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Transient Analysis of Third-Grade Viscoelastic Nanofluid Flow External to a Heated Cylinder with Buoyancy Effects

  • Research Article--Mechanical Engineering
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Abstract

Nanotechnology is rapidly embracing numerous areas of manufacturing and process engineering. New types of nanomaterials are being exploited to improve, for example, coating integrity, anti-corrosion characteristics and other features of fabricated components. Motivated by these developments, in the current study a mathematical model is developed for unsteady free-convective laminar flow of third-grade viscoelastic fluid (doped with nanoparticles) from a semi-infinite vertical isothermal cylinder, as a model of thermal coating flow of a pipe geometry. Non-Newtonian behavior is simulated with the thermodynamically robust third-grade Reiner–Rivlin model which accurately represents polymer fluids. Nanoscale effects are analyzed with the Buongiorno two-component nanofluid model. The governing equations comprise a set of highly coupled, nonlinear, multi-degree partial differential equations featuring viscoelastic and nanofluid parameters. An implicit Crank–Nicolson numerical scheme is implemented to solve the emerging nonlinear problem with appropriate initial and boundary conditions. Detailed graphical plots for velocity, temperature and nanoparticle volume fraction are presented for a range of different parameters (i.e., third-grade fluid parameter, Brownian motion parameter, thermophoretic parameter, buoyancy ratio parameter, Lewis number). Additionally, distributions of the heat transfer coefficient, skin friction and Sherwood number at the cylinder surface are visualized. Furthermore, streamlines, isotherms and nanoparticle volume fraction contour plots are included for variation of the third-grade parameter. Contour plots for the third-grade nanofluid flow are found to deviate significantly from those corresponding to Newtonian nanofluids. Validation of the numerical solutions with earlier studies is also included.

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Abbreviations

\( g^{\prime} \) :

Acceleration due to gravity

\( Gr \) :

Grashof number

\( Pr \) :

Prandtl number

\( C_{\text{p}} \) :

Specific heat at constant pressure

\( Sh \) :

Sherwood number

\( \bar{C}_{f} \) :

Dimensionless average momentum transport coefficient

\( \overline{Nu} \) :

Average heat transport coefficient

k :

Thermal conductivity

r o :

Radius of the cylinder

t :

Dimensionless time

t′:

Time

P :

Fluid pressure

T′:

Temperature

D B :

Coefficient of Brownian diffusion

D T :

Coefficient of thermophoresis diffusion

T :

Dimensionless temperature

tr:

Trace

x, r :

Axial and radial coordinates, respectively

u, v :

Velocity components in (x, r) coordinate system

X, R :

Dimensionless axial and radial coordinate

U, V :

Dimensionless velocity components in X, R directions, respectively

Nr:

Buoyancy ratio parameter

Nb:

Brownian motion parameter

Le :

Lewis number

Nt:

Thermophoretic parameter

β T :

Volumetric thermal expansion coefficient

α :

Thermal diffusivity

β :

Non-dimensional third-grade fluid parameter

ρ f :

Density of base fluid (i.e., third-grade fluid)

ρ p :

Density of nanoparticles

ψ :

Stream function

φ :

Dimensional volume fraction

μ :

Viscosity of the nanofluid

Θ:

Dimensionless volume fraction (nanoparticle species concentration)

ϑ :

Kinematic viscosity

f, g :

Grid levels in (X, R) coordinate system

w:

Wall conditions

∞:

Ambient conditions

h :

Time level

References

  1. Yang, Y.; Cheng, Y.F.: Mechanistic aspects of electrodeposition of Ni–Co–SiC composite nano-coating on carbon steel. Electrochim. Acta 109, 638–644 (2013)

    Article  Google Scholar 

  2. Dheeraj, P.R.; Patra, A.; Sengupta, S.; Das, S.; Das, K.: Synergistic effect of peak current density and nature of surfactant on microstructure, mechanical and electrochemical properties of pulsed electrodeposited Ni–Co–SiC nanocomposites. J. Alloys Compd. 729, 1093–1107 (2017)

