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Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

Part of: Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  08 July 2016

Vatsala Mathur  and
Kavita Khandelwal
Vatsala Mathur*
Affiliation:
Department of Mathematics, Malaviya National Institute of Technology, Jaipur 302017, India
Kavita Khandelwal*
Affiliation:
Department of Mathematics, Malaviya National Institute of Technology, Jaipur 302017, India
*
*Corresponding author. Email:[email protected] (V. Mathur), [email protected] (K. Khandelwal)
*Corresponding author. Email:[email protected] (V. Mathur), [email protected] (K. Khandelwal)
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Abstract

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress are also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.

Keywords

Fractional Maxwell fluidvelocity fieldshear stressfractional calculusHankel transformLaplace transform

MSC classification

Secondary: 76A05: Non-Newtonian fluids
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 5 , October 2016 , pp. 784 - 794
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Rajagopal, K. R. and Srinivasa, A. R., A thermodynamical frame-work for rate type fluid models, J. Non-Newtonian Fluid Mech., 88 (2000), pp. 207–227.CrossRefGoogle Scholar
[2]Dunn, J. E. and Rajagopal, K. R., Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci., 33 (1995), pp. 689–729.CrossRefGoogle Scholar
[3]Ting, T. W., Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), pp. 1–23.CrossRefGoogle Scholar
[4]Srivastava, P. N., Non-steady Helical flow of a viscoelastic liquid, Arch. Mech. Stos., 18 (1966), pp. 145–150.Google Scholar
[5]Waters, N. D. and King, M. J., The unsteady flow of an elastico-viscous liquid in a straight pipe of circular cross section, J. Phys. D Appl. Phys., 4 (1971), pp. 204–211.CrossRefGoogle Scholar
[6]Bandelli, R. and Rajagopal, K. R., Start-up flows of second grade fluids in domains with one finite dimension, Int. J. Non-Linear Mech., 30 (1995), pp. 817–839.CrossRefGoogle Scholar
[7]Bandelli, R., Rajagopal, K. R. and Galdi, G. P., On some unsteady motions of fluids of second grade, Arch. Mech., 47 (1995), pp. 661–676.Google Scholar
[8]Waters, N. D. and King, M. J., Unsteady flow of an elastico-viscous liquid, Rheol. Acta, 9 (1970), pp. 345–355.CrossRefGoogle Scholar
[9]Tong, D., Wang, R. and Yang, H., Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe, Science in China Ser. G Physics, Mechanics & Astronomy, 48 (2005), pp. 485–495.CrossRefGoogle Scholar
[10]Fetecau, C., Imran, M., Fetecau, C. and Burdujan, I., Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses, Zangew Math. Phys., 61 (2010), pp. 959–969.Google Scholar
[11]Imran, M., Kamran, M., Athar, M. and Zafar, A. A., Taylor-Couette flow of a fractional second grade fluid in an annulus due to a time-dependent couple, Nonlinear Anal. Model. Control, 16 (2011), pp. 47–58.CrossRefGoogle Scholar
[12]Athar, M., Awan, A. U. and Fetecau, C., A note on the unsteady flow of a fractional Maxwell fluid through a circular cylinder, Acta Mech. Sin., 28 (2012), pp. 308–314.CrossRefGoogle Scholar
[13]Athar, M., Kamran, M. and Fetecau, C., Taylor-Couette flow of a generalized second grade fluid due to a constant couple, Nonlinear Anal. Model. Control, 15 (2010), pp. 3–13.CrossRefGoogle Scholar
[14]Athar, M., Fetecau, C., Kamran, M., Sohail, A. and Imran, M., Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear, Nonlinear Anal. Model. Control, 16 (2011), pp. 135–151.CrossRefGoogle Scholar
[15]Fetecau, C., Analytical solutions for non-Newtonian fluid flows in pipe-like domains, Int. J. Non-Linear Mech., 39 (2004), pp. 225–231.CrossRefGoogle Scholar
[16]Rubbab, Q., Husnine, S. M. and Mahmood, A., Exact solutions of generalized Oldroyd-B fluid subject to a time-dependent shear stress in a pipe, J. Prime Research Math., 5 (2009), pp. 139–148.Google Scholar
[17]Friedrich, C. H. R., Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheologica Acta, 30 (1991), pp. 151–158.CrossRefGoogle Scholar
[18]Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific Press, Singapore, 2000.CrossRefGoogle Scholar
[19]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[20]Shen, F., Tan, W., Zhao, Y. and Masuoka, T., The Rayleigh-stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Anal. Real World Appl., 7 (2006), pp. 1072–1080.CrossRefGoogle Scholar
[21]Kamran, M., Imran, M., Athar, M. and Imran, M. A., On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains, Meccanica, 47 (2012), pp. 573–584.CrossRefGoogle Scholar
[22]Amir, M., Fetecau, C. and Imran, S., Exact solutions for some unsteady flows of generalized second grade fluids in cylindrical domains, J. Prime Research Math., 4 (2008), pp. 171–180.Google Scholar
[23]Kamran, M., Imran, M. and Athar, M., Exact solutions for the unsteady rotational flow of a generalized second grade fluid through a circular cylinder, Nonlinear Anal. Model. Control, 15 (2010), pp. 437–444.CrossRefGoogle Scholar
[24]Kamran, M., Athar, M. and Imran, M., On the unsteady linearly accelerating flow of a fractional second grade fluid through a circular cylinder, Int. J. Nonlinear Science, 11 (2011), pp. 317–324.Google Scholar
[25]Qi, H. and Jin, H., Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta Mech. Sinica, 22 (2006), pp. 301–305.CrossRefGoogle Scholar
[26]Wang, S. and Xu, M., Axial Coutte flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Anal. Real Word Appl., 10 (2009), pp. 1087–1096.CrossRefGoogle Scholar
[27]Lorenzo, C. F. and Hartley, T. T., Generalized functions for the fractional calculus, NASA/TP-1999-209424/REV1, 1999.Google Scholar
[28]Debnath, L. and Bhatta, D., Integral Transforms and Their Applications, Chapman & Hall/CRC, 2007.Google Scholar