THERMAL SCIENCE
International Scientific Journal
THE ANALYTIC SOLUTIONS FOR THE UNSTEADY ROTATING FLOWS OF THE GENERALIZED MAXWELL FLUID BETWEEN COAXIAL CYLINDERS
ABSTRACT
In this paper, we consider the unsteady rotating flow of the generalized Maxwell fluid with fractional derivative model between two infinite straight circular cylinders, where the flow is due to an infinite straight circular cylinder rotating and oscillating pressure gradient. The velocity field is determined by means of the combine of the Laplace and finite Hankel transforms. The analytic solutions of the velocity and the shear stress are presented by series form in terms of the generalized G and R functions. The similar solutions can be also obtained for ordinary Maxwell and Newtonian fluids as limiting cases.
KEYWORDS
PAPER SUBMITTED: 2019-08-03
PAPER REVISED: 2020-01-15
PAPER ACCEPTED: 2020-01-20
PUBLISHED ONLINE: 2020-11-27
THERMAL SCIENCE YEAR
2020, VOLUME
24, ISSUE
Issue 6, PAGES [4041 - 4048]
- Sneddon, I. N., Functional Analysis, Encyclopedia of Physics, Springer, Berlin, 1955
- Fridrich, C., Relaxation and Retardation Functions of the Maxwell Model with Fractional Derivatives, Rheological Acta, 30 (1999), 1, pp. 151-158
- Bagley R. L., A Theoretical Basis for The Application of Fractional Calculus to Viscoelasticity, Journal of Rheology, 27 (1983), 2, pp. 201-210
- Glockle W. G., et al., Fractional Relaxation and the Time-temperature Superposition Principle, Rheological Acta, 33 (1994), 3, pp. 337-343
- Kavita K., et al., Exact Solutions for an Unsteady Flow of Viscoelastic Fluid in Cylindrical Domains Using the Fractional Maxwell Model, International Journal of Applied and Computational Mathematics, 1 (2015), 2, pp. 143-156
- Rossikhin Y. A., et al., A new Method for Solving Dynamic Problems of Fractional Derivative Viscoelasticity, International Journal of Engineering Science, 39 (2000), 3, pp. 149-176
- Rossikhin Y. A., et al., Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders, Zeitschrift fur Angewandte Mathematik und Mechanik, 81 (2001), 2, pp. 363-376
- Mainardi, F., Fractional Relaxation-oscillation and Fractional Diffusion-wave Phenomena, Chaos Solitons & Fractals, 7 (1996), 9, pp. 1461-1477
- Song, D. Y., et al., Study on the Constitutive Equation with Fractional Derivative for the Viscoelastic Fluid-modified Jeffreys Model and its Applications, Rheological Acta, 37 (1998), 6, pp. 512-517
- Mahmooda, A., et al., Exact Analytic Solutions for the Unsteady Flow of a Non-Newtonian Fluid Between Two Cylinders with Fractional Derivative Model, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 8, pp. 3309-3319
- Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands, 2006
- Khan, M., et al., Exact Solution for MHD Flow of a Generalized Oldroyd-B Fluid with Modified Darcy's Law, International Journal of Engineering Science, 44 (2006), 3, pp. 333-339
- Yang, X. J. New Rheological Problems Involving General Fractional Derivatives with Nonsingular Power-Law Kernels, Proceedings of the Proceedings of the Romanian Academy Series A, 19 (2018), 1, pp. 45-52
- Yang, X. J. New General Fractional-Order Rheological Models with Kernels of Mittag-Leffler Functions, Romanian Reports in Physics, 69 (2017), 4, pp. 1-15
- Yang, X. J., General Fractional Derivatives: Theory, Methods and Applications, CRC Press, New York, USA, 2019
- Yang, X. J., et al., General Fractional Derivatives with Applications in Viscoelasticity, Academic Press, New York, USA, 2020