    Article  Google Scholar 

  3. Song, G.; Xu, G.; Quan, Y.; Yuan, Q.; Davies, P.A.: Uniform design for the optimization of Al2O3 nanofilms produced by electrophoretic deposition. Surf. Coat. Technol. 286, 268–278 (2016)

    Article  Google Scholar 

  4. Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, D.A., Wang, H.P. (eds.) Developments and Applications of Non-Newtonian flows. American Society of Mechanical Engineers, New York (1995)

    Google Scholar 

  5. Zhang, H.; Zhang, H.; Tang, L.; Zhang, Z.; Gu, L.; Xu, Y.; Eger, C.: Wear-resistant and transparent acrylate-based coating with highly filled nanosilica particles. Tribol. Int. 43, 83–91 (2010)

    Article  Google Scholar 

  6. Satish, S.; Chandar Shekar, B.; Sathyamoorthy, R.: Nano polymer films by fast dip coating method for field effect transistor applications. Phys. Procedia 49, 166–176 (2013)

    Article  Google Scholar 

  7. Masuda, H.; Ebata, A.; Teramae, K.; Hishinuma, N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7(4), 227–233 (1993)

    Article  Google Scholar 

  8. Xuan, Y.; Li, Q.: Heat transfer enhancement of nanofluids. Int. J. Heat Fluid Flow 21(1), 58–64 (2000)

    Article  Google Scholar 

  9. Xuan, Y.; Roetzel, W.: Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf. 43(19), 3701–3707 (2000)

    Article  MATH  Google Scholar 

  10. Putra, N.; Roetzel, W.; Das, S.K.: Natural convection of nanofluids. Heat Mass Transf. 39(8–9), 775–784 (2003)

    Article  Google Scholar 

  11. Timofeeva, E.V.; Yu, W.; France, D.M.; Singh, D.; Routbort, J.L.: Nanofluids for heat transfer: an engineering approach. Nanoscale Res. Lett. 6, 182 (2011)

    Article  Google Scholar 

  12. Yu, W.; Xie, H.: A review on nanofluids: preparation, stability mechanisms, and applications. J. Nanomater. 2012, 1–17 (2012)

    Google Scholar 

  13. Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128(3), 240–250 (2006)

    Article  Google Scholar 

  14. Tiwari, R.K.; Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50(9–10), 2002–2018 (2007)

    Article  MATH  Google Scholar 

  15. Bég, O.A.: Nonlinear multi-physical laminar nanofluid bioconvection flows: models and computation. In: Sohail, A., Li, Z. (eds.) Computational Approaches in Biomedical Nano-engineering, pp. 113–145. Wiley, New York (2018)

    Google Scholar 

  16. Kuznetsov, A.V.; Nield, D.A.: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 49(2), 243–247 (2010)

    Article  Google Scholar 

  17. Sheremet, M.A.; Pop, I.; Shenoy, A.: Unsteady free convection in a porous open wavy cavity filled with a nanofluid using Buongiorno’s mathematical model. Int. Commun. Heat Mass Transf. 67, 66–72 (2015)

    Article  Google Scholar 

  18. Rohni, A.M.; Ahmad, S.; Ismail, A.I.M.; Pop, I.: Flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluid using Buongiorno’s model. Int. Commun. Heat Mass Transf. 43, 75–80 (2013)

    Article  Google Scholar 

  19. Rajesh, V.; Bég, O.A.; Mallesh, M.P.: Transient nanofluid flow and heat transfer from a moving vertical cylinder in the presence of thermal radiation: numerical study. Proc. Inst. Mech. Eng. Part N J. Nanoeng. Nanosyst. 230, 3–16 (2014)

    Google Scholar 

  20. Khan, M.I.; Ullah, S.; Hayata, T.; Waqas, M.; Khan, M.I.; Alsaedi, A.: Salient aspects of entropy generation optimization in mixed convection nanomaterial flow. Int. J. Heat Mass Transf. 126, 1337–1346 (2018)

