Analytical and numerical study for oscillatory flow of viscoelastic fluid in a tube with isosceles right triangular cross section
-
Yi Li
Abstract
A theoretical investigation is carried out to analyze the oscillatory flow of second-grade fluid under the periodic pressure gradient in a long tube of isosceles right triangular cross section in the present study. The analytical expressions for the velocity profile and phase difference are obtained. The numerical solutions are calculated by using the finite difference method with Crank–Nicolson (C–N) scheme. In comparison with the Newtonian fluid (λ = 0), the effects of retardation time, Deborah number and Womersley number on the velocity profile and phase difference are discussed numerically and graphically. For smaller Womersley number, the behavior of second-grade fluid is dominated by viscosity. For larger Womersley number α = 20, the flow becomes more difficult to be generated under periodic pressure gradient with increasing retardation time. Furthermore, the analytical expressions of the mean velocity amplitude and phase difference are given explicitly for discussing.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11672164
Award Identifier / Grant number: 12072177
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 12072177, 11672164).
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: This work is funded by National Natural Science Foundation of China (Grant no. 11672164 and 12072177).
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] B. D. Coleman and W. Noll, “An approximation theorem for functionals, with applications in continuum mechanics,” Arch. Ration. Mech. Anal., vol. 6, no. 1, pp. 355–370, 1960. https://doi.org/10.1007/BF00276168.Suche in Google Scholar
[2] D. W. Beard, K. Walters, and J. G. Oldroyd, “Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point,” Math. Proc. Camb. Phil. Soc., vol. 60, no. 03, pp. 667–674, 1964. https://doi.org/10.1017/S0305004100038147.Suche in Google Scholar
[3] R. B. Bird, “Useful non-Newtonian models,” Annu. Rev. Fluid Mech., vol. 8, no. 1, pp. 13–34, 1976. https://doi.org/10.1146/annurev.fl.08.010176.000305.Suche in Google Scholar
[4] H. Giesekus, “Several comments of the paper “some remarks on ‘useful theorems for the second order fluid’” by P. N. Kaloni,” J. Non-Newtonian Fluid Mech., vol. 33, no. 3, pp. 343–348, 1989. https://doi.org/10.1016/0377-0257(89)80006-0.Suche in Google Scholar
[5] T. W. Ting, “Certain non-steady flows of second-order fluids,” Arch. Ration. Mech. Anal., vol. 14, no. 1, pp. 1–26, 1963. https://doi.org/10.1007/BF00250690.Suche in Google Scholar
[6] A. C. Pipkin, “Alternating flow of non-Newtonian fluids in tubes of arbitrary cross-section,” Arch. Ration. Mech. Anal., vol. 15, no. 1, pp. 1–13, 1964. https://doi.org/10.1007/BF00257401.Suche in Google Scholar
[7] C. Fetecau and J. Zierep, “On a class of exact solutions of the equations of motion of a second grade fluid,” Acta Mech., vol. 150, no. 1, pp. 135–138, 2001. https://doi.org/10.1007/BF01178551.Suche in Google Scholar
[8] I. C. Christov and C. I. Christov, “Comment on “On a class of exact solutions of the equations of motion of a second grade fluid” by C. Fetecau and J. Zierep (Acta Mech. 150, 135-138, 2001),” Acta Mech., vol. 215, no. 1, pp. 25–28, 2010. https://doi.org/10.1007/s00707-010-0300-2.Suche in Google Scholar
[9] I. C. Christov, “Stokes’ first problem for some non-Newtonian fluids: results and mistakes,” Mech. Res. Commun., vol. 37, no. 8, pp. 717–723, 2010. https://doi.org/10.1016/j.mechrescom.2010.09.006.Suche in Google Scholar
