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On the Motion of Non-Newtonian Eyring–Powell Fluid Conveying Tiny Gold Particles Due to Generalized Surface Slip Velocity and Buoyancy

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Abstract

In the painting industry, space science and biomedical science, the nature of relaxation in the flow of non-Newtonian fluid (i.e. blood) containing gold (Ag) suits the characteristics of Eyring–Powell fluid flow induced by generalized surface slip velocity and buoyancy. However, flow of various non-Newtonian fluids on the horizontal surface of a slanted paraboloid of revolution objects (i.e. rocket, as in space science), over a bonnet of a car and over a pointed surface of an aircraft is of importance to experts in all these fields. In this article, the analysis of the motion within the thin layer formed on a horizontal object which is neither a perfect horizontal nor vertical and neither an inclined surface nor a cone/wedge is presented. The transformed governing equations which model the flow was non-dimenzionalized, parameterized and solved numerically using a well-known Runge–Kutta integration procedure along with shooting technique. The influence of increasing the magnitude of major parameters on the temperature distribution, local heat transfer rate, concentration of the fluid, local skin friction coefficient and velocity of the flow are illustrated graphically and discussed. Velocity slip parameter is found to be a decreasing function of temperature distribution across the flow. Heat transfer rate \((Nu_{x}Re_{x}^{-1/2})\) at the wall (\(\xi = 0\)) is an increasing function of velocity slip parameter. Maximum coefficient of concentration of homogeneous bulk fluid at the wall exists at larger values of the emerged velocity slip and volume fraction parameters.

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Abbreviations

a :

Dimensional concentration of the bulk fluid

b and A :

Thickness parameters

c :

Characteristics of Eyring–Powell

\(C_{f}\) :

Skin friction coefficient

\(D_{A},D_{B}\) :

Mass diffusivities

\(f(\eta )\) and \(F(\varsigma )\) :

Dimensionless velocity

\(g(\eta )\) and \(G(\xi )\) :

Concentration of the bulk fluid

\(G_r\) :

Buoyancy parameter

g :

Acceleration due to gravity

\(h(\eta )\) and \(H(\xi )\) :

Concentration of the catalyst

K :

Strength of homogeneous reaction

\(\kappa _{nf}\) :

Thermal conductivity of nanofluid

\(\kappa _{bf}\) :

Thermal conductivity of the base fluid

\(k^{*}\) :

Rosseland mean absorption coefficient

\(k_{1},k_{s}\) :

Chemical rate coefficients

m :

Velocity power index

\(Nu_{x}\) :

Local Nusselt number

\(P_{r}\) :

Prandtl number

\(q_{r}\) :

Radiative heat flux

R :

Radiation parameter

T :

Temperature of the fluid

\(T_{w}\) :

Wall temperature

\(T_{\infty }\) :

Free stream temperature

u :

Velocity component in x-direction

v :

Velocity component in y-direction

x :

Distance along the surface

y :

Distance normal to the surface

\(\alpha , \aleph \) :

Eyring–Powell fluid parameters

\(\beta _j\) :

Eyring–Powell parameter

\(\beta _{ep}\) :

Vol. coefficients of thermal expansion

\(\delta \) :

Ratio of diffusion coefficients

\(\eta \) :

Similarity variable \([\chi ,\infty )\)

\(\theta (\eta )\) and \(\Theta (\xi )\) :

Dimensionless temperature

\(\vartheta _{nf}\) :

Kinematics viscosity of the nanofluid

\(\hbar \) :

Generalized slip parameter

\(\Lambda \) :

Strength of the heterogeneous reaction

\(\theta _w\) :

Temperature parameter

\(\mu _{nf}\) :

Viscosity of Erying–Powell nanofluid

\(\mu _{bf}\) :

Viscosity of basefluid (blood)

\(\rho \) :

Fluid density

\(\ell \) :

Dimensional concentration of the catalyst

\(\sigma ^*\) :

Stefan–Boltzmann constant

\(\chi \) :

Wall thickness parameter

\(\psi (x,y)\) :

Stream function

\(\xi \) :

Dimensionless distance \([0,\infty )\)