    Article  Google Scholar 

  21. Prasad, V.R.; Gaffar, S.A.; Bég, O.A.: Non-similar computational solutions for free convection boundary-layer flow of a nanofluid from an isothermal sphere in a non-Darcy porous medium. J. Nanofluids 4, 203–213 (2015)

    Article  Google Scholar 

  22. Bég, O.A.; Mabood, F.; Islam, M.N.: Homotopy simulation of nonlinear unsteady rotating nanofluid flow from a spinning body. Int. J. Eng. Math. 2015, 1–15 (2015)

    Article  MATH  Google Scholar 

  23. Chamkha, A.J.; Rashad, A.M.; Aly, A.M.: Transient natural convection flow of a nanofluid over a vertical cylinder. Meccanica 48(1), 71–81 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Y.; Wu, X.; Yang, W.; Zhai, Y.; Xie, B.; Yang, M.: Aggregate of nanoparticles: rheological and mechanical properties. Nanoscale Res. Lett. 6, 114 (2011)

    Article  Google Scholar 

  25. Aoki, Y.; Hatano, A.; Watanabe, H.: Rheology of carbon black suspensions. I. Three types of viscoelastic behaviour. Rheol. Acta 42(3), 209–216 (2003)

    Google Scholar 

  26. Du, F.; Scogna, R.C.; Zhou, W.; Brand, S.; Fischer, J.E.; Winey, K.I.: Nanotube networks in polymer nanocomposites: rheology and electrical conductivity. Macromolecules 37(24), 9048–9055 (2004)

    Article  Google Scholar 

  27. Elias, L.; Fenouillot, F.; Majeste, J.C.; Alcouffe, P.; Cassagnau, P.: Immiscible polymer blends stabilized with nano-silica particles: rheology and effective interfacial tension. Polymer 49(20), 4378–4385 (2008)

    Article  Google Scholar 

  28. Chang, H.; Jwo, C.S.; Lo, C.H.; Tsung, T.T.; Kao, M.J.; Lin, H.M.: Rheology of CuO nanoparticle suspension prepared by ASNSS. Rev. Adv. Mater. Sci. 10, 128–132 (2005)

    Google Scholar 

  29. Latiff, N.A.; Uddin, M.J.; Bég, O.A.; Ismail, A.I.M.: Unsteady forced bioconvection slip flow of a micropolar nanofluid from a stretching/shrinking sheet. Proc Inst. Mech. Eng. Part N J. Nanomater. Nanoeng. Nanosyst. 230(4), 177–187 (2016)

    Google Scholar 

  30. Hayata, T.; Kiyani, M.Z.; Alsaedi, A.; Khan, M.I.; Ahmad, I.: Mixed convective three-dimensional flow of Williamson nanofluid subject to chemical reaction. Int. J. Heat Mass Transf. 127, 422–429 (2018)

    Article  Google Scholar 

  31. Uddin, M.J.; Bég, O.A.; Ghose, P.K.; Ismael, A.I.M.: Numerical study of non-Newtonian nanofluid transport in a porous medium with multiple convective boundary conditions and nonlinear thermal radiation effects. Int. J. Numer. Methods Heat Fluid Flow 26(5), 1–25 (2016)

    Article  Google Scholar 

  32. Rana, P.; Bhargava, R.; Bég, O.A.; Kadir, A.: Finite element analysis of viscoelastic nanofluid flow with energy dissipation and internal heat source/sink effects. Int. J. Appl. Comput. Math. 3, 1421–1447 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Amanulla, C.H.; Nagendra, N.; Reddy, M.S.N.; Rao, A.S.; Bég, O.A.: Mathematical study of non-Newtonian nanofluid transport phenomena from an isothermal sphere. Front. Heat Mass Transf. 8(29), 1–13 (2017)

    Google Scholar 

  34. Prakash, J.; Siva, E.P.; Tripathi, D.; Kuharat, S.; Bég, O.A.: Peristaltic pumping of magnetic nanofluids with thermal radiation and temperature-dependent viscosity effects: modelling a solar magneto-biomimetic nanopump. Renew. Energy 133, 1308–1326 (2019)