[10] P. S. Rao, B. Murmu, and S. Agarwal, “Effects of surface roughness and non-Newtonian micropolar fluid squeeze film between conical bearings,” Z. Naturforsch. A., vol. 72, no. 12, pp. 1151–1158, 2017. https://doi.org/10.1515/zna-2017-0257.Suche in Google Scholar
[11] I. M. Eldesoky, R. M. Abumandour, and E. T. Abdelwahab, “Analysis for various effects of relaxation time and wall properties on compressible maxwellian peristaltic slip flow,” Z. Naturforsch. A., vol. 74, no. 4, pp. 317–331, 2019. https://doi.org/10.1515/zna-2018-0479.Suche in Google Scholar
[12] K. Boubaker and Y. Khan, “Study of the phan-thien–tanner equation of viscoelastic blood non-Newtonian flow in a pipe-shaped artery under an emotion-induced pressure gradient,” Z. Naturforsch. A., vol. 67, nos 10–11, pp. 628–632, 2012. https://doi.org/10.5560/ZNA.2012-0069.Suche in Google Scholar
[13] S. Thohura, M. M. Molla, and M. Sarker, “Bingham fluid flow simulation in a lid-driven skewed cavity using the finite volume method,” Int. J. Comput. Math., vol. 97, no. 3, pp. 1–34, 2019. https://doi.org/10.1080/00207160.2019.1613527.Suche in Google Scholar
[14] M. M. Molla, P. Nag, S. Thohura, and A. Khan, “A graphics process unit-based multiple-relaxation-time lattice Boltzmann simulation of non-newtonian fluid flows in a backward facing step,” Computation, vol. 8, no. 83, pp. 1–24, 2020. https://doi.org/10.3390/computation8030083.Suche in Google Scholar
[15] A. Rahman, P. Nag, M. Molla, and S. Hassan, “Magnetic field effects on natural convection and entropy generation of non-Newtonian fluids using multiple-relaxation-time lattice Boltzmann method,” Int. J. Mod. Phys. C, vol. 32, no. 01, pp. 2150015, 2020. https://doi.org/10.1142/S0129183121500157.Suche in Google Scholar
[16] E. G. Richardson and E. Tyler, “The transverse velocity gradient near the mouths of pipes in which an alternating or continuous flow of air is established,” Proc. Phys. Soc., vol. 42, no. 1, pp. 1–15, 1929. https://doi.org/10.1088/0959-5309/42/1/302.Suche in Google Scholar
[17] S. Uchida, “The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe,” Z. Angew. Math. Phys., vol. 7, no. 5, pp. 403–422, 1956. https://doi.org/10.1007/BF01606327.Suche in Google Scholar
[18] Y. Wang, Y. L. He, G. H. Tang, and W. Q. Tao, “Simulation of two-dimensional oscillating flow using the lattice Boltzmann method,” Int. J. Mod. Phys. C, vol. 17, no. 05, pp. 615–630, 2006. https://doi.org/10.1142/S0129183106009023.Suche in Google Scholar
[19] S. W. Wang, P. L. Li, and M. L. Zhao, “Analytical study of oscillatory flow of Maxwell fluid through a rectangular tube,” Phys. Fluids, vol. 31, no. 6, p. 063102, 2019. https://doi.org/10.1063/1.5100220.Suche in Google Scholar
[20] C. Fetecau, A. Rauf, T. M. Qureshi, and M. Khan, “Permanent solutions for some oscillatory motions of fluids with power-law dependence of viscosity on the pressure and shear stress on the boundary,” Z. Naturforsch. A., vol. 75, no. 9, pp. 757–769, 2020. https://doi.org/10.1515/zna-2020-0135.Suche in Google Scholar
[21] H. Xu, S. W. Wang, and M. L. Zhao, “Oscillatory flow of second grade fluid in a straight rectangular duct,” J. Non-Newtonian Fluid Mech., vol. 279, p. 104245, 2020. https://doi.org/10.1016/j.jnnfm.2020.104245.Suche in Google Scholar
[22] B. D. Aggarwala and M. K. Gangal, “Laminar flow development in triangular ducts,” Trans. Can. Soc. Mech. Eng., vol. 3, no. 4, pp. 231–233, 1975. https://doi.org/10.1139/tcsme-1975-0031.Suche in Google Scholar
[23] S. Tsangaris and N. W. Vlachakis, “Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle,” Z. Angew. Math. Phys., vol. 54, no. 6, pp. 1094–1100, 2003. https://doi.org/10.1007/s00033-003-2013-z.Suche in Google Scholar
