transfer and Lorentz Force Effects on the Flow of MHD Casson Fluid over an Upper Horizontal Surface of a Thermally Stratified Melting Surface of a Paraboloid of Revolution

O. D. Makinde
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  • Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa
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/ N. Sandeep / T. M. Ajayi / I. L. Animasaun
Published Online: 2018-03-07 | DOI: https://doi.org/10.1515/ijnsns-2016-0087

Abstract

Considering the recent aspiration of experts dealing with the painting of aircraft and bonnet of cars to further understand the relevance of skin friction and heat transfer while painting all these objects that are neither horizontal nor vertical, neither a cone/wedge or cylinder but upper horizontal surface of a paraboloid of revolution; a two-dimensional electrically conducting Casson fluid flow on an upper horizontal thermally stratified surface of a paraboloid of revolution is analyzed. The influence of melting heat transfer and thermal stratification are properly accounted for by modifying classical boundary condition of temperature. Plastic dynamic viscosity and thermal conductivity of the fluid are assumed to vary linearly with temperature. In view of this, all necessary models were modified to suit the case Tm<T. It is assumed that natural convection is driven by buoyancy; hence the suitable model of Boussinesq approximation is adopted. A suitable similarity transformation is applied to reduce the governing equations to coupled ordinary differential equations. These equations along with the boundary conditions are solved numerically by using Runge–Kutta technique along with shooting method. Effects of the magnetic field, temperature-dependent plastic dynamic viscosity and buoyancy parameters on the velocity and temperature are showed graphically and discussed. Normal influence of Lorentz force exists on Casson fluid flow when the thickness of the surface is small. Scientists and experts are urge to note an adverse effect of this force occurs on the fluid flow when the thickness of the surface is large.

Keywords: melting surface; variable plastic dynamic viscosity; casson fluid; variable thermal conductivity; magnetohydrodynamic; paraboloid of revolution

MSC 2010: 76D10

References

  • [1]

    L. Prandtl, “Über Flüssigkeitsbewegung bei sehr kleiner Reibung” translated to “Motion of fluids with very little viscosity”, Internationalen Mathematiker-Kongresses in Heidelberg 8 (13) (1904), 1–8.Google Scholar

  • [2]

    L. L. Lee, Boundary layer over a thin needle, Phys Fluids. 10 (4) (1967), 822–828. doi:.CrossrefGoogle Scholar

  • [3]

    D. R. Miller, The Boundary-layer on a paraboloid of revolution, Proc. Cambridge Philos. Soc. 65 (1969), 285–298.Google Scholar

  • [4]

    R. T. Davi , M. J. Werle , Numerical solutions for laminar incompressible flow past a paraboloid of revolution, AIAA J. 10 (9) (1972), 1224–1230. doi: .CrossrefGoogle Scholar

  • [5]

    S. Ahmad, R. Nazar , Pop L., Mathematical modeling of Boundary layer flow over a moving thin needle with variable heat flux, in: Proceedings of the 12th WSEAS International Conference on Applied Mathematics, pp. 48–53, World Scientific and Engineering Academy and Society (WSEAS) Stevens point Wisconsin, USA., December 29–31.Google Scholar

  • [6]

    A. Ishak, R. Nazar, Pop I., Boundary layer flow over a continuously moving thin needle in a parallel free stream, Chin. Phys. Lett. 24 (10) (2007), 2895–2897. doi:.CrossrefGoogle Scholar

  • [7]

    T. Fang, J. Zhang, Y.Zhong, Boundary layer flow over a stretching sheet with variable thickness, Appl. Math. Comput. 218 (2012), 7241–7252. doi:.CrossrefGoogle Scholar

  • [8]

    S. P.Anjali Devi, M.Prakash, Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet, J. Nigerian Math. Soc. 34 (2015), 318–330. doi:.CrossrefGoogle Scholar

  • [9]

    G. K. Ramesh, B.C. Prasannakumara, Gireesha B. J., Rashidi M. M., Casson fluid flow near the stagnation point over a stretching sheet with variable thickness and radiation, J. Appl. Fluid Mech. 9 (3) (2016), 1115–1122.Google Scholar

  • [10]

    I. L. Animasaun, 47nm alumina-water nanofluid flow within boundary layer formed on upper horizontal surface of paraboloid of revolution in the presence of quartic autocatalysis chemical reaction, Eng Alexandria. J. 55 (3) (2016), 2375–2389. doi: .CrossrefGoogle Scholar

  • [11]

    L. Roberts, On the melting of a semi-infinite body of ice placed in a hot stream of air, Fluid Mech J. 4 (1958), 505–528. doi:.CrossrefGoogle Scholar

  • [12]

    M. Epstein, D. H. Cho, Melting heat transfer in steady laminar flow over a flat plate, Heat Trans J. 98 (1976), 531–533. doi:.CrossrefGoogle Scholar