    Article  Google Scholar 

  35. Fosdick, R.L.; Rajagopal, K.R.: Thermodynamics and stability of fluids of third grade. Proc. R. Soc. Lond. Ser. A 369, 351–377 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Bird, R.B.; Armstrong, R.C.; Hassager, O.: Dynamics of Polymeric Liquids. Fluid Mechanics, vol. 1, 2nd edn. Wiley, New York (1987)

    Google Scholar 

  37. Javed, T.; Mustafa, I.: Slip effects on a mixed convection flow of a third-grade fluid near the orthogonal stagnation point on a vertical surface. J. Appl. Mech. Tech. Phys. 57, 527–536 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bég, O.A.; Takhar, H.S.; Bhargava, R.; Rawat, S.; Prasad, V.R.: Numerical study of heat transfer of a third-grade viscoelastic fluid in a non-Darcy porous media with thermophysical effects. Phys. Scr. 77, 065402–065413 (2008)

    Article  MATH  Google Scholar 

  39. Sahoo, B.: Hiemenz flow and heat transfer of a third grade fluid. Commun. Nonlinear Sci. Numer. Simul. 14(3), 811–826 (2009)

    Article  Google Scholar 

  40. Sahoo, B.; Do, Y.: Effect of slip on sheet-driven flow and heat transfer of a third grade fluid past a stretching sheet. Int. Commun. Heat Mass Transf. 37(8), 1064–1071 (2010)

    Article  Google Scholar 

  41. Aiyesimi, Y.M.; Jiya, M.; Olayiwola, R.O.; Wachin, A.A.: Mathematical analysis of convective flow of an unsteady magnetohydrodynamic (MHD) third grade fluid in a cylindrical channel. Am. J. Comput. Appl. Math. 6(2), 103–108 (2016)

    Google Scholar 

  42. Reddy, G.J.; Hiremath, A.; Kumar, M.: Computational modeling of unsteady third-grade fluid flow over a vertical cylinder: a study of heat transfer visualization. Results Phys. 8, 671–682 (2018)

    Article  Google Scholar 

  43. Reddy, G.J.; Hiremath, A.; Basha, H.; Narayanan, N.S.V.: Transient flow and heat transfer characteristics of non-Newtonian supercritical third-grade fluid (CO2) past a vertical cylinder. Int. J. Chem. React. Eng. 16(8), 1542–6580 (2018)

    Google Scholar 

  44. Gaffar, S.A.; Prasad, V.R.; Bég, O.A.; Khan, M.H.H.; Venkatadri, K.: Radiative and magnetohydrodynamics flow of third grade viscoelastic fluid past an isothermal inverted cone in the presence of heat generation/absorption. J. Braz. Soc. Mech. Sci. Eng. 40, 127–146 (2018)

    Article  Google Scholar 

  45. Farooq, U.; Hayat, T.; Alsaedi, A.; Liao, S.: Heat and mass transfer of two-layer flows of third-grade nano-fluids in a vertical channel. Appl. Math. Comput. 242, 528–540 (2014)

    MathSciNet  MATH  Google Scholar 

  46. Nadeem, S.; Saleem, S.: Analytical study of third grade fluid over a rotating vertical cone in the presence of nanoparticles. Int. J. Heat Mass Transf. 85, 1041–1048 (2015)

    Article  Google Scholar 

  47. Khan, W.A.; Culham, J.R.; Makinde, O.D.: Combined heat and mass transfer of third-grade nanofluids over a convectively-heated stretching permeable surface. Can. J. Chem. Eng. 93, 1880–1888 (2015)

    Article  Google Scholar 

  48. Qayyum, S.; Hayat, T.; Alsaedi, A.: Thermal radiation and heat generation/absorption aspects in third grade magneto-nanofluid over a slendering stretching sheet with Newtonian conditions. Physica B 537, 139–149 (2018)