[24] C. Y. Wang, “Ritz method for oscillatory flow in ducts,” Int. J. Numer. Methods Fluid., vol. 67, no. 5, pp. 609–615, 2011. https://doi.org/10.1002/fld.2379.Suche in Google Scholar
[25] R. Kumar, Varun, and A. Kumar, “Thermal and fluid dynamic characteristics of flow through triangular cross-sectional duct: a review,” Renew. Sustain. Energy Rev., vol. 61, pp. 123–140, 2016. https://doi.org/10.1016/j.rser.2016.03.011.Suche in Google Scholar
[26] X. Y. Sun, S. W. Wang, and M. L. Zhao, “Oscillatory flow of Maxwell fluid in a tube of isosceles right triangular cross section,” Phys. Fluids, vol. 31, no. 12, p. 123101, 2019. https://doi.org/10.1063/1.5128764.Suche in Google Scholar
[27] K. R. Rajagopal, “On the creeping flow of the second-order fluid,” J. Non-Newtonian Fluid Mech., vol. 15, no. 2, pp. 239–246, 1984. https://doi.org/10.1016/0377-0257(84)80008-7.Suche in Google Scholar
[28] R. S. Rivlin and J. L. Ericksen, “Stress deformation relation for isotropic materials,” Indiana Univ. Math. J., vol. 4, no. 2, pp. 323–425, 1955. https://doi.org/10.1512/iumj.1955.4.54011.Suche in Google Scholar
[29] J. E. Dunn and R. L. Fosdick, “Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade,” Arch. Ration. Mech. Anal., vol. 56, no. 3, pp. 191–252, 1974. https://doi.org/10.1007/BF00280970.Suche in Google Scholar
[30] J. R. Womersley, “Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known,” J. Physiol., vol. 127, no. 3, pp. 553–563, 1955. https://doi.org/10.1113/jphysiol.1955.sp005276.Suche in Google Scholar PubMed PubMed Central
[31] C. S. Yih, Fluid Mechanics: A Concise Introduction to the Theory, New York, McGraw-Hill Book Company, 1969.Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Dynamical Systems & Nonlinear Phenomena
- Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves
- 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
- Ion-acoustic solitary structures at the acoustic speed in a collisionless magnetized nonthermal dusty plasma
- Exact Beltrami flows in a spherical shell
- Hydrodynamics
- Insight into the dynamics of non-Newtonian carboxy methyl cellulose conveying CuO nanoparticles: significance of channel branch angle and pressure drop
- Analytical and numerical study for oscillatory flow of viscoelastic fluid in a tube with isosceles right triangular cross section
- Solid State Physics & Materials Science
- Numerical study of highly efficient tin-based perovskite solar cell with MoS2 hole transport layer
- An improved photocatalytic activity of H2 production: a hydrothermal synthesis of TiO2 nanostructures in aqueous triethanolamine
Artikel in diesem Heft
- Frontmatter
- Dynamical Systems & Nonlinear Phenomena
- Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves
- 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
- Ion-acoustic solitary structures at the acoustic speed in a collisionless magnetized nonthermal dusty plasma
- Exact Beltrami flows in a spherical shell
- Hydrodynamics
- Insight into the dynamics of non-Newtonian carboxy methyl cellulose conveying CuO nanoparticles: significance of channel branch angle and pressure drop
- Analytical and numerical study for oscillatory flow of viscoelastic fluid in a tube with isosceles right triangular cross section
- Solid State Physics & Materials Science
- Numerical study of highly efficient tin-based perovskite solar cell with MoS2 hole transport layer
- An improved photocatalytic activity of H2 production: a hydrothermal synthesis of TiO2 nanostructures in aqueous triethanolamine
