About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Physics
Volume 2012 (2012), Article ID 831063, 17 pages
http://dx.doi.org/10.5402/2012/831063
Research Article

First Problem of Stokes for Generalized Burgers' Fluids

1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan

Received 11 October 2011; Accepted 31 October 2011

Academic Editors: I. Radinschi and W.-H. Steeb

Copyright © 2012 Muhammad Jamil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The velocity field and the adequate shear stress corresponding to the first problem of Stokes for generalized Burgers’ fluids are determined in simple forms by means of integral transforms. The solutions that have been obtained, presented as a sum of steady and transient solutions, satisfy all imposed initial and boundary conditions. They can be easily reduced to the similar solutions for Burgers, Oldroyd-B, Maxwell, and second-grade and Newtonian fluids. Furthermore, as a check of our calculi, for small values of the corresponding material parameters, their diagrams are almost identical to those corresponding to the known solutions for Newtonian and Oldroyd-B fluids. Finally, the influence of the rheological parameters on the fluid motions, as well as a comparison between models, is graphically illustrated. The non-Newtonian effects disappear in time, and the required time to reach steady-state is the lowest for Newtonian fluids.

1. Introduction

There is evidence that the interest of the workers in non-Newtonian fluids is on the leading edge during the last few years. Many researchers have the opinion that flows of such fluids are important in industry and technology. Several investigations in the field cite a wide variety of applications in rheological problems in biological sciences, geophysics, and chemical and petroleum industries [1]. It is an established fact that unlike the Newtonian fluids, the flows of non-Newtonian fluids cannot be analyzed by a single constitutive equation. This is due to the rheological properties of non-Newtonian fluids. The understanding of flows of such fluids has progressed via a number of theoretical, computational, and experimental efforts. The resulting equations of such fluids are in general of higher order than the Navier-Stokes equation and one needs additional conditions for a unique solution [2, 3]. Specifically to obtain an analytic solution for such flows is not an easy task. In spite of several challenges, many investigations regarding the analytic solutions for flows of non-Newtonian fluids have been performed [419].

Many models are accorded to describe the rheological behavior of non-Newtonian fluids [20, 21]. 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 one-dimensional rate type model known as Burgers’ model [22] to the frame-indifferent three-dimensional form by Murali Krishnan and Rajagopal [23]. 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 [24]. This model is mostly used to model other geological structures, such as Olivine rocks [25] and the propagation of seismic waves in the interior of the earth [26]. 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 [2733] made by using Burgers’ model.

The purpose of this work is to established exact solutions corresponding to the first problem of Stokes for generalized Burgers’ fluids. Actually, we determine the velocity and the adequate shear stress corresponding to the motion of such a fluid over a plane wall, which initially is at rest and is suddenly moved in its own plane withe a constant velocity. The general solutions, obtained by means of Fourier sine and Laplace transforms, are presented under integral form in terms of the elementary functions and can be reduced to the similar solutions for Burgers fluids. As a check of their correctness, we also showed that for small values of the rheological parameters 𝜆1, 𝜆2, 𝜆3, and 𝜆4 or 𝜆2 and 𝜆4 only, the diagrams of the general solutions are almost identical to those corresponding to the known solutions for Newtonian, respectively, Oldroyd-B fluids. The influence of the material parameters on the fluid motion, as well as a comparison between some models, is also underlined by graphical illustrations. The non-Newtonian effects disappear in time and the Newtonian fluid flows faster.

2. Basic Governing Equations

The Cauchy stress tensor 𝐓 for an incompressible generalized Burgers’ fluid is characterized by the following constitutive equations [3033]:𝐓=𝑝𝐈+𝐒,𝐒+𝜆1𝛿𝐒𝛿𝑡+𝜆2𝛿2𝐒𝛿𝑡2=𝜇𝐀+𝜆3𝛿𝐀𝛿𝑡+𝜆4𝛿2𝐀𝛿𝑡2,(2.1) where 𝑝𝐈 denotes the indeterminate spherical stress, 𝐒 is the extra-stress tensor, 𝐀=𝐋+𝐋𝑇 is the first Rivlin-Ericksen tensor (𝐋 being the velocity gradient), 𝜇 is the dynamic viscosity, 𝜆1 and 𝜆3 (<𝜆1) are relaxation and retardation times, 𝜆2 and 𝜆4 are new material parameters of the generalized Burgers’ fluid (having the dimension of 𝑡2), and 𝛿/𝛿𝑡 denotes the upper convected derivative defined in [3033].