  • [13]

    B. C. Prasannnakumara, B. J. Gireesha, P. T. Manjunatha , Melting phenomenon in MHD stagnation point flow of dusty fluid over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink, Int. J. Comput. Methods Eng. Sci. Mech. 16 (5) (2015), 265–274. doi: .CrossrefGoogle Scholar

  • [14]

    K. S.Adegbie, A. J. Omowaye, A. B. Disu, I. L. Animasaun, Heat and mass transfer of upper convected Maxwell fluid flow with variable thermo-physical properties over a horizontal melting surface, Appl. Math. 6 (2015), 1362–1379. doi: .CrossrefGoogle Scholar

  • [15]

    A. J. Omowaye, I. L. Animasaun, Upper-convected Maxwell fluid flow with variable thermo-physical properties over a melting surface situated in hot environment subject to thermal stratification, Appl J. Mech Fluid. 9 (4) (2016), 1777–1790.Google Scholar

  • [16]

    T. Hayat, Z. Hussain, A. Alsaedi, B. Ahmad, Heterogeneous-homogeneous reactions and melting heat transfer effects in flow with carbon nanotubes, Molecul Liquid. J., 220 (2016), 200–207. doi: .CrossrefGoogle Scholar

  • [17]

    I. L.Animasaun, Casson fluid flow of variable viscosity and thermal conductivity along exponentially stretching sheet embedded in a thermally stratified medium with exponentially heat generation, Heat Mass J. Trans. Res. (JHMTR) 2 (2) (2015), 63–78. doi: .CrossrefGoogle Scholar

  • [18]

    H. Alfven, Existence of electromagnetic-hydrodynamic waves. Nature Publishing Group 150 (3805) (1942), 405–406. doi .CrossrefGoogle Scholar

  • [19]

    V. J.Rossow, On flow of electrically conducting fluid over a flat plate in the presence of a transverse magnetic field, Technical Report NACA, 3071. Report/Patent Number: NACA-TR-1358, 1957.Google Scholar

  • [20]

    N. Liron, Wilhelm H. E., Integration of the magnetohydrodynamic boundary-layer equations by Meksin’s method, ZAMM - J. Appl. Math. Mech. 54 (1) (1974), 27–37. doi: .CrossrefGoogle Scholar

  • [21]

    K.Das, Radiation and melting effect on MHD boundary layer flow over a moving surface, Ain Shams Eng. J. 5 (4) (2014), 1207–1214. doi:.CrossrefGoogle Scholar

  • [22]

    S. S. Motsa, Animasaun I. L., A new numerical investigation of some thermo-physical properties on unsteady MHD non-Darcian flow past an impulsively started vertical surface, Thermal Sci. 19 (Suppl. 1) (2015), S249–S258. doi: .CrossrefGoogle Scholar

  • [23]

    C. S. K. Raju , N.Sandeep, Heat and mass transfer in MHD non-Newtonian bio-convection flow over a rotating cone/plate with cross diffusion, J. Molecul. Liquid. 215 (2016), 115–126. doi:CrossrefGoogle Scholar

  • [24]

    O. D.Makinde, F. Mabood, W. A. Khan , M. S. Tshehla, MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat, J. Molecul. Liquid 219 (2016), 624–630. doi: .CrossrefGoogle Scholar

  • [25]

    N. Sandeep, C. S. K. Raju , C. Sulochana, V.Sugunamma. Effects of aligned magnetic field and radiation on the flow of ferrofluids over a flat plate with non-uniform heat source/sink, Int. J. Sci. Eng. 8(2) (2015), 151–158. doi: .CrossrefGoogle Scholar

  • [26]

    W. A.Khan, O. D. Makinde, Z. H. Khan, MHD boundary layer flow of a nanofluid containing gyrotactic microorganisms past a vertical plate with Navier slip, Int. J. Heat Mass Trans. 74 (2014), 285–291.Google Scholar

  • [27]

    C. S. K. Raju, N. Sandeep, V. Sugunamma, M. Jayachandra Babu, J. V. Ramana Reddy, Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface, Eng. Sci. Technol. Int. J. 19 (1) (2016), 45–52. doi:.CrossrefGoogle Scholar

  • [28]

    W. A.Khan, O. D. Makinde, MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet, Int. J. Thermal Sci. 81 (2014), 118–124. doi:.CrossrefGoogle Scholar

  • [29]

    J. W. Smith, N. Epstein , Effect of wall roughness on convective heat transfer in commercial pipes, Am. Ins. Chem. Eng. AIChE J.() 3 (2) (1957), 242–248. doi: .CrossrefGoogle Scholar

  • [30]

