Starting solutions for the motion of a generalized Burgers' fluid between coaxial cylinders
 Muhammad Jamil^{1, 2}Email author and
 Constantin Fetecau^{1, 3, 4}
DOI: 10.1186/16872770201214
© Jamil and Fetecau; licensee Springer. 2012
Received: 28 March 2011
Accepted: 10 February 2012
Published: 10 February 2012
Abstract
The unsteady flow of a generalized Burgers' fluid, between two infinite coaxial circular cylinders, is studied by means of the Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, after the initial moment, applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained, presented in series form in terms of usual Bessel functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for Burgers' fluids appear as special cases of present results. For large values of t, these solutions are going to the steady solutions that are the same for both kinds of fluids. Finally, the influence of the material parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.
Mathematics Subject Classification (2010). 76A05; 76A10.
Keywords
Generalized Burgers' fluid coaxial cylinders velocity field timedependent shear stress Laplace and Hankel transforms1 Introduction
Recently, considerable attention has been focused to study the behavior of nonNewtonian fluids. Many substances of multiphase nature and/or of high molecular weight are frequently encountered in disciplinary fields such as chemical engineering, food stuff, biomedicine and so forth, and are also closely related to industrial processes. Typical nonNewtonian characteristics include shearthinning, shearthickening, viscoelasticity, viscoplasticity (i.e., the exhibition of an apparent yield stress) and so on. In particular, polymer melts and solutions, liquid crystals or biological fluids exhibit such properties which lead to nonlinear viscoelastic behavior that cannot be simply described by the classical NavierStokes theory. NonNewtonian fluids form a broad class of fluids in which the relation connecting the shear stress and shear rate is nonlinear and hence there is no universal constitutive model available which exhibits the characteristics of all nonNewtonian fluids. Moreover, due to the flow behavior of these fluids, the governing equations become more complex to handle as additional nonlinear terms appear in the equations of motion. Numerous models have been proposed to describe the response corresponding to nonNewtonian fluids. They are usually classified as fluids of differential, rate and integral type. Amongst the nonNewtonian fluids, the rate type fluids are those which take into account the elastic and memory effects. The simplest subclasses of rate type fluids are those of Maxwell and OldroydB fluids. But these fluid models do not exhibit rheological properties of many real fluids such as asphalt in geomechanics and cheese in food products.
Recently, a thermodynamic framework has been put into place to develop the 1D rate type model known as Burgers' model [1] to the frameindifferent 3D form by Krishnan and Rajagopal [2]. This model has been successfully used to describe the motion of the earth's mantle. The Burgers' model is the preferred model to describe the response of asphalt and asphalt concrete [3]. This model is mostly used to model other geological structures, such as Olivine rocks [4] and the propagation of seismic waves in the interior of the earth [5]. In the literature, the vast majority of the flows of the rate type models has been discussed using Maxwell and OldroydB models. However, the Burgers' model has not received much attention in spite of its diverse applications. We here mention some of the studies [6–12] made by using Burgers' model. The most of the above mentioned studies dealt with problems in which the velocity is given on the boundary. The motion of a fluid due to translating or rotating cylinder is of great interest to both theoretical and practical domains. The first exact solutions for motions of nonNewtonian fluids due to a circular cylinder that applies a longitudinal or rotational shear stress to the fluid are those of Bandelli and Rajagopal [13, Sects. 4 and 5]. These solutions, for rotational shear stresses, have been recently extended to generalized Burgers fluids [11, 12].
The aim of this note is to extend the results of Bandelli and Rajagopal [13, Sect. 4] to Burgers and generalized Burgers fluids. More exactly, we study the motion of a generalized Burgers' fluid between two infinite coaxial circular cylinders. The motion of the fluid is induced by the inner cylinder that applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained satisfy both the governing equations and all imposed initial and boundary conditions. They can immediately be reduced to the similar solutions for Burgers' fluids. As a check of the results, the equivalence of general solutions (for small values of the material parameters λ_{2} and λ_{4}) with the known solutions for OldroydB fluids is shown by graphical illustrations. Furthermore, in order to reveal some relevant physical aspects of these results, the diagrams of the velocity and shear stress are depicted against r for different values of pertinent parameters.
2 Basic governing equations
where ρ and V are, respectively, the fluid density and velocity vector and ∇ represents the gradient operator.
Into above relation, d/dt is the usual material time derivative.
This model includes as special cases the Burgers' model (for λ_{4} = 0), OldroydB model (for λ_{2} = λ_{4} = 0), Maxwell model (for λ_{2} = λ_{3} = λ_{4} = 0) and the linearly viscous fluid model when λ_{1} = λ_{2} = λ_{3} = λ_{4} = 0. In some special flows, like that to be here considered, the governing equations corresponding to generalized Burgers' fluids also resemble those for second grade fluids. However, closed form expressions for the similar solutions corresponding to OldroydB, Maxwell, second grade and Newtonian fluids cannot be obtained as limiting cases of general solutions.
where e_{ z }is the unit vector in the zdirection of a cylindrical coordinate system r, θ, z. For these flows the constraint of incompressibility is automatically satisfied.
where ν = μ/ρ is the kinematic viscosity of the fluid and ρ is its constant density.
3 Starting flow due to a time dependent shear stress
In order to solve the partial differential equations (7) and (5), with the initial and boundary conditions (9) and (10), we shall use the Laplace and finite Hankel transforms.
3.1 Calculation of the velocity field
From the RouthHurwitz's principle [19], we get Re(q_{ in }) < 0, if ${\lambda}_{1}{\lambda}_{3}{\lambda}_{2}+{\lambda}_{4}>2\sqrt{{\lambda}_{1}{\lambda}_{3}{\lambda}_{4}}$, provided λ_{1}, λ_{2}, λ_{3}, λ_{4} > 0.
3.2 Calculation of the shear stress
corresponding to Newtonian fluids are recovered. They correspond to a constant shear stress τ(R_{1},t) = f on the inner cylinder.
4 Steady and transient solutions
5 Numerical results and discussion
In the previous sections, we have presented exact analytical solutions for a flow problem of a generalized Burgers' fluid. In order to verify and capture relevant physical effects of the obtained results, several graphs are depicted in this section. The numerical results illustrate the velocity as well as shear stress profiles for the axial flow induced by the inner cylinder. We interpret these results with respect to the variations of emerging parameters of interest.
For large values of t, all solutions tend to the steady solutions v_{ S }(r,t) and τ_{ S }(r,t) which are the same for all kinds of fluids although the motion of rate type fluids (generalized Burgers, Burgers, OldroydB, and Maxwell) is due to a time dependent shear stress on the boundary. However, this is not a surprise because, for large times, the boundary shear stress corresponding to rate type fluids, as it results from Eqs. (8) and (29), tends to the same constant f corresponding to Newtonian and second grade fluids.
6 Concluding remarks
In this article, the velocity v(r,t) and the shear stress τ(r,t) corresponding to the flow of an incompressible generalized Burgers' fluid, between two infinite coaxial circular cylinders, have been determined by means of Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, after the initial moment, applies a time dependent longitudinal shear to the fluid. The solutions that have been obtained, written in series form in terms of Bessel functions J_{0}(•), J_{1}(•), Y_{0}(•), and Y_{1}(•), satisfy all imposed initial and boundary conditions. They can easily be reduced to the similar solutions for Burgers fluids. For large values of t, all solutions are going to the steady solutions v_{ S }(r) and τ_{ S }(r), which are the same for all kinds of fluids. The following conclusions may be extracted from graphical results.

