Thermal entry flow problem for Giesekus fluid inside an axis-symmetric tube through isothermal wall condition: a comparative numerical study between exact and approximate solution

Muhammad Waris Saeed Khan; Nasir Ali

doi:10.1515/zna-2021-0098

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Thermal entry flow problem for Giesekus fluid inside an axis-symmetric tube through isothermal wall condition: a comparative numerical study between exact and approximate solution

Muhammad Waris Saeed Khan and Nasir Ali

Published/Copyright: August 31, 2021

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From the journal Zeitschrift für Naturforschung A Volume 76 Issue 11

Abstract

The thermal entry flow problem also known as the Graetz problem is investigated for a Giesekus fluid model. Both analytical (exact) and approximate solutions for velocity are obtained. The nondimensional pressure gradient is numerically obtained via the mean flow rate relation. The energy equation along with the Giesekus fluid velocity is analytically solved for the constant wall temperature case by using the classical separation of variable method. This method transforms the energy equation into a Sturm–Liouville (SL) boundary value problem. The MATLAB solver bvp5c is employed to compute the eigenvalues and the related eigenfunctions numerically. The impact of mobility parameter and Weissenberg number on local Nusselt number, mean temperature, and average Nusselt number is discussed and displayed graphically. It is also found that the presence of the Weissenberg number elevates the Nusselt numbers. Further, the presence of the mobility parameter of the Giesekus fluid model delays the prevalence fully developed conditions in both entrance and fully developed regions. The comparison between approximate and exact solution is also presented. It reveals that both solutions have an exact match with each other for smaller values of mobility parameter and Weissenberg number. However, there is a deviation for larger values of both parameters.

Keywords: bvp5c; Giesekus fluid; Graetz problem; mobility parameter; Sturm–Liouville boundary value problem; Weissenberg number

Corresponding author: Muhammad Waris Saeed Khan, Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan, E-mail: waris.saeed88@gmail.com

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors have no conflict of interest regarding this manuscript.

References

[1] H. Giesekus, “A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility,” J. Non-Newtonian Fluid Mech., vol. 11, pp. 69–109, 1982. https://doi.org/10.1016/0377-0257(82)85016-7.Search in Google Scholar

[2] H. Giesekus, “J Stressing behaviour in simple shear flow as predicted by a new constitutive model for polymer fluids,” Non-Newtonian Fluid Mech., vol. 12, pp. 367–374, 1983. https://doi.org/10.1016/0377-0257(83)85009-5.Search in Google Scholar

[3] L. M. Quinzani, G. H. McKinley, R. A. Brown, and R. C. Armstrong, “Modeling the rheology of polyisobutylene solutions,” J. Rheol., vol. 34, no. 5, p. 705, 1990. https://doi.org/10.1122/1.550148.Search in Google Scholar

[4] T. Holz, P. Fischer, and H. Rehage, “Shear relaxation in the nonlinear-viscoelastic regime of a Giesekus fluid,” J. Non-Newtonian Fluid Mech., vol. 88, nos 1-2, pp. 133–148, 1999. https://doi.org/10.1016/s0377-0257(99)00016-6.Search in Google Scholar

[5] G. H. McKinley, “Using filament stretching rheometry to predict strand formation and processability in adhesives and other non-Newtonian fluids,” J. Non-Newtonian Fluid Mech., vol. 39, pp. 321–337, 2000.10.1007/s003970000072Search in Google Scholar

[6] R. B. Bird and J. M. Weist, “Constitutive equations for polymeric liquids,” Annu. Rev. Fluid Mech., vol. 27, pp. 169–193, 1995. https://doi.org/10.1146/annurev.fl.27.010195.001125.Search in Google Scholar

[7] M. Turkyilmazoglu, “Exact solutions for two-dimensional laminar flow over a continuously stretching or shrinking sheet in an electrically conducting quiescent couple stress fluid,” Int. J. Heat Mass Tran., vol. 72, pp. 1–8, 2014. https://doi.org/10.1016/j.ijheatmasstransfer.2014.01.009.Search in Google Scholar

[8] M. Turkyilmazoglu, “Thermal radiation effects on the time-dependent MHD permeable flow having variable viscosity,” Int. J. Thermal sciences, vol. 50, pp. 88–96, 2011. https://doi.org/10.1016/j.ijthermalsci.2010.08.016.Search in Google Scholar