References

  1. Anderson, J.D.: Ludwig Prandtl’s boundary layer. Phys. Today AIP 58, 42–48 (2005)

    Article  Google Scholar 

  2. Sakiadis, B.C.: Boundary layer behaviour on continuous solid surfaces. AIChE J. 7(1), 26–28 (1961)

    Article  Google Scholar 

  3. Sakiadis, B.C.: Boundary layer behavior on continuous solid surfaces: II, the boundary layer on a continuous flat surface. AIChE J. 17, 221–225 (1961)

    Article  Google Scholar 

  4. Boundary layer (2015). https://www.grc.nasa.gov/WWW/k-12/airplane/boundlay.html

  5. Sakiadis, B.C.: Boundary layer behavior on continuous solid surfaces: the boundary layer on a continuous flat surface. Am. Inst. Chem. Eng. (AIChE) 7, 221–225 (1961)

    Article  Google Scholar 

  6. Moore, D.W.: The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161 (1963). https://doi.org/10.1017/S0022112063000665

    Article  MathSciNet  MATH  Google Scholar 

  7. Murphy, J.S.: Some effects of surface curvature on laminar boundary-layer flow. J. Aeronaut. Sci. 20(5), 338–344 (1953)

    Article  MathSciNet  Google Scholar 

  8. Naramgari, S., Sulochana, C.: MHD flow over a permeable stretching/shrinking sheet of a nanofluid with suction/injection. Alex. Eng. J. 55(2), 819–827 (2016). https://doi.org/10.1016/j.aej.2016.02.001

    Article  Google Scholar 

  9. Sowerby, L., Cooke, J.: The flow of fluid along corners and edges. Q. J. Mech. Appl. Math. 6(1), 50–70 (1953). https://doi.org/10.1093/qjmam/6.1.50

    Article  MathSciNet  MATH  Google Scholar 

  10. Sawchuk, S.P., Zamir, M.: Boundary layer on a circular cylinder in axial flow. Int. J. Heat Fluid Flow 13(2), 184–188 (1992). https://doi.org/10.1016/0142-727x(92)90026-6

    Article  Google Scholar 

  11. Sulochanaa, C., Ashwinkumara, G.P., Sandeep, N.: Transpiration effect on stagnation-point flow of a Carreau nanofluid in the presence of thermophoresis and Brownian motion. Alex. Eng. J. 55(2), 1151–1157 (2016). https://doi.org/10.1016/j.aej.2016.03.031

    Article  Google Scholar 

  12. Animasaun, I.L.: Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity. J. Egypt. Math. Soc. 25(1), 79–85 (2016). https://doi.org/10.1016/j.joems.2016.06.007

    Article  MathSciNet  MATH  Google Scholar 

  13. Benazir, A.J., Sivaraj, R., Rashidi, M.M.: Comparison between Casson fluid flow in the presence of heat and mass transfer from a vertical cone and flat plate. J. Heat Transf. ASME 138(11), 112005 (2016). https://doi.org/10.1115/1.4033971

    Article  Google Scholar 

  14. McLachlan, R.I.: The boundary layer on a finite flat plate. Phys. Fluids A 3(2), 341–348 (1991). https://doi.org/10.1063/1.858143

    Article  MATH  Google Scholar 

  15. Lakshmi, K.B., Kumar, K.A., Reddy, J.V.R., Sugunamma, V.: Influence of nonlinear radiation and cross diffusion on MHD flow of Casson and Walters-B nanofluids past a variable thickness sheet. J. Nanofluids 8(1), 73–83 (2019). https://doi.org/10.1166/jon.2019.1564

    Article  Google Scholar 

  16. Kumara, K.A., Reddy, J.V.R., Sugunamma, V., Sandeep, N.: Magnetohydrodynamic Cattaneo–Christov flow past a cone and a wedge with variable heat source/sink. Alex. Eng. J. 57(1), 435–443 (2018). https://doi.org/10.1016/j.aej.2016.11.013