This model includes as special cases the Burgers’ model (for𝜆4=0), Oldroyd-B model (for𝜆2=𝜆4=0), Maxwell model (for𝜆2=𝜆3=𝜆4=0), and the Newtonian fluid model when 𝜆1=𝜆2=𝜆3=𝜆4=0. In some special flows, like those to be here considered, the governing equations corresponding to generalized Burgers’ fluids resemble those for second-grade fluids.

For the problem under consideration, we assume a velocity field 𝐕 and an extra-stress tensor 𝐒 of the form𝐕=𝐕(𝑦,𝑡)=𝑢(𝑦,𝑡)𝐢,𝐒=𝐒(𝑦,𝑡),(2.2) where i is the unit vector along the 𝑥-coordinate direction. For these flows, the constraint of incompressibility is automatically satisfied. If the fluid is at rest up to the moment 𝑡=0, then𝐕(𝑦,0)=𝟎,𝐒(𝑦,0)=𝜕𝐒(𝑦,0)𝜕𝑡=𝟎.(2.3) Equations (2.1) and (2.3) imply 𝑆𝑦𝑦=𝑆𝑦𝑧=𝑆𝑧𝑧=𝑆𝑥𝑧=0, and the meaningful equation1+𝜆1𝜕𝜕𝑡+𝜆2𝜕2𝜕𝑡2𝜏(𝑦,𝑡)=𝜇1+𝜆3𝜕𝜕𝑡+𝜆4𝜕2𝜕𝑡2𝜕𝑢(𝑦,𝑡)𝜕𝑦,(2.4) where 𝜏(𝑦,𝑡)=𝑆𝑥𝑦(𝑦,𝑡) is the nonzero shear stress. In the absence of body forces, the balance of linear momentum reduces to𝜕𝜏(𝑦,𝑡)𝜕𝑦𝜕𝑝𝜕𝑥=𝜌𝜕𝑢(𝑦,𝑡),𝜕𝑡𝜕𝑝=𝜕𝑦𝜕𝑝𝜕𝑧=0.(2.5) Eliminating 𝜏 between (2.4) and (2.5) and assuming that there is no pressure gradient in the flow direction, we find the governing equation under the form1+𝜆1𝜕𝜕𝑡+𝜆2𝜕2𝜕𝑡2𝜕𝑢(𝑦,𝑡)𝜕𝑡=𝜈1+𝜆3𝜕𝜕𝑡+𝜆4𝜕2𝜕𝑡2𝜕2𝑢(𝑦,𝑡)𝜕𝑦2;𝑦,𝑡>0.(2.6)

Consider an incompressible generalized Burgers’ fluid occupying the space above a flat plate perpendicular to the 𝑦-axis. Initially, the fluid is at rest and at the moment 𝑡=0+ the plate is impulsively brought to the constant velocity 𝑈 in its plane. Due to the shear, the fluid above the plate is gradually moved. Its velocity is of the form (2.3)1, while the governing equations are given by (2.6) and (2.4). The relevant problem under initial and boundary conditions [3436] is𝑢(𝑦,0)=𝜕𝑢(𝑦,0)=𝜕𝜕𝑡2𝑢(𝑦,0)𝜕𝑡2=0,𝑦>0,(2.7)𝑢(0,𝑡)=𝑈𝐻(𝑡),𝑡0,(2.8) where 𝐻(𝑡) is the Heaviside function. Moreover, the natural conditions𝑢(𝑦,𝑡),𝜕𝑢(𝑦,𝑡)𝜕𝑦0as𝑦,𝑡>0(2.9) have to be also satisfied. They are consequences of the fact that the fluid is at rest at infinity and there is no shear in the free stream.