    H. E. Zellnik, S. WChurchill,, Convective heat transfer from hightemperature air inside a tube, Am. Inst. Chem. Eng. (AIChE J.) 4 (1) (1958), 37–42. doi: .CrossrefGoogle Scholar

  • [31]

    I. L. Animasaun, Double diffusive unsteady convective micropolar flow past a vertical porous plate moving through binary mixture using modified Boussinesq approximation, Ain Shams Eng. J. 7 (2) (2016), 755–765. doi: .CrossrefGoogle Scholar

  • [32]

    N. Casson, A flow equation for Pigment oil-suspensions of the printing ink type, in: Mill CC (Ed.), Rheology of Disperse Systems, p. 84, Pergamon Press, Oxford, UK, 1959.Google Scholar

  • [33]

    N. T. M.Eldabe, G. Saddeck, A. F. El-Sayed, Heat transfer of MHD non-Newtonian Casson fluid flow between two rotating cylinder, Mech. Mech. Eng. 5 (2) (2001), 237–251.Google Scholar

  • [34]

    C. S. K.Raju, N. Sandeep, V. Sugunamma, M. Jayachandra Babu and J.V. Ramana Reddy , Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface, Eng. Sci. Technol., Int. J. 19 (1) (2016), 45–52. doi: .CrossrefGoogle Scholar

  • [35]

    W. Ibrahim, O. D. Makinde, Magnetohydrodynamic stagnation point flow and heat transfer of casson nanofluid past a stretching sheet with slip and convective boundary condition, J. Aerospace Eng. 29 (2) (2016), 04015037. doi: .CrossrefGoogle Scholar

  • [36]

    I. L. Animasaun, Effects of thermophoresis, variable viscosity and thermal conductivity on free convective heat and mass transfer of non-darcian MHD dissipative Casson fluid flow with suction and nth order of chemical reaction, J. Nigerian Math. Soc. 34 (1) (2015), 11–31. doi:.CrossrefGoogle Scholar

  • [37]

    O. D.Makinde, I. L. Animasaun, Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution, Thermal Sci Int. J. 109 (2016), 159–171. doi:.CrossrefGoogle Scholar

  • [38]

    G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Press, (1967), ISBN: 0-521-66396-2.Google Scholar

  • [39]

    T. Y. Na, Computational Methods in Engineering Problems Boundary Value, Press Academic, York New, 1979.Google Scholar

  • [40]

    A. Pantokratoras, A common error made in investigation of boundary layer flows, Appl. Math. Modell. 33 (2009), 413–422.Google Scholar

  • [41]

    F. S. Gökhan, Effect of the Function Guess & Continuation Method on the Run Time of Solvers MATLAB BVP. Ionescu Clara M. (Ed.), 1, (2011).Google Scholar

  • [42]

    J. Kierzenka, L. F. Shampine, A BVP solver based on residual control and the MATLAB PSE, ACM Trans. Math. Software (TOMS). 27(3) (2001), 299–316.CrossrefGoogle Scholar

  • [43]

    M. Mustafa, T. Hayat, I. Pop, A. Aziz, Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate, Heat Trans. Asian Res. 40 (6) (2011), 563–576. doi: .CrossrefGoogle Scholar

  • [44]

    A. J. Chamkha, MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, Int. Commun. Heat Mass Trans. 30 (3) (2003), 413–422. doi:.CrossrefGoogle Scholar

  • [45]

    T. M. Agbaje, S. Mondal, Z. G. Makukula, S. S. Motsa, Sibanda P., A new numerical approach to MHD stagnation point flow and heat transfer towards a stretching sheet, Ain Shams Eng. J. (2015) in-press. doi:.CrossrefGoogle Scholar

  • [46]

    E. A. Adebile, I. L. Animasaun, A. I. Fagbade, Casson fluid flow with variable thermo-physical property along exponentially stretching sheet with suction and exponentially decaying internal heat generation using the homotopy analysis method, J. Nigerian Math. Soc. 35 (1) (2016), 1–17. doi:.CrossrefGoogle Scholar

  • [47]

    O. K. Koriko, I. L.Animasaun, New similarity solution of micropolar fluid flow problem over an uhspr in the presence of quartic kind of autocatalytic chemical reaction, Frontiers Transfer Heat Mass (FHMT) 8 (2017), 26. doi: .CrossrefGoogle Scholar

  • [48]

    T.Hayat, M. Hussain, M. Awais, S. Obaidat, Melting heat transfer in a boundary layer flow of a second grade fluid under soret and dufour effects, Numerical Methods Int. J. Flow Heat Fluid. 23 (2013), 1155–1168. doi:.CrossrefGoogle Scholar

About the article

Received: 2016-06-14

Accepted: 2018-02-02

Published Online: 2018-03-07

Published in Print: 2018-04-25


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 2, Pages 93–106, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0087.

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