[9] L. Graetz, “Uber die Warmeleitungsfahigheit von Flus- singkeiten, part 1,” Annalen der Physik und Chemie, vol. 18, pp. 79–94, 1883, part 2, vol. 25, pp. 337–357, 1885.10.1002/andp.18852610702Search in Google Scholar

[10] W. Nusselt, “Die abhangigkeit der warmeubergangszahl von der rohrlange (The dependence of the heat transfer coefficient on the tube length),” Z. ver. Deut. Ing., vol. 54, pp. 1154–1158, 1910.Search in Google Scholar

[11] J. R. Sellars, M. Tribus, and J. S. Klein, “Heat transfer to laminar flow in a round tube or flat conduit the Graetz problem extended,” Transactions of the ASME, vol. 78, pp. 441–448, 1956.10.21236/ADA280848Search in Google Scholar

[12] B. C. Lyche and R. B. Bird, “The Graetz–Nusselt problem for a power-law non-Newtonian fluid,” Chem. Eng. Sci., vol. 6, p. 35, 1956. https://doi.org/10.1016/0009-2509(56)80008-0.Search in Google Scholar

[13] R. D. Cess and E. C. Schaffer, “Heat transfer to laminar flow between parallel plates with a prescribed heat flux,” Appl. Sci. Res., vol. A8, p. 339, 1959. https://doi.org/10.1007/bf00411758.Search in Google Scholar

[14] R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, 3rd ed, New York, Academic Press, 1978.Search in Google Scholar

[15] R. K. Shah and M. S. Bhatti, “Laminar convective heat transfer in ducts,” in Handbook of Single-Phase Convective Heat Transfer, 2nd ed, S. Kakac, R. K. Shah, and W. Aung, Eds., New York, Wiley, 1987. Chapter 3.Search in Google Scholar

[16] P. J. Oliveira, P. M. Coelho, and F. T. Pinho, “The Graetz problem with viscous dissipation for FENE-P fluids,” J. Non-Newtonian Fluid Mech., vol. 121, pp. 69–72, 2004. https://doi.org/10.1016/j.jnnfm.2004.04.005.Search in Google Scholar

[17] P. M. Coelho, F. T. Pinho, and P. J. Oliveira, “Thermal entry flow for a viscoelastic fluid, the Graetz problem for the PTT model,” Int. J. Heat Mass Tran., vol. 46, pp. 3865–3880, 2003. https://doi.org/10.1016/s0017-9310(03)00179-0.Search in Google Scholar

[18] M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.Search in Google Scholar

[19] A. Filali, L. Khezzar, D. Siginer, and Z. Nemouchi, “Graetz problem with non-linear viscoelastic fluids in non-circular tubes,” Int. J. Therm. Sci., vol. 61, p. 50e60, 2012. https://doi.org/10.1016/j.ijthermalsci.2012.06.011.Search in Google Scholar

[20] M. Barışık, A. G. Yazıcıoğlu, B. Çetin, and S. Kakaç, “Analytical solution of thermally developing microtube heat transfer including axial conduction, viscous dissipation, and rarefaction effects,” Int. Commun. Heat Mass Tran., vol. 67, pp. 81–88, 2015.10.1016/j.icheatmasstransfer.2015.05.004Search in Google Scholar

[21] M. F. Letelier, C. B. Hinojosa, and D. A. Siginer, “Analytical solution of the Graetz problem for non-linear viscoelastic fluids in tubes of arbitrary cross-section,” Int. J. Therm. Sci., vol. 111, pp. 369–378, 2017. https://doi.org/10.1016/j.ijthermalsci.2016.05.034.Search in Google Scholar

[22] N. Ali and M. W. S. Khan, “The Graetz problem for the Ellis fluid model,” Z. Naturforsch., A, vol. 74, pp. 15–24, 2019. https://doi.org/10.1515/zna-2018-0410.Search in Google Scholar

[23] M. Norouzi, S. Z. Daghighi, and O. Anwer Beg, “Exact analysis of heat convection of viscoelastic FENE-P fluids through isothermal slits and tubes,” Mecanica, vol. 53, pp. 817–831, 2018. https://doi.org/10.1007/s11012-017-0782-2.Search in Google Scholar

[24] M. W. S. Khan and N. Ali, “Thermal entry flow of power-law fluid through ducts with homogeneous slippery wall (s) in the presence of viscous dissipation,” Int. Commun. Heat Mass Tran., vol. 120, p. 105041, 2021.10.1016/j.icheatmasstransfer.2020.105041Search in Google Scholar

