Mixed convection flow of Newtonian fluids over an upper horizontal thermally stratified melting surface of a paraboloid of revolution

  • I. L. AnimasaunEmail author
  • O. D. Makinde
  • S. Saleem
Technical Paper


This article presents the combined effects of melting heat transfer and linear stratification of not only the heat energy, but also stratified concentration on the motion of an electrically conducting fluid over an object with a non-uniform thickness. Owing to the melting heat transfer and stretching of fluid layers at the free stream, the case of mixed convection is considered to be more appropriate than neither free nor forced convection. Consequently, the energy and concentration equations which model the flow and satisfy the free stream conditions are presented. Suitable thermophoresis model and Boussinesq approximation for the case \(T_m(x)<T_\infty (x)\) and \(C_m(x)<C_\infty (x)\) were adopted. A suitable similarity transformation is applied to reduce the governing equations to coupled ordinary differential equations. These equations along with the boundary conditions were solved numerically using Runge–Kutta technique along with shooting technique. Maximum variations in the local skin friction coefficients \(Re_{x}^{1/2}C_{fx}\) with thermal stratification occur at larger values of temperature-dependent viscosity parameter. Temperature distribution, local skin friction coefficients \(Re_{x}^{1/2}C_{fx}\) and mass transfer rate \(S_{hx}Re_{x}^{-1/2}\) are decreasing properties of thermal stratification and thermophoresis parameter.


Melting surface Thermal stratification Thermophoresis Magnetohydrodynamic Paraboloid of revolution 



The authors would like to appreciate the support of the Editor-in-Chief, all the reviewers of BMSE-D-17-01017, BMSE-D-17-01593, and BMSE-D-18-01535 for their valuable comments and useful suggestions. Also, the authors would like to express their gratitude to King Khalid University, Abha 61413, Saudi Arabia, for providing administrative and technical support.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.


