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Exact Solution on Unsteady Couette Flow of
Generalized Maxwell Fluid with Fractional
Derivative
1
Shaowei Wang Mingyu Xu
I
nstitute of Applied Math, School of Math. and System Science
Shandong University, Jinan, PRC 250100
wangshaowei@math.sdu.edu.cn xumingyu@163169.net
Abstract
In this paper the unsteady Couette flow of a generalized Maxwell fluid with fractional
derivative (GMF) is studied. The exact solution is obtained with the help of integral
transforms (Laplace transform and Weber transform) and generalized Mittag-Leffler
function. It was shown that the distribution and establishment of the velocity is
governed by two non-dimensional parameters
η
, b and fractional derivative
α
of
the model. The result of classical (Newtonian fluid) Couette flow can be contained as
a special case of the result given by this paper, and the decaying of unsteady part of
GMF displays power law behavior, which has scale invariance.
Keywords: unsteady Couette flow; generalized Maxwell fluid; fractional calculus;
exact solution
1 Introduction
The non-Newtonian fluids are increasing being considered more important and appropriate in
technological applications than the Newtonian fluids. Strictly speaking, the linear relation between
stress and the rate of strain does not exist for a lot of real fluids, such as blood, oils, paints and
polymeric solution. In general, the analysis of the behavior of the fluid motion of the non-Newtonian
fluids tends to be much more complicated and subtle in comparison with that of the Newtonian fluids.
There has been fairly large number of flows of Newtonian fluids for which a closed form analytical
1
This work was supported by the National Natural Science Foundation of China (Grant No. 10272067) and the
Doctoral Program Foundation of the Education Ministry of P.R. China (Grant No.20030422046).
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