    Article  Google Scholar 

  17. Ramadevi, B., Sugunamma, V., Kumar, K.A., Reddy, J.V.R.: MHD flow of Carreau fluid over a variable thickness melting surface subject to Cattaneo–Christov heat flux. Multidiscip. Model. Mater. Struct. (2018). https://doi.org/10.1108/mmms-12-2017-0169

  18. Kumar, K.A., Reddy, J.V.R., Sugunamma, V., Sandeep, N.: Impact of cross diffusion on MHD viscoelastic fluid flow past a melting surface with exponential heat source. Multidiscip. Model. Mater. Struct. (2018). https://doi.org/10.1108/mmms-12-2017-0151

    Article  Google Scholar 

  19. Taylor, R., Coulombe, S., Otanicar, T., Phelan, P., Gunawan, A., Lv, W., Tyagi, H.: Small particles, big impacts: a review of the diverse applications of nanofluids. J. Appl. Phys. 113(1), 1 (2013). https://doi.org/10.1063/1.4754271

    Article  Google Scholar 

  20. Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. Am. Soc. Mech. Eng. 128(3), 240 (2006). https://doi.org/10.1115/1.2150834

    Article  Google Scholar 

  21. Haroun, N.A., Sibanda, P., Mondal, S., Motsa, S.S., Rashidi, M.M.: Heat and mass transfer of nanofluid through an impulsively vertical stretching surface using the spectral relaxation method. Bound. Value Probl. 2015(1), 161 (2015)

    Article  MathSciNet  Google Scholar 

  22. Oyelakin, I.S., Mondal, S., Sibanda, P.: Unsteady Casson nanofluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions. Alex. Eng. J. 55(2), 1025–1035 (2016)

    Article  Google Scholar 

  23. Sithole, H.M., Mondal, S., Sibanda, P., Motsa, S.S.: An unsteady MHD Maxwell nanofluid flow with convective boundary conditions using spectral local linearization method. Open Phys. 15(1), 637–646 (2017)

    Article  Google Scholar 

  24. Koriko, O.K., Animasaun, I.L., Mahanthesh, B., Saleem, S., Sarojamma, G., Sivaraj, R.: Heat transfer in the flow of blood-gold Carreau nanofluid induced by partial slip and buoyancy. Heat Transf. Asian Res. 47(6), 806–823 (2018). https://doi.org/10.1002/htj.21342

    Article  Google Scholar 

  25. Powell, R.E., Eyring, H.: Mechanisms for the relaxation theory of viscosity. Nature 154(3909), 427–428 (1944). https://doi.org/10.1038/154427a0

    Article  Google Scholar 

  26. Ziegenhagen, A.: The very slow flow of a Powell–Eyring fluid around a sphere. Appl. Sci. Res. Sect. A 14(1), 43–56 (1965). https://doi.org/10.1007/bf00382230

    Article  Google Scholar 

  27. Sirohi, V., Timol, M.G., Kalthia, N.L.: Powell–Eyring model flow near an accelerated plate. Fluid Dyn. Res. 2(3), 193–204 (1987). https://doi.org/10.1016/0169-5983(87)90029-3

    Article  Google Scholar 

  28. Malek, J.: Some Frequently Used Models for Non-Newtonian Fluids. Mathematical Institute Charles University, Prague (2011)

    Google Scholar 

  29. Hayat, T., Farooq, M., Alsaedi, A., Iqbal, Z.: Melting heat transfer in the stagnation point flow of Powell–Eyring fluid. J. Thermophys. Heat Transf. 27(4), 761–766 (2013). https://doi.org/10.2514/1.T4059

    Article  Google Scholar 

  30. Khan, N.A., Aziz, S., Khan, N.A.: MHD flow of Powell–Eyring fluid over a rotating disk. J. Taiwan Inst. Chem. Eng. 45(6), 2859–2867 (2014). https://doi.org/10.1016/j.jtice.2014.08.018

    Article  Google Scholar 

  31. Nadeem, S., Saleem, S.: Mixed convection flow of Erying–Powell fluid along a rotating cone. Results Phys. 4, 54–62 (2014). https://doi.org/10.1016/j.rinp.2014.03.004