The required time to reach the steadystate increases with respect to λ_{1} and decreases with regard to λ_{3} and ν. Consequently, the required time to reach the steady state for Newtonian fluids is lower/higher in comparison with Maxwell, respectively, second grade fluids.

For small values of λ_{3} and λ_{4} or λ_{1}, λ_{2}, λ_{3}, and λ_{4} the general solutions (20) and (24), as expected, are equivalent to those corresponding to OldroydB, respectively, Newtonian fluids.

The relaxation parameters λ_{1} and λ_{2}, as well as the retardation parameters λ_{3} and λ_{4}, have opposite effects on the fluid motion. More exactly, the velocity v(r,t) and the shear stress τ(r,t) in absolute value are decreasing functions with respect to λ_{1} and λ_{3} and increasing ones with regard to λ_{2} and λ_{4}.

It is observed that the increase of the kinematic viscosity ν leads to a decay of velocity and grown up the shear stress.

The Newtonian fluid is the swiftest and the generalized Burgers' fluid is the slowest.

The nonNewtonian effects disappear in time.
Appendix
Declarations
Acknowledgements
The authors would like to express their sincere gratitude to the referees for their careful assessment and fruitful remarks and suggestions regarding the initial version of the manuscript. The author Muhammad Jamil highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering & Technology, Karachi75270, Pakistan and also Higher Education Commission of Pakistan for generous supporting and facilitating this research study.
Authors’ Affiliations
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