3. Solution of the Problem

In order to determine the exact solution, we shall use the Fourier sine transforms [37]. Multiplying both sides of (2.6) by 2/𝜋sin(𝑦𝜉), integrating the result with respect to 𝑦 from 0 to infinity, and taking into account the boundary conditions (2.8) and (2.9), we find that1+𝜆1𝜕𝜕𝑡+𝜆2𝜕2𝜕𝑡2𝜕𝑢𝑠𝜕𝑡+𝜈𝜉21+𝜆3𝜕𝜕𝑡+𝜆4𝜕2𝜕𝑡2𝑢𝑠=𝜈𝜉𝑈2𝜋𝐻(𝑡)+𝜆3𝛿(𝑡)+𝜆4𝛿(𝑡),(3.1) where 𝛿(𝑡) and 𝛿(𝑡) are delta function and its derivative and the Fourier sine transform 𝑢𝑠=𝑢𝑠(𝜉,𝑡) of 𝑢(𝑦,𝑡) defined by𝑢𝑠(𝜉,𝑡)=2𝜋0𝑢(𝑦,𝑡)sin(𝑦𝜉)𝑑𝑦,(3.2) has to satisfy the initial conditions𝑢𝑠(𝜉,0)=𝜕𝑢𝑠(𝜉,0)=𝜕𝜕𝑡2𝑢𝑠(𝜉,0)𝜕𝑡2=0,𝜉>0.(3.3) By applying the Laplace transform to (3.1) and having in mind the initial conditions (2.7), we find that𝑢𝑠(𝜉,𝑞)=𝜈𝜉𝑈2𝜋𝜆4𝑞2+𝜆3𝑞+1𝑞𝜆2𝑞3+𝜆1+𝜆4𝜈𝜉2𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2.(3.4) Now, for a more suitable presentation of the final results, we rewrite (3.4) in the following equivalent form:𝑢𝑠(𝜉,𝑞)=𝑈2𝜋1𝜉1𝑞𝜆2𝑞2+𝜆1𝑞+1𝜆2𝑞3+𝜆1+𝜆4𝜈𝜉2𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2.(3.5) Inverting (3.5) by means of the Fourier sine formula, we can write 𝑢(𝑦,𝑞) as𝑢(𝑦,𝑞)=2𝑈𝜋0sin(𝑦𝜉)𝜉1𝑞𝜆2𝑞2+𝜆1𝑞+1𝜆2𝑞3+𝜆1+𝜆4𝜈𝜉2𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2𝑑𝜉.(3.6) Finally, in order to obtain the velocity field 𝑢(𝑦,𝑡)=1{𝑢(𝑦,𝑞)}, we apply the inverse Laplace transform to (3.6) and use (A.1) from the Appendix. As a result, we find for the velocity field, the following simple expression:𝑢(𝑦,𝑡)=𝑈𝐻(𝑡)2𝑈𝐻(𝑡)𝜋𝜆20sin(𝑦𝜉)𝜉×𝜆2𝑞21+𝜆1𝑞1𝑒+1𝑞1𝑡𝑞1𝑞2𝑞1𝑞3+𝜆2𝑞22+𝜆1𝑞2𝑒+1𝑞2𝑡𝑞2𝑞1𝑞2𝑞3+𝜆2𝑞23+𝜆1𝑞3𝑒+1𝑞3𝑡𝑞3𝑞1𝑞3𝑞2𝑑𝜉,(3.7) where𝑞𝑖=𝑠𝑖𝜆1+𝜆4𝜈𝜉23𝜆2,𝑖=1,2,3,(3.8) are the roots of the algebraic equation 𝜆2𝑞3+(𝜆1+𝜆4𝜈𝜉2)𝑞2+(1+𝜆3𝜈𝜉2)𝑞+𝜈𝜉2=0. In above relations (see the Cardano’s formulae [38]),𝑠1=3𝛽12+𝛽214+𝛼31+273𝛽12𝛽214+𝛼31,𝑠272=𝑍3𝛽12+𝛽214+𝛼3127+𝑍23𝛽12𝛽214+𝛼31,𝑠273=𝑍23𝛽12+𝛽214+𝛼3127+𝑍3𝛽12𝛽214+𝛼3127(3.9) are the roots of the algebraic equation 𝑋3+𝛼1𝑋+𝛽1=0, where𝛼1=1+𝜆3𝜈𝜉2𝜆2𝜆1+𝜆4𝜈𝜉223𝜆22,𝛽1=𝜈𝜉2𝜆2+2𝜆1+𝜆4𝜈𝜉2327𝜆32𝜆1+𝜆4𝜈𝜉21+𝜆3𝜈𝜉23𝜆22,𝑍=1+𝑖32.(3.10) From Routh-Hurwitz’s principle [39], we get Re(𝑞𝑖)<0 if 𝜆1𝜆3𝜆2+𝜆4>2𝜆1𝜆3𝜆4, provided 𝜆1,𝜆2,𝜆3,𝜆4>0. The corresponding shear stress (see also (A.2))𝜏(𝑦,𝑡)=2𝜇𝑈𝐻(𝑡)𝜋𝜆20×𝜆cos(𝑦𝜉)4𝑞21+𝜆3𝑞1𝑒+1𝑞1𝑡𝑞1𝑞2𝑞1𝑞3+𝜆4𝑞22+𝜆3𝑞2𝑒+1𝑞2𝑡𝑞2𝑞1𝑞2𝑞3+𝜆4𝑞23+𝜆3𝑞3𝑒+1𝑞3𝑡𝑞3𝑞1𝑞3𝑞2𝑑𝜉,(3.11) is obtained in the same way from (2.4).