    Article  Google Scholar 

  32. Malik, M.Y., Khan, I., Hussain, A., Salahuddin, T.: Mixed convection flow of MHD Eyring–Powell nanofluid over a stretching sheet: a numerical study. AIP Adv. 5, 117118 (2015). https://doi.org/10.1063/1.4935639

    Article  Google Scholar 

  33. Sugunamma, V., Sandeep, N., Ramana Reddy, J.V., Mohan Krishna, P.: Influence of non uniform heat source/sink on Powell–Erying fluid past an inclined stretching sheet with suction/injection. Math. Theory Model. 6(3), 51–60 (2016)

    Google Scholar 

  34. Agbaje, T.M., Mondal, S., Motsa, S.S., Sibanda, P.: A numerical study of unsteady non-Newtonian Powell–Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation. Alex. Eng. J. 56(1), 81–91 (2017). https://doi.org/10.1016/j.aej.2016.09.006

    Article  Google Scholar 

  35. Abegunrin, O.A., Animasaun, I.L., Sandeep, N.: Insight into the boundary layer flow of non-Newtonian Eyring–Powell fluid due to catalytic surface reaction on an upper horizontal surface of a paraboloid of revolution. Alex. Eng. J. (2017). https://doi.org/10.1016/j.aej.2017.05.018

    Article  Google Scholar 

  36. Chaudhary, M.A., Merkin, J.H.: A simple isothermal model for homogeneous–heterogeneous reactions in boundary layer flow. I Equal diffusivities. Fluid Dyn. Res. 16, 311–333 (1995). https://doi.org/10.1016/0169-5983(95)00015-6

    Article  MathSciNet  MATH  Google Scholar 

  37. Animasaun, I.L., Raju, C.S.K., Sandeep, N.: Unequal diffusivities case of homogeneous–heterogeneous reactions within viscoelastic fluid flow in the presence of induced magnetic-field, and nonlinears thermal radiation. Alex. Eng. J. 55(2), 1595–1606 (2016). https://doi.org/10.1016/j.aej.2016.01.018

    Article  Google Scholar 

  38. Makinde, O.D., Animasaun, I.L.: Bioconvection in MHD nanofluidflow 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 (2016). https://doi.org/10.1016/j.ijthermalsci.2016.06.003

    Article  Google Scholar 

  39. Imtiaz, M., Hayat, T., Alsaedi, A.: MHD convective flow of Jeffry fluid due to a curved stretching surface with homogeneous–heterogeneous reactions. PLoS ONE 11(9), e0161641 (2016). https://doi.org/10.1371/journal.pone.0161641

    Article  Google Scholar 

  40. Koriko, O.K., Animasaun, I.L.: 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. 8(26), 1–13 (2017). https://doi.org/10.5098/hmt.8.26

    Article  Google Scholar 

  41. Lee, L.L.: Boundary layer over a thin Needle. Phys. Fluids 10, 820 (1967). https://doi.org/10.1063/1.1762194

    Article  MATH  Google Scholar 

  42. Davis, R.T., Werle, M.J.: Numerical solutions for laminar incompressible flow past a paraboloid of revolution. AIAA J. 10(9), 1224–1230 (1972). https://doi.org/10.2514/3.50354

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Animasaun, I.L.: 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. Alex. Eng. J. 55(3), 2375–2389 (2016). https://doi.org/10.1016/j.aej.2016.04.030

    Article  Google Scholar 

  45. Ajayi, T.M., Omowaye, A.J., Animasaun, I.L.: 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 (2017). https://doi.org/10.1155/2017/1697135

    Article  MathSciNet  Google Scholar 

  46. Abegunrin, O.A., Okhuevbie, S.O., Animasaun, I.L.: Comparison between the flow of two non-Newtonian fluids over an upper horizontal surface of paraboloid of revolution: boundary layer analysis. Alex. Eng. J. 55(3), 1915–1929 (2016). https://doi.org/10.1016/j.aej.2016.08.002