    Article  Google Scholar 

  49. Hayat, T.; Ahmad, S.; Khan, M.I.; Alsaedi, A.: Modeling and analyzing flow of third grade nanofluid due to rotating stretchable disk with chemical reaction and heat source. Physica B 537, 116–126 (2018)

    Article  Google Scholar 

  50. Dunn, J.E.; Rajagopal, K.R.: Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci. 33, 689–729 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Fosdick, R.L.; Straughan, B.: Catastrophic instabilities and related results in a fluid of third grade. Int. J. Non Linear Mech. 16, 191 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sahoo, B.; Poncet, S.: Flow and heat transfer of a third-grade fluid past an exponentially stretching sheet with partial slip boundary condition. Int. J. Heat Mass Transf. 54(23–24), 5010–5019 (2011)

    Article  MATH  Google Scholar 

  53. Keimanesh, M.; Rashidi, M.M.; Chamkha, A.J.; Jafari, R.: Study of a third-grade non-Newtonian fluid flow between two parallel plates using the multi-step deferential transform method. Comput. Math. Appl. 62(8), 2871–2891 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Sajid, M.; Hayat, T.; Asghar, S.: Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet. Int. J. Heat Mass Transf. 50, 1723–1736 (2007)

    Article  MATH  Google Scholar 

  55. Hayat, T.; Nazar, H.; Imtiaz, M.; Alsaedi, A.; Ayub, M.: Axisymmetric squeezing flow of third grade fluid in presence of convective conditions. Chin. J. Phys. 55, 738–754 (2017)

    Article  Google Scholar 

  56. Hayat, T.; Mustafa, M.; Asghar, S.: Unsteady flow with heat and mass transfer of a third grade fluid over a stretching surface in the presence of chemical reaction. Nonlinear Anal. Real World Appl. 11, 3186–3197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Carnahan, B.; Luther, H.A.; Wilkes, J.O.: Applied Numerical Methods. Wiley, New York (1969)

    MATH  Google Scholar 

  58. Von Rosenberg, D.U.: Methods for the Numerical Solution of Partial Differential Equations. American Elsevier Publishing Company, New York (1969)

    MATH  Google Scholar 

  59. Rani, H.P.; Kim, C.N.: A numerical study on unsteady natural convection of air with variable viscosity over an isothermal vertical cylinder. Korean J. Chem. Eng. 27(3), 759–765 (2010)

    Article  Google Scholar 

  60. Vasu, B.; Gorla, R.S.R.; Bég, O.A.; Murthy, P.V.S.N.; Prasad, V.R.; Kadir, A.: Unsteady flow of a nanofluid over a sphere with non-linear Boussinesq approximation. AIAA J. Thermophys. Heat Transf. 33(2), 343–355 (2018)

    Article  Google Scholar 

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Acknowledgements

The first author Ashwini Hiremath wishes to thank DST-INSPIRE (Code No. IF160409) for the grant of research fellowship and to Central University of Karnataka for providing the research facilities. The authors appreciate greatly the comments of the reviewers which have served to improve the present work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Central University of Karnataka, Kalaburagi, India

    Ashwini Hiremath & G. Janardhana Reddy

  2. Fluid Mechanics, Aeronautical and Mechanical Engineering Department, School of Computing, Science and Engineering, University of Salford, Manchester, M54WT, UK

    O. Anwar Bég

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  1. Ashwini Hiremath
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  2. G. Janardhana Reddy
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  3. O. Anwar Bég
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Corresponding author

Correspondence to G. Janardhana Reddy.