4. Special Cases

4.1. Burgers’ Fluid

Making 𝜆40 into (3.7) and (3.11), we obtain the velocity field and the associated shear stress corresponding to a Burgers’ fluid performing the same motion.

4.2. Oldroyd-B Fluid

Making 𝜆2 and 𝜆4=0 into (3.6) and following the same way as before, we get the velocity field (see also (A.3))𝑢OB(𝑦,𝑡)=𝑈𝐻(𝑡)2𝑈𝐻(𝑡)𝜋𝜆10sin(𝑦𝜉)𝜉𝜆1𝑞8𝑒+1𝑞8𝑡𝜆1𝑞7𝑒+1𝑞7𝑡𝑞8𝑞7𝑑𝜉,(4.1) and the shear stress𝜏OB(𝑦,𝑡)=2𝜇𝑈𝐻(𝑡)𝜋𝜆10𝜆cos(𝑦𝜉)3𝑞8𝑒+1𝑞8𝑡𝜆3𝑞7𝑒+1𝑞7𝑡𝑞8𝑞7𝑑𝜉,(4.2) corresponding to an Oldroyd-B fluid. In the above relations,𝑞7,𝑞8=1+𝜆3𝜈𝜉2±1+𝜆3𝜈𝜉224𝜈𝜆1𝜉22𝜆1(4.3) and (4.1) is identical to (15) from [40].

4.3. Maxwell Fluid

Making the limit of (4.1) and (4.2) as 𝜆30, we obtain the solutions𝑢𝑀(𝑦,𝑡)=𝑈𝐻(𝑡)2𝑈𝐻(𝑡)𝜋𝜆10sin(𝑦𝜉)𝜉𝜆1𝑞10𝑒+1𝑞10𝑡𝜆1𝑞9𝑒+1𝑞9𝑡𝑞10𝑞9𝜏𝑑𝜉,𝑀(𝑦,𝑡)=2𝜇𝑈𝐻(𝑡)𝜋𝜆10𝑒cos(𝑦𝜉)𝑞10𝑡𝑒𝑞9𝑡𝑞10𝑞9𝑑𝜉,(4.4) corresponding to a Maxwell fluid. The new roots 𝑞9 and 𝑞10 are given by𝑞9,𝑞10=1±14𝜈𝜆1𝜉22𝜆1.(4.5)

4.4. Second-Grade Fluid

It is worthwhile pointing out that the similar solutions for second-grade fluids can be also obtained as limiting case of our solutions. Indeed, if we do not take into consideration the restriction 𝜆𝜆𝑟 and make 𝜆10 into (4.1) and (4.2), we recover the expressions𝑢SG2(𝑦,𝑡)=𝑈𝐻(𝑡)1𝜋0sin(𝑦𝜉)𝜉1+𝜆3𝜈𝜉2exp𝜈𝜉2𝑡1+𝜆3𝜈𝜉2𝜏𝑑𝜉,(4.6)SG(𝑦,𝑡)=2𝜇𝑈𝐻(𝑡)𝜋0cos(𝑦𝜉)1+𝜆3𝜈𝜉2exp𝜈𝜉2𝑡1+𝜆3𝜈𝜉2𝑑𝜉,(4.7) corresponding to a second-grade fluid. The solution (4.6) is identical to that from [40, equation (16)] or [36, equation (14)].

4.5. Newtonian Fluid

Finally, making 𝜆10 into (4.4) or 𝜆30 into (4.6) and (4.7), the solutions for a Newtonian fluid𝑢𝑁2(𝑦,𝑡)=𝑈𝐻(𝑡)1𝜋0sin(𝑦𝜉)𝜉𝑒𝜈𝜉2𝑡𝜏𝑑𝜉,(4.8)𝑁(𝑦,𝑡)=2𝜇𝑈𝐻(𝑡)𝜋0cos(𝑦𝜉)𝑒𝜈𝜉2𝑡𝑑𝜉,(4.9) are achieved. The above equations for 𝑢𝑁(𝑦,𝑡) and 𝜏𝑁(𝑦,𝑡) can be written under classical forms𝑢𝑁𝑦(𝑦,𝑡)=𝑈𝑒𝑟𝑓𝑐2𝜈𝑡,𝜏𝑁(𝑦,𝑡)=𝜇𝑈𝑦𝜋𝜈𝑡exp24𝜈𝑡,(4.10) corresponding to the first problem of Stokes.