    Article  Google Scholar 

  47. Steff, J.F.: Rheological Methods in Food Process Engineering, 2nd edn. Freeman Press, East Lansing (1996)

    Google Scholar 

  48. Ara, A., Khan, N.A., Khan, H., Sultan, F.: Radiation effect on boundary layer flow of an Erying–Powell fluid over an exponentially shrinking sheet. Ain Shams Eng. J. 5, 1337–1342 (2014)

    Article  Google Scholar 

  49. Lynch, D.T.: Chaotic behavior of reaction systems: mixed cubic and quadratic autocatalysis. Chem. Eng. Sci. 47(17–18), 4435–4444 (1992). https://doi.org/10.1016/0009-2509(92)85121-Q

    Article  Google Scholar 

  50. Mintsa, H.A., Nguyen, C.T., Roy, G.: New temperature dependent thermal conductivity data of water based nanofluids. In: Proceedings of the 5th IASME/WSEAS int. conference on heat transfer, thermal engineering and environment, vol 290, Athens, Greece, pp. 25–27 (2007)

  51. Michaelides, E.E.: Transport properties of nanofluids. A critical review. J. Non Equilib. Thermodyn. 38(1), 1–79 (2013). https://doi.org/10.1515/jnetdy-2012-0023

    Article  MATH  Google Scholar 

  52. Wang, X., Xu, X., Choi, S.U.S.: Thermal conductivity of nanoparticle-fluid mixture. J. Thermophys. Heat Transf. 13(4), 474–480 (1999). https://doi.org/10.2514/2.6486

    Article  Google Scholar 

  53. Motsa, S.S., Haroun, N.A., Sibanda, P., Mondal, S.: On unsteady MHD mixed convection in a nanofluid due to a stretching/shrinking surface with suction/injection using the spectral relaxation method. Bound. Value Probl. 24 (2015). https://doi.org/10.1186/s13661-015-0289-5

  54. Hatami, M., Hatami, J., Ganji, D.D.: Computer simulation of MHD blood conveying gold nanoparticles as a third grade non-Newtonian nanofluid in a hollow porous vessel. Comput. Methods Programs Biomed. 113(2), 632–641 (2014). https://doi.org/10.1016/j.cmpb.2013.11.001

    Article  Google Scholar 

  55. Thompson, P.A., Troian, S.M.: A general boundary condition for liquid flow at solid surfaces. Nature 389(6649), 360–362 (1997)

    Article  Google Scholar 

  56. Aziz, A.A.: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1064–1068 (2009)

    Article  Google Scholar 

  57. Grosan, T., Revnic, C., Pop, I.: Blasius problem with generalized surface slip velocity. J. Appl. Fluid Mech. 9(4), 1641–1644 (2016)

    Google Scholar 

  58. Na, T.Y.: Computational Methods in Engineering Boundary Value Problems, p. 1979. Academic Press, New York (2009)

    Google Scholar 

  59. Aljoufi, M.D., Ebaid, A.: Effect of a convective boundary condition on boundary layer slip flow and heat transfer over a stretching sheet in view of the exact solution. J. Theor. Appl. Mech. 46(4), 85–95 (2016). https://doi.org/10.1515/jtam-2016-0022

    Article  MathSciNet  Google Scholar 

  60. Koriko, O.K., Animasaun, I.L., Gnaneswara Reddy, M., Sandeep, N.: Scrutinization of thermal stratification, nonlinear thermal radiation and quartic autocatalytic chemical reaction effects on the flow of three-dimensional Eyring-Powell alumina-water nanofluid. Multidiscip. Model. Mater. Struct. 14(2), 261–283 (2018). https://doi.org/10.1108/MMMS-08-2017-0077

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

    I. L. Animasaun & O. K. Koriko

  2. Department of Mathematics, Christ University, Bangalore, 560058, India

    B. Mahanthesh

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Eyring–Powell nanofluid due to generalized surface slip velocity.

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Animasaun, I.L., Mahanthesh, B. & Koriko, O.K. On the Motion of Non-Newtonian Eyring–Powell Fluid Conveying Tiny Gold Particles Due to Generalized Surface Slip Velocity and Buoyancy. Int. J. Appl. Comput. Math 4, 137 (2018). https://doi.org/10.1007/s40819-018-0571-1

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