Starting solutions for the motion of a generalized Burgers' fluid between coaxial cylinders
© Jamil and Fetecau; licensee Springer. 2012
Received: 28 March 2011
Accepted: 10 February 2012
Published: 10 February 2012
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.
Recently, considerable attention has been focused to study the behavior of non-Newtonian fluids. Many substances of multi-phase nature and/or of high molecular weight are frequently encountered in disciplinary fields such as chemical engineering, food stuff, bio-medicine and so forth, and are also closely related to industrial processes. Typical non-Newtonian characteristics include shear-thinning, shear-thickening, 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 non-linear viscoelastic behavior that cannot be simply described by the classical Navier-Stokes theory. Non-Newtonian fluids form a broad class of fluids in which the relation connecting the shear stress and shear rate is non-linear and hence there is no universal constitutive model available which exhibits the characteristics of all non-Newtonian fluids. Moreover, due to the flow behavior of these fluids, the governing equations become more complex to handle as additional non-linear terms appear in the equations of motion. Numerous models have been proposed to describe the response corresponding to non-Newtonian fluids. They are usually classified as fluids of differential, rate and integral type. Amongst the non-Newtonian 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 Oldroyd-B 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  to the frame-indifferent 3D form by Krishnan and Rajagopal . 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 . This model is mostly used to model other geological structures, such as Olivine rocks  and the propagation of seismic waves in the interior of the earth . In the literature, the vast majority of the flows of the rate type models has been discussed using Maxwell and Oldroyd-B 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 non-Newtonian 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 λ2 and λ4) with the known solutions for Oldroyd-B 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 λ4 = 0), Oldroyd-B model (for λ2 = λ4 = 0), Maxwell model (for λ2 = λ3 = λ4 = 0) and the linearly viscous fluid model when λ1 = λ2 = λ3 = λ4 = 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 Oldroyd-B, Maxwell, second grade and Newtonian fluids cannot be obtained as limiting cases of general solutions.
where e z is the unit vector in the z-direction 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
3.1 Calculation of the velocity field
From the Routh-Hurwitz's principle , we get Re(q in ) < 0, if , provided λ1, λ2, λ3, λ4 > 0.
3.2 Calculation of the shear stress
corresponding to Newtonian fluids are recovered. They correspond to a constant shear stress τ(R1,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, Oldroyd-B, 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 J0(•), J1(•), Y0(•), and Y1(•), 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 steady-state increases with respect to λ1 and decreases with regard to λ3 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 λ3 and λ4 or λ1, λ2, λ3, and λ4 the general solutions (20) and (24), as expected, are equivalent to those corresponding to Oldroyd-B, respectively, Newtonian fluids.
The relaxation parameters λ1 and λ2, as well as the retardation parameters λ3 and λ4, 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 λ1 and λ3 and increasing ones with regard to λ2 and λ4.
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 non-Newtonian effects disappear in time.
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, Karachi-75270, Pakistan and also Higher Education Commission of Pakistan for generous supporting and facilitating this research study.
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