5. Numerical Results and Discussion

In order to reveal some relevant physical aspects of the obtained results, several graphs are sketched in this section. A series of diagrams of the velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) against 𝑦 were performed for different situations with typical values. For example, we chose 𝑈=1, 𝜌=1, and 𝜈=1 for simplicity, and different values for 𝜆1, 𝜆2, 𝜆3, and 𝜆4 were chosen to illustrate their effects on the fluid motion. From Figure 1, it is clear that the velocity is an increasing function with respect to 𝑡, while the shear stress in absolute value decreases with regard to 𝑡. Both are decreasing functions with respect to 𝑦. Figure 2 shows the variations of the two physical entities with respect to the kinematic viscosity 𝜈. As it was to be expected, both the velocity and the shear stress (of course, in absolute value) are increasing functions with respect to 𝜈.

fig1
Figure 1: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆1=3,𝜆2=4,𝜆3=2,𝜆4=8, and different values of 𝑡.
fig2
Figure 2: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆1=3,𝜆2=4,𝜆3=2,𝜆4=8,𝑡=2s, and different values of 𝜈.

The influence of the relaxation and retardation times 𝜆1 and 𝜆3 on the fluid motion is underlined by Figures 3 and 4. Their effects, as expected, are opposite. More exactly, both the velocity and the shear stress are decreasing functions with respect to 𝜆1 and increasing ones with regard to 𝜆3. Figures 5 and 6 show the influence of the other two material parameters on the fluid motion. From these figures, it is clear that 𝜆2 and 𝜆4 have opposite effects upon velocity on the whole domain and shear stress on a part only. More exactly, the velocity of the fluid is everywhere a decreasing function with respect to 𝜆2 and an increasing one with regard to 𝜆4 exist. The shear stress is a decreasing function of 𝜆2 on the whole domain and of 𝜆4 near the plate. The effects of 𝜆1 and 𝜆2 on the fluid motion and of 𝜆3 and 𝜆4 upon velocity are qualitatively the same.

fig3
Figure 3: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆2=4,𝜆3=2,𝜆4=8,𝑡=2s, and different values of 𝜆1.
fig4
Figure 4: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆1=8,𝜆2=2,𝜆4=8,𝑡=2s, and different values of 𝜆3.
fig5
Figure 5: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆1=5.5,𝜆3=2,𝜆4=6,𝑡=2s, and different values of 𝜆2.
fig6
Figure 6: Profiles of the velocity field 𝑢(𝑦,𝑡) and the shear stress and 𝜏(𝑦,𝑡) given by (3.7) and (3.11), for 𝜆1=3,𝜆2=4,𝜆3=2,𝑡=2s, and different values of 𝜆4.

Finally, for comparison, the profiles of the velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) corresponding to three models (Newtonian, Oldroyd-B, and generalized Burgers’) are together depicted in Figure 7 for the same values of 𝑡 and the common material constants. It is clearly seen from these figures that the Newtonian fluid is the swiftest and the generalized Burgers’ fluid is the slowest. Furthermore, the non-Newtonian effects disappear in time and the behavior of Oldroyd-B and generalized Burgers’ fluids, as it results from Figure 8, can be well enough approximated by that of the Newtonian fluids. From the expressions (4.6) and (4.8) of the velocity field 𝑢(𝑦,𝑡), it results that the required time to reach the steady state is lower for Newtonian fluids in comparison with second-grade fluids. A comparison with other types of fluids has been also realized by graphical illustrations.

fig7
Figure 7: Profiles of the velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) for generalized Burgers’, Oldroyd-B and Newtonian fluids, for 𝜆1=4,𝜆2=2,𝜆3=3,𝜆4=3, and 𝑡=3s.
fig8
Figure 8: Profiles of the velocity 𝑢(𝑦,𝑡) for generalized Burgers’, Oldroyd-B, and Newtonian fluids, for 𝜆1=4,𝜆2=2,𝜆3=3,𝜆4=3, and 𝑡=7s and 𝑡=20s.