Appendix

Appendix

The discretized finite difference equations for Eqs. (13)–(16) are as follows:

$$\begin{aligned} &\frac{{U_{f,g}^{h + 1} - U_{f - 1,g}^{h + 1} + U_{f,g}^{h} - U_{f - 1,g}^{h} }}{2\Delta X} \\& \quad + \frac{{V_{f,g}^{h + 1} - V_{f,g - 1}^{h + 1} + V_{f,g}^{h} - V_{f,g - 1}^{h} }}{2\Delta R} + V_{f,g}^{h + 1} \left( {JR} \right) = 0 \end{aligned}$$
(A.1)
$$ \begin{aligned} & \frac{{U_{f,g}^{h + 1} - U_{f,g}^{h} }}{\Delta t} + U_{f,g}^{h} \frac{{\left( {U_{f,g}^{h + 1} - U_{f - 1,g}^{h + 1} + U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{2\Delta X} + V_{f,g}^{h} \frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{4\Delta R} \\ & \quad = \frac{{T_{f,g}^{h} + T_{f,g}^{h + 1} }}{2} - {\text{Nr}}\frac{{\Theta_{f,g}^{h} + \Theta_{f,g}^{h + 1} }}{2} + JR\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{4\Delta R} \\ & \quad \quad + \frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{2\left( {\Delta R} \right)^{2} }} + \alpha_{1} \left[ {\frac{{\left( {U_{f,g - 2}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 2}^{h + 1} - U_{f,g - 2}^{h} + 2U_{f,g}^{h} - U_{f,g + 2}^{h} } \right)}}{{4\left( {\Delta R} \right)^{2} \left( {\Delta t} \right)}}} \right. \\ & \quad \quad + \frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g + 1}^{h } - U_{f,g - 1}^{h+1} + U_{f,g - 1}^{h} } \right)}}{{2\left( {\Delta R} \right)\left( {\Delta t} \right)}} + V_{f,g}^{h} \frac{{\left( {U_{f,g + 2}^{h + 1} - 2U_{f,g + 1}^{h + 1} + 2U_{f,g - 1}^{h + 1} - U_{f,g - 2}^{h + 1} } \right.\left. { + U_{f,g + 2}^{h} - 2U_{f,g + 1}^{h} + 2U_{f,g - 1}^{h} - U_{f,g - 2}^{h} } \right)}}{{4\left( {\Delta R} \right)^{3} }} \\ & \quad \quad + \frac{{\left( {V_{f,g + 1}^{h} - V_{f,g - 1}^{h} } \right)}}{2\Delta R}\frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }} \\ & \quad \quad + 3\frac{{\left( {U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)}}\frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{2\left( {\Delta R} \right)^{2} }} + \frac{{\left( {V_{f,g - 1}^{h} - 2V_{f,g}^{h} + V_{f,g + 1}^{h} } \right)}}{{4\left( {\Delta R} \right)^{2} }} \\ & \quad \quad \frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} + \frac{{\left( {U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{\Delta R}\frac{{\left( {U_{f,g + 1}^{h + 1} + U_{f,g + 1}^{h } - U_{f-1,g + 1}^{h+1} - U_{f-1,g + 1}^{h} - U_{f,g- 1}^{h+1} - U_{f,g - 1}^{h} + U_{f-1,g - 1}^{h+1} +U_{f-1,g - 1}^{h} } \right)}}{{2\left( {\Delta X} \right)\left( {\Delta R} \right)}} \\ & \quad \quad + \left( {JR} \right)V_{f,g}^{h} \frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{2\left( {\Delta R} \right)^{2} }} \\ & \quad \quad + \left( {JR} \right)U_{f,g}^{h} \frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f - 1,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f - 1,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f - 1,g + 1}^{h} - U_{f,g - 1}^{h} + U_{f - 1,g - 1}^{h} } \right) }}{{4\left( {\Delta R} \right)\left( {\Delta X} \right)}} \\ & \quad \quad + U_{f,g}^h\frac{{\left( {U_{f,g + 1}^{h + 1} + U_{f,g + 1}^h - U_{f - 1,g + 1}^{h + 1} - U_{f - 1,g + 1}^h - 2U_{f,g}^{h + 1} - 2U_{f,g}^h + 2U_{f - 1,g}^{h + 1} + 2U_{f - 1,g}^h + U_{f,g - 1}^{h + 1} + U_{f,g - 1}^h - U_{f - 1,g - 1}^{h + 1} - U_{f - 1,g - 1}^h} \right)}}{{2{{\left( {\Delta R} \right)}^2}\Delta X}} \\ & \quad \quad + 3\left( {JR} \right)\frac{{\left( {U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)}}\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{{4\left( {\Delta R} \right)}}\left. { + \frac{{ {JR} }}{8}\frac{{\left( {V_{f,g + 1}^{h} - V_{f,g - 1}^{h} } \right)}}{\Delta R}\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }}} \right] \\ & \quad \quad + \alpha_{2} \left[ {\left( {JR} \right)\frac{{\left( {V_{f,g + 1}^{h} - V_{f,g - 1}^{h} } \right)}}{\Delta R}\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{\Delta R}} \right. + \left( {JR} \right)\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{{2\left( {\Delta R} \right)}} \\ & \quad \quad \frac{{\left( {U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)}} + \frac{{\left( {U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)}}\frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }} + \frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{2\Delta R} \\ & \quad \quad \frac{{\left( {V_{f,g - 1}^{h} - 2V_{f,g}^{h} + V_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }} \\ & \quad \quad + \frac{{\left( {U_{f,g + 1}^h - U_{f,g - 1}^h} \right)}}{{2\left( {\Delta R} \right)}}\frac{{\left( {U_{f,g + 1}^{h + 1} + U_{f,g + 1}^h - U_{f - 1,g + 1}^{h + 1} - U_{f - 1,g + 1}^h - U_{f,g - 1}^{h + 1} - ~U_{f,g - 1}^h + U_{f - 1,g - 1}^{h + 1} + U_{f - 1,g - 1}^h} \right)}}{{\left( {\Delta X} \right)\left( {\Delta R} \right)}} + \frac{{\left( {V_{f,g + 1}^{h} - V_{f,g - 1}^{h} } \right)}}{{2\left( {\Delta R} \right)}} \\ & \quad \quad \left. {\frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }}} \right] \\ & \quad \quad + \beta \left[ {\left( {JR} \right)\left( {Gr} \right)^{2} \frac{{\left( {U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)^{3} }}{{4\left( {\Delta R} \right)^{3} }} + 3\left( {Gr} \right)^{2} \frac{{\left( {U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)^{2} }}{{4\left( {\Delta R} \right)^{2} }}\frac{{\left( {U_{f,g - 1}^{h + 1} - 2U_{f,g}^{h + 1} + U_{f,g + 1}^{h + 1} + U_{f,g - 1}^{h} - 2U_{f,g}^{h} + U_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }}} \right. \\ & \quad \quad + \frac{{\left( {U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)^{2} }}{{\left( {\Delta R} \right)^{2} }}\frac{{\left( {U_{f - 1,g}^{h} - 2U_{f,g}^{h} + U_{f + 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)^{2} }} + \frac{1}{8}\frac{{\left( {U_{f,g}^{h} - U_{f - 1,g}^{h} } \right)}}{{\left( {\Delta X} \right)}}\frac{{\left( {U_{f,g + 1}^{h} - U_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ & \quad \quad \left. {\frac{{\left( {U_{f,g + 1}^{h + 1} - U_{f - 1,g + 1}^{h + 1} - U_{f,g - 1}^{h + 1} + U_{f - 1,g - 1}^{h + 1} + U_{f,g + 1}^{h} - U_{f - 1,g + 1}^{h} - U_{f,g - 1}^{h} + U_{f - 1,g - 1}^{h} } \right)}}{{\left( {\Delta X} \right)\left( {\Delta R} \right)}}} \right] \\ \end{aligned} $$