  1. 1.
    Prandtl L (1904) “\(\ddot{U}\)ber Flussigkeitsbewegung bei sehr kleiner Reibung” translated to “Motion of fluids with very little viscosity”. Int Math Kongr Heidelb 8(13):1–8Google Scholar
  2. 2.
    Mohamed AMO, Shooshpasha I, Yong RN (1996) Boundary layer transport of metal ions in frozen soil. Int J Numer Anal Methods Geomech 20(10):693–713CrossRefGoogle Scholar
  3. 3.
    Van Ingen JL (1998) Looking back at forty years of teaching and research in Ludwig Prandtl’s heritage of boundary layer flows. ZAMM J Appl Math Mech/Z Angew Math Mech 78(1):3–20MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lee LL (1967) Boundary layer over a thin needle. Phys Fluids 10(4):822–828. CrossRefzbMATHGoogle Scholar
  5. 5.
    Miller DR (1969) The boundary-layer on a paraboloid of revolution. Proc Camb Philos Soc 65:285–298MathSciNetCrossRefGoogle Scholar
  6. 6.
    Davis RT, Werle MJ (1972) Numerical solutions for laminar incompressible flow past a paraboloid of revolution. AIAA J 10(9):1224–1230. CrossRefzbMATHGoogle Scholar
  7. 7.
    Ahmad S, Nazar R, Pop L (2007) 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. World Scientific and Engineering Academy and Society (WSEAS) Stevens point Wisconsin, New York, December 29–31, pp 48–53Google Scholar
  8. 8.
    Ishak A, Nazar R, Pop I (2007) Boundary layer flow over a continuously moving thin needle in a parallel free stream. Chin Phys Lett 24(10):2895–2897. CrossRefGoogle Scholar
  9. 9.
    Fang T, Zhang J, Zhong Y (2012) Boundary layer flow over a stretching sheet with variable thickness. Appl Math Comput 218:7241–7252. MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Animasaun IL (2016) 47 nm alumina–water nanofluid flow within boundary layer formed on upper horizontal surface of paraboloid of revolution in the presence of quartic autocatalysis chemical reaction. Alex Eng J 55(3):2375–2389. CrossRefGoogle Scholar
  11. 11.
    Animasaun IL, Koriko OK (2017) New similarity solution of micropolar fluid flow problem over an uhspr in the presence of quartic kind of autocatalytic chemical reaction. Front Heat Mass Transf (FHMT) 8:26. CrossRefGoogle Scholar
  12. 12.
    Roberts L (1958) On the melting of a semi-infinite body of ice placed in a hot stream of air. J Fluid Mech 4:505–528. MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Epstein M, Cho DH (1976) Melting heat transfer in steady laminar flow over a flat plate. J Heat Transf 98:531–533. CrossRefGoogle Scholar
  14. 14.
    Prasannnakumara BC, Gireesha BJ, Manjunatha PT (2015) 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):265–274. CrossRefGoogle Scholar
  15. 15.
    Adegbie KS, Omowaye AJ, Disu AB, Animasaun IL (2015) Heat and mass transfer of upper convected Maxwell fluid flow with variable thermo-physical properties over a horizontal melting surface. Appl Math 6:1362–1379. CrossRefGoogle Scholar
  16. 16.
    Animasaun IL (2015) Casson fluid flow of variable viscosity and thermal conductivity along exponentially stretching sheet embedded in a thermally stratified medium with exponentially heat generation. J Heat Mass Transf Res (JHMTR) 2(2):63–78MathSciNetGoogle Scholar
  17. 17.
    Ajayi TM, Omowaye AJ, Animasaun IL (2017) Viscous dissipation effects on the motion of Casson fluid over an upper horizontal thermally stratified melting surface of a paraboloid of revolution: boundary layer analysis. J Appl Math. Article ID 1697135.
  18. 18.
    Alfven H (1942) Existence of electromagnetic–hydrodynamic waves. Nat Publ Group 150(3805):405–406. CrossRefGoogle Scholar
  19. 19.
    Rossow VJ (1957) On flow of electrically conducting fluid over a flat plate in the presence of a transverse magnetic field. NACA technical report, 3071. Report/patent no. NACA-TR-1358Google Scholar
  20. 20.
    Liron N, Wilhelm HE (1974) Integration of the magnetohydrodynamic boundary-layer equations by Meksin’s method. ZAMM J Appl Math Mech 54(1):27–37. CrossRefzbMATHGoogle Scholar
  21. 21.
    Das K (2014) Radiation and melting effect on MHD boundary layer flow over a moving surface. Ain Shams Eng J 5(4):1207–1214. CrossRefGoogle Scholar
  22. 22.
    Motsa SS, Animasaun IL (2015) A new numerical investigation of some thermo-physical properties on unsteady MHD non-Darcian flow past an impulsively started vertical surface. Therm Sci 19(Suppl. 1):S249–S258. CrossRefGoogle Scholar
  23. 23.
    Raju CSK, Sandeep N (2016) Heat and mass transfer in MHD non-Newtonian bio-convection flow over a rotating cone/plate with cross diffusion. J Mol Liquids 215:115–126. CrossRefGoogle Scholar
  24. 24.
    Makinde OD, Animasaun IL (2016) Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution. Int J Therm Sci 109:159–171. CrossRefGoogle Scholar
  25. 25.
    Makinde OD, Animasaun IL (2016) Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. J Mol Liq 221:733–743. CrossRefGoogle Scholar
  26. 26.
    Makinde OD, Sandeep N, Ajayi TM, Animasaun IL (2018) 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. Int J Nonlinear Sci Numer Simul 19(2–3):93–106. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Babu MJ, Sandeep N, Saleem S (2017) Free convective MHD Cattaneo–Christov flow over three different geometries with thermophoresis and Brownian motion. Alex Eng J 56(4):659–669. CrossRefGoogle Scholar