6. Concluding Remarks

In this paper, the velocity field 𝑢(𝑦,𝑡) and the adequate shear stress 𝜏(𝑦,𝑡) corresponding to the first problem of Stokes for generalized Burgers’ fluids are determined using the Fourier sine and Laplace transforms. The solutions that have been obtained are presented under integral form in terms of the elementary functions sin(), cos(), and exp() and satisfy all imposed initial and boundary conditions. They are written as a sum of steady and transient solutions and can be easily reduced to give the similar solutions for Burgers’ fluids. The steady solutions𝑢𝑆(𝑦)=𝑢𝑆(𝑦,)=𝑈,𝜏𝑆(𝑦)=𝜏(𝑦,)=0(6.1) are the same for both types of fluids if the conditions 𝜆1𝜆3𝜆2+𝜆4>2𝜆1𝜆3𝜆4 and 𝜆1𝜆3𝜆2>0 are satisfied. Furthermore, they are also identical to the steady solutions corresponding to Oldroyd-B, Maxwell, and second-grade and Newtonian fluids. The required time to reach the steady-state can be easily determined by graphical illustrations. It depends of the material constants and differs from a fluid to another one.

The general solutions (3.7) and (3.11) presented in the simplest forms, and their correctness has been graphically verified by comparison with the known solutions for Oldroyd-B and Newtonian fluids. More exactly, from Figures 9 and 10, it clearly results that for small values of the material constants 𝜆2 and 𝜆4 or 𝜆1, 𝜆2, 𝜆3, and 𝜆4, as expected, the diagrams of these solutions are almost identical to those corresponding to Oldroyd-B and Newtonian fluids, respectively. Finally, in order to bring light on some relevant physical aspects of the obtained results, the influence of the material parameters on the fluid motion is underlined by graphical illustrations. A comparison between the Newtonian, Oldrotd-B, and generalized Burgers’ fluid is also realized. The main outcomes of this study are as follows.(i)The general solutions (3.7) and (3.11) are presented under simple forms as a sum of steady and transient solutions. They have been immediately particularized to give the similar solutions for Burgers’ fluids.(ii)As a check of our calculi, we showed that for small values of the material constants 𝜆2 and 𝜆4 or 𝜆1, 𝜆2, 𝜆3, and 𝜆4 the diagrams of these solutions as it was to be expected, are almost identical to those corresponding to Oldroyd-B and Newtonian fluids.(iii)The velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) (in absolute value) are increasing functions with respect to 𝜈.(iv)The relaxation and retardation times, 𝜆1 and 𝜆3, as expected, have opposite effects on the fluid motion. Both the velocity and the shear stress (in absolute value) are decreasing functions with respect to 𝜆1 and increasing ones with regard to 𝜆3.(v)The other two material constants 𝜆2 and 𝜆4 have opposite effects on the velocity on the whole flow domain. Their effect on the shear stress is qualitatively the same near the plate and different in rest. Roughly speaking, the effects of 𝜆2 and 𝜆4 on the fluid velocity are qualitatively the same as those of 𝜆1 and 𝜆3.(vi)The Newtonian fluid is the swiftest, and the generalized Burgers’ fluid is the slowest. The non-Newtonian effects disappear in time, and the required time to reach the steady-state is the lowest for Newtonian fluid.

fig9
Figure 9: Profiles of the velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) for generalized Burgers’ and Oldroyd-B fluids, for 𝜆1=3,𝜆2=0.00001,𝜆3=2.5,𝜆4=0.00001, and different values of 𝑡.
fig10
Figure 10: Profiles of the velocity 𝑢(𝑦,𝑡) and the shear stress 𝜏(𝑦,𝑡) for generalized Burgers’ and Newtonian fluids, for 𝜆1=0.00001, 𝜆2=0.00001, 𝜆3=0.00001, 𝜆4=0.00001, and different values of 𝑡.