[25] M. W. S. Khan and N. Ali, “Theoretical analysis of thermal entrance problem for blood flow: an extension of classical Graetz problem for Casson fluid model using generalized orthogonality relations,” Int. Commun. Heat Mass Tran., vol. 108, p. 104314, 2019. https://doi.org/10.1016/j.icheatmasstransfer.2019.104314.Search in Google Scholar

[26] M. Azari, A. Sadeghi, and S. Chakraborty, “Graetz problem for combined pressure-driven and electroosmotic flow in microchannels with distributed wall heat,” Flux, vol. 128, pp. 150–160, 2019. https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.106.Search in Google Scholar

[27] M. W. S. Khan, N. Ali, and Z. Asghar, “Thermal and rheological effects in a classical Graetz problem using a nonlinear Robertson‐Stiff fluid model,” Heat Transfer, vol. 50, pp. 2321–2338, 2021. https://doi.org/10.1002/htj.21980.Search in Google Scholar

[28] N. Suzzi and M. Lorenzini, “Viscous heating of a laminar flow in the thermal entrance region of a rectangular channel with rounded corners and uniform wall temperature,” Int. J. Therm. Sci., vol. 145, p. 106032, 2019. https://doi.org/10.1016/j.ijthermalsci.2019.106032.Search in Google Scholar

[29] N. Ali, M. W. S. Khan, and M. Sajid, “The Graetz-Nusselt problem for the curved channel using spectral collocation method,” Phys. Scripta, vol. 96, 2021, Art no. 055204. https://doi.org/10.1088/1402-4896/abe586.Search in Google Scholar

[30] M. W. S. Khan, N. Ali, and Z. Asghar, “Mathematical modelling of classical Graetz–Nusselt problem for axisymmetric tube and flat channel using the Carreau fluid model: a numerical benchmark study,” Z. Naturforsch., vol. 76, 2021. https://doi.org/10.1515/zna-2021-0042.Search in Google Scholar

[31] N. Ali, M. A. Javed, O. A. Bég, and T. Hayat, “Mathematical model for isothermal wire-coating from a bath of Giesekus viscoelastic fluid,” Chem. Eng. Commun., vol. 203, no. 10, pp. 1336–1348, 2016. https://doi.org/10.1080/00986445.2016.1194272.Search in Google Scholar

[32] A. K. Abbasi, N. Ali, M. Sajid, I. Ahmad, and S. Hussain, “Peristaltic tube flow of a Giesekus fluid,” Nihon Reoroji Gakkaishi, vol. 44, no. 2, pp. 99–108, 2016. https://doi.org/10.1678/rheology.44.99.Search in Google Scholar

[33] J. Y. Yoo and H. C. Choi, “On the steady simple shear flows of the one-mode Giesekus fluid,” Rheol. Acta, vol. 28, pp. 13–24, 1989. https://doi.org/10.1007/bf01354764.Search in Google Scholar

[34] G. Schleiniger and R. J. Weinacht, “Steady Poiseuille flows for a Giesekus fluid,” J. Non-Newtonian Fluid Mech., vol. 40, pp. 79–102, 1991. https://doi.org/10.1016/0377-0257(91)87027-u.Search in Google Scholar

[35] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, New York, John Wiley & Sons, 2006, pp. 249–262.Search in Google Scholar

[36] M. M. Mohseni and F. Rashidi, “Viscoelastic fluid behaviour in annulus using Giesekus model,” J. Non-Newtonian Fluid Mech., vol. 165, pp. 1550–1553, 2010. https://doi.org/10.1016/j.jnnfm.2010.07.012.Search in Google Scholar

[37] M. Mostafaiyan, K. Khodabandehlou, and F. Sharif, “Analysis of viscoelastic fluid in an annulus using Giesekus model,” J. Non-Newtonian Fluid Mech., vol. 118, pp. 49–55, 2004. https://doi.org/10.1016/j.jnnfm.2004.01.007.Search in Google Scholar

Received: 2021-04-16

Accepted: 2021-08-08

Published Online: 2021-08-31

Published in Print: 2021-11-25

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Articles in the same Issue

https://doi.org/10.1515/zna-2021-0098

Keywords for this article

bvp5c; Giesekus fluid; Graetz problem; mobility parameter; Sturm–Liouville boundary value problem; Weissenberg number