  28. 28.
    Sheikholeslami M (2018) Influence of magnetic field on Al2O\(_{3}\)-H\(_{2}\)O nanofluid forced convection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM. J Mol Liq 263:472–488. CrossRefGoogle Scholar
  29. 29.
    Awais M, Saleem S, Hayat T, Irum S (2016) Hydromagnetic couple-stress nanofluid flow over a moving convective wall: OHAM analysis. Acta Astronaut 129:271–276. CrossRefGoogle Scholar
  30. 30.
    Sheikholeslami M, Li Z, Shafee A (2018) Lorentz forces effect on NEPCM heat transfer during solidification in a porous energy storage system. Int J Heat Mass Transf 127:665–674. CrossRefGoogle Scholar
  31. 31.
    Animasaun IL, Mahanthesh B, Jagun AO, Bankole TD, Sivaraj R, Shah NA, Saleem S (2019) Significance of Lorentz force and thermoelectric on the flow of 29 nm CuO–water nanofluid on an upper horizontal surface of a paraboloid of revolution. J Heat Transf 141(2):022402CrossRefGoogle Scholar
  32. 32.
    Sheikholeslami M, Zeeshan A (2017) Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM. Comput Methods Appl Mech Eng 320:68–81. MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sheikholeslami M, Rokni HB (2017) Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force. Comput Methods Appl Mech Eng 317:419–430MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sheikholeslami M, Rokni HB (2017) Simulation of nanofluid heat transfer in presence of magnetic field: a review. Int J Heat Mass Transf 115:1203–1233CrossRefGoogle Scholar
  35. 35.
    Raju CSK, Sandeep N, Saleem S (2016) Effects of induced magnetic field and homogeneous-heterogeneous reactions on stagnation flow of a Casson fluid. Int J Eng Sci Technol 19(2):875–887. CrossRefGoogle Scholar
  36. 36.
    Nadeem S, Saleem S (2014) Theoretical investigation of MHD nanofluid flow over a rotating cone: an optimal solutions. Inf Sci Lett 3(2):55–62CrossRefGoogle Scholar
  37. 37.
    Agbaje TM, Mondal S, Makukula ZG, Motsa SS, Sibanda P (2018) A new numerical approach to MHD stagnation point flow and heat transfer towards a stretching sheet. Ain Shams Eng J 9(2):233–243. CrossRefGoogle Scholar
  38. 38.
    Huppert HE, Turner JS (1981) Double-diffusive convection. J Fluids Mech 106:299–329MathSciNetCrossRefGoogle Scholar
  39. 39.
    Makinde OD, Olanrewaju PO, Charles WM (2011) Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture. Afrika Mat 21:65–78. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Makinde OD, Olanrewaju PO (2011) Unsteady mixed convection with Soret and Dufour effects past a porous plate moving through a binary mixture of chemically reacting fluid. Chem Eng Commun 198(7):920–38. CrossRefGoogle Scholar
  41. 41.
    Makinde OD (2005) Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int Commun Heat Mass Transf 32:1411–1419. CrossRefGoogle Scholar
  42. 42.
    Animasaun IL (2015) Dynamics of unsteady MHD convective flow with thermophoresis of particles and variable thermo-physical properties past a vertical surface moving through binary mixture. Open J Fluid Dyn 5:106–120. CrossRefGoogle Scholar
  43. 43.
    Animasaun IL (2015) 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):755–765. CrossRefGoogle Scholar
  44. 44.
    Sandeep N, Raju CSK, Sulochana C, Sugunamma V (2015) Effects of aligned magneticfield and radiation on the flow of ferrofluids over a flat plate with non-uniform heat source/sink. Int J Sci Eng 8(2):151–158. CrossRefGoogle Scholar
  45. 45.
    Raju CSK, Sandeep N, Sugunamma V, Jayachandra Babu M, Ramana Reddy JV (2016) Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface. Int J Eng Sci Technol 19(1):45–52. CrossRefGoogle Scholar
  46. 46.
    Adebile EA, Animasaun IL, Fagbade AI (2015) 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 Niger Math Soc 35(1):1–17. MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Batchelor GK (1967) An introduction to fluid dynamics. Cambridge Press, Cambridge ISBN 0-521-66396-2zbMATHGoogle Scholar
  48. 48.
    Na TY (1979) Computational methods in engineering boundary value problems. Academic, New YorkzbMATHGoogle Scholar
  49. 49.
    Pantokratoras A (2009) A common error made in investigation of boundary layer flows. Appl Math Model 33:413–422CrossRefGoogle Scholar
  50. 50.
    Gökhan FS (2011) Effect of the guess function and continuation method on the run time of MATLAB BVP Solvers. Clara M. Ionescu (Ed.) 1Google Scholar
  51. 51.
    Kierzenka J, Shampine LF (2001) A BVP solver based on residual control and the MATLAB PSE. ACM TOMS 27(3):299–316MathSciNetCrossRefGoogle Scholar
  52. 52.
    Anjali Devi SP, Prakash M (2015) Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. J Niger Math Soc 34(3):318–330. MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Shah NA, Animasaun IL, Ibraheem RO, Babatunde HA, Sandeep N, Pop I (2018) Scrutinization of the effects of Grashof number on the flow of different fluids driven by convection over various surfaces. J Mol Liq 249:980–990. CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesFederal University of TechnologyAkureNigeria
  2. 2.Faculty of Military ScienceStellenbosch UniversitySaldanhaSouth Africa
  3. 3.Department of Mathematics, College of SciencesKing Khalid UniversityAbhaSaudi Arabia

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