Appendix

1𝜆2𝑞2+𝜆1𝑞+1𝜆2𝑞3+𝜆1+𝜆4𝜈𝜉2𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2=1𝜆21𝜆2𝑞2+𝜆1𝑞+1𝑞𝑞1𝑞𝑞2𝑞𝑞3=1𝜆21𝜆2𝑞21+𝜆1𝑞1+1𝑞1𝑞2𝑞1𝑞3𝑞𝑞1+𝜆2𝑞22+𝜆1𝑞2+1𝑞2𝑞1𝑞2𝑞3𝑞𝑞2+𝜆2𝑞22+𝜆1𝑞2+1𝑞3𝑞1𝑞3𝑞2𝑞𝑞3=1𝜆2𝜆2𝑞21+𝜆1𝑞1𝑒+1𝑞1𝑡𝑞1𝑞2𝑞1𝑞3+𝜆2𝑞22+𝜆1𝑞2𝑒+1𝑞2𝑡𝑞2𝑞1𝑞2𝑞3+𝜆2𝑞23+𝜆1𝑞3𝑒+1𝑞3𝑡𝑞3𝑞1𝑞3𝑞2,0sin(𝑦𝜉)𝜉𝜋𝑑𝜉=2,𝑦>0,(A.1)1𝜆4𝑞2+𝜆3𝑞+1𝜆2𝑞3+𝜆1+𝜆4𝜈𝜉2𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2=1𝜆21𝜆2𝑞2+𝜆1𝑞+1𝑞𝑞1𝑞𝑞2𝑞𝑞3=1𝜆21𝜆4𝑞21+𝜆3𝑞1+1𝑞1𝑞2𝑞1𝑞3𝑞𝑞1+𝜆4𝑞22+𝜆3𝑞2+1𝑞2𝑞1𝑞2𝑞3𝑞𝑞2+𝜆4𝑞22+𝜆3𝑞2+1𝑞3𝑞1𝑞3𝑞2𝑞𝑞3=1𝜆2𝜆4𝑞21+𝜆3𝑞1𝑒+1𝑞1𝑡𝑞1𝑞2𝑞1𝑞3+𝜆4𝑞22+𝜆3𝑞2𝑒+1𝑞2𝑡𝑞2𝑞1𝑞2𝑞3+𝜆4𝑞23+𝜆3𝑞3𝑒+1𝑞3𝑡𝑞3𝑞1𝑞3𝑞2,(A.2)1𝜆1𝑞+1𝜆1𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2=1𝜆11𝜆1𝑞+1𝑞𝑞7𝑞𝑞8=1𝜆11𝜆1𝑞+1𝑞7𝑞8𝑞𝑞7+𝜆1𝑞+1𝑞8𝑞7𝑞𝑞8=1𝜆1𝜆1𝑒𝑞+1𝑞7𝑡𝑞7𝑞8+𝜆1𝑒𝑞+1𝑞8𝑡𝑞8𝑞7,1𝜆3𝑞+1𝜆1𝑞2+1+𝜆3𝜈𝜉2𝑞+𝜈𝜉2=1𝜆11𝜆3𝑞+1𝑞𝑞7𝑞𝑞8=1𝜆11𝜆3𝑞+1𝑞7𝑞8𝑞𝑞7+𝜆3𝑞+1𝑞8𝑞7𝑞𝑞8=1𝜆1𝜆3𝑒𝑞+1𝑞7𝑡𝑞7𝑞8+𝜆3𝑒𝑞+1𝑞8𝑡𝑞8𝑞7.(A.3)

Acknowledgments

The author Muhammad Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering and Technology, Karachi-75270, Pakistan; also Higher Education Commission of Pakistan for generous support, facilitating this research work.