(A.2)
$$ \begin{aligned} & \frac{{T_{f,g}^{h + 1} - T_{f,g}^{h} }}{\Delta t} + U_{f,g}^{h} \frac{{\left( {T_{f,g}^{h + 1} - T_{f - 1,g}^{h + 1} + T_{f,g}^{h} - T_{f - 1,g}^{h} } \right)}}{2\Delta X} \\ & \quad+ V_{f,g}^{h} \frac{{\left( {T_{f,g + 1}^{h + 1} - T_{f,g - 1}^{h + 1} + T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{4\Delta R} \\ & \quad = \frac{1}{2Pr }\frac{{\left( {T_{f,g - 1}^{h + 1} - 2T_{f,g}^{h + 1} + T_{f,g + 1}^{h + 1} + T_{f,g - 1}^{h} - 2T_{f,g}^{h} + T_{f,g + 1}^{h} } \right)}}{{\left( {\Delta R} \right)^{2} }} \\ & \quad \quad + \frac{JR}{4Pr }\frac{{\left( {T_{f,g + 1}^{h + 1} - T_{f,g - 1}^{h + 1} + T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ & \quad+ \frac{{\text{Nb}}}{8}\frac{{\left( {\Theta_{f,g + 1}^{h} - \Theta_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ & \quad \quad \times\frac{{\left( {T_{f,g + 1}^{h + 1} - T_{f,g - 1}^{h + 1} + T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ & \quad+ \frac{{\text{Nt}}}{8}\frac{{\left( {T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}}\frac{{\left( {T_{f,g + 1}^{h + 1} - T_{f,g - 1}^{h + 1} + T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ \end{aligned} $$
(A.3)
$$ \begin{aligned} & \frac{{\Theta_{f,g}^{h + 1} - \Theta_{f,g}^{h} }}{\Delta t} + U_{f,g}^{h} \frac{{\left( {\Theta_{f,g}^{h + 1} - \Theta_{f - 1,g}^{h + 1} + \Theta_{f,g}^{h} - \Theta_{f - 1,g}^{h} } \right)}}{2\Delta X} \\ &\quad + V_{f,g}^{h} \frac{{\left( {\Theta_{f,g + 1}^{h + 1} - \Theta_{f,g - 1}^{h + 1} + \Theta_{f,g + 1}^{h} - \Theta_{f,g - 1}^{h} } \right)}}{4\Delta R} \\ & \quad = \frac{1}{{\left( {Le} \right)}}\frac{{\left( {\Theta_{f,g - 1}^{h + 1} - 2\Theta_{f,g}^{h + 1} + \Theta_{f,g + 1}^{h + 1} + \Theta_{f,g - 1}^{h} - 2\Theta_{f,g}^{h} + \Theta_{f,g + 1}^{h} } \right)}}{{2\left( {\Delta R} \right)^{2} }} \\ &\quad + \frac{JR}{{\left( {Le} \right)}}\frac{{\left( {\Theta_{f,g + 1}^{h + 1} - \Theta_{f,g - 1}^{h + 1} + \Theta_{f,g + 1}^{h} - \Theta_{f,g - 1}^{h} } \right)}}{{4\left( {\Delta R} \right)}} \\ & \quad \quad + \frac{{\left( {{\text{Nt}}} \right)}}{{\left( {Le} \right)\left( {{\text{Nb}}} \right)}}\frac{{\left( {T_{f,g - 1}^{h + 1} - 2T_{f,g}^{h + 1} + T_{f,g + 1}^{h + 1} + T_{f,g - 1}^{h} - 2T_{f,g}^{h} + T_{f,g + 1}^{h} } \right)}}{{2\left( {\Delta R} \right)^{2} }} \\ & \quad \quad + \frac{{\left( {JR} \right)\left( {{\text{Nt}}} \right)}}{{4\left( {Le} \right)\left( {{\text{Nb}}} \right)}} \\ &\quad \times \frac{{ \left( {T_{f,g + 1}^{h + 1} - T_{f,g - 1}^{h + 1} + T_{f,g + 1}^{h} - T_{f,g - 1}^{h} } \right)}}{{\left( {\Delta R} \right)}} \\ \end{aligned} $$
(A.4)

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Hiremath, A., Reddy, G.J. & Bég, O.A. Transient Analysis of Third-Grade Viscoelastic Nanofluid Flow External to a Heated Cylinder with Buoyancy Effects. Arab J Sci Eng 44, 7875–7893 (2019). https://doi.org/10.1007/s13369-019-03933-4

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