References

  1. Z. S. Yu and J. Z. Lin, “Numerical research on the coherent structure in the viscoelastic second-order mixing layers,” Applied Mathematics and Mechanics, vol. 19, no. 8, pp. 717–723, 1998. View at Scopus
  2. K. R. Rajagopal, “On the boundary conditions for fluids of differential type,” in Navier-Stokes Equations and Related Non-Linear Problems, A. Sequeira, Ed., Plenum Press, New York, NY, USA, 1995.
  3. K. R. Rajagopal and P. N. Kaloni, “Some remarks on boundary conditions for fluids of differential type,” in Continuum Mechanics and its Applications, G. A. C. Gram and S. K. Malik, Eds., Hemisphere, Washington, DC, USA, 1989.
  4. R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, NY, USA, 1999.
  5. K. R. Rajagopal, “A note on unsteady unidirectional flows of a non-Newtonian fluid,” International Journal of Non-Linear Mechanics, vol. 17, no. 5-6, pp. 369–373, 1982. View at Scopus
  6. K. R. Rajagopal, “On the creeping flow of the second-order fluid,” Journal of Non-Newtonian Fluid Mechanics, vol. 15, no. 2, pp. 239–246, 1984. View at Scopus
  7. T. Hayat, R. Ellahi, and F. M. Mahomed, “Exact solutions for thin film flow of a third grade fluid down an inclined plane,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1336–1341, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. W. Tan and T. Masuoka, “Stability analysis of a Maxwell fluid in a porous medium heated from below,” Physics Letters, Section A, vol. 360, no. 3, pp. 454–460, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. C. Fetecau, T. Hayat, and C. Fetecau, “Starting solutions for oscillating motions of Oldroyd-B fluids in cylindrical domains,” Journal of Non-Newtonian Fluid Mechanics, vol. 153, no. 2-3, pp. 191–201, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. Z. Zhang, C. Fu, and W. Tan, “Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below,” Physics of Fluids, vol. 20, no. 8, Article ID 084103, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. M. A. Rana, A. M. Siddiqui, and R. Qamar, “Hall effects on hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 204–213, 2009. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Fetecau, M. Jamil, C. Fetecau, and I. Siddique, “A note on the second problem of Stokes for Maxwell fluids,” International Journal of Non-Linear Mechanics, vol. 44, no. 10, pp. 1085–1090, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Hyder Ali Muttaqi Shah, M. Khan, and H. Qi, “Exact solutions for a viscoelastic fluid with the generalized Oldroyd-B model,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2590–2599, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Fetecau, M. Jamil, C. Fetecau, and D. Vieru, “The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 60, no. 5, pp. 921–933, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. T. Hayat, S. Najam, M. Sajid, M. Ayub, and S. Mesloub, “On exact solutions for oscillatory flows in a generalized Burgers fluid with slip condition,” Zeitschrift fur Naturforschung Section A, vol. 65, no. 5, pp. 381–391, 2010.
  16. C. Fetecau, J. Zierep, R. Bohning, and C. Fetecau, “On the energetic balance for the flow of an Oldroyd-B fluid due to a flat plate subject to a time-dependent shear stress,” Computers and Mathematics with Applications, vol. 60, no. 1, pp. 74–82, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. M. Jamil and C. Fetecau, “Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4302–4311, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. M. Jamil, C. Fetecau, and M. Imran, “Unsteady helical flows of Oldroyd-B fluids,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1378–1386, 2011. View at Publisher · View at Google Scholar
  19. M. Jamil, A. Rauf, C. Fetecau, and N. A. Khan, “Helical flows of second grade fluid due to constantly accelerated shear stresses,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1959–1969, 2011. View at Publisher · View at Google Scholar
  20. J. E. Dunn and K. R. Rajagopal, “Fluids of differential type: critical review and thermodynamic analysis,” International Journal of Engineering Science, vol. 33, no. 5, pp. 689–729, 1995. View at Scopus
  21. K. R. Rajagopal, “Mechanics of non-Newtonian fluids in recent development in theoretical fluid Mechanics,” Pitman Research Notes in Mathematics Series, vol. 291, pp. 129–162, 1993.
  22. J. M. Burgers, “Mechanical considerations-model systems-phenomenological theories of relaxation and of viscosity,” in First Report on Viscosity and Plasticity, J. M. Burgers, Ed., Nordemann Publishing Company, New York, NY, USA, 1935.
  23. J. Murali Krishnan and K. R. Rajagopal, “Thermodynamic framework for the constitutive modeling of asphalt concrete: theory and applications,” Journal of Materials in Civil Engineering, vol. 16, no. 2, pp. 155–166, 2004. View at Publisher · View at Google Scholar · View at Scopus
  24. A. R. Lee and A. H. D. Markwick, “The mechanical properties of bituminous surfacing materials under constant stress,” Journal of the Indian Chemical Society, vol. 56, pp. 146–156, 1937.
  25. B. H. Tan, I. Jackson, and J. D. Fitz Gerald, “High-temperature viscoelasticity of fine-grained polycrystalline olivine,” Physics and Chemistry of Minerals, vol. 28, no. 9, pp. 641–664, 2001. View at Publisher · View at Google Scholar · View at Scopus
  26. W. R. Peltier, P. Wu, and D. A. Yuen, “The viscosities of the earth mantle,” in Anelasticity in the Earth, F. D. Stacey, M. S. Paterson, and A. Nicholas, Eds., American Geophysical Union, Colo, USA, 1981.
  27. P. Ravindran, J. M. Krishnan, and K. R. Rajagopal, “A note on the flow of a Burgers' fluid in an orthogonal rheometer,” International Journal of Engineering Science, vol. 42, no. 19-20, pp. 1973–1985, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. M. Khan, S. Hyder Ali, and C. Fetecau, “Exact solutions of accelerated flows for a Burgers' fluid. I. The case γ < λ2 / 4,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 881–894, 2008. View at Publisher · View at Google Scholar · View at Scopus
  29. T. Hayat, S. B. Khan, and M. Khan, “Influence of hall current on rotating flow of a burgers' fluid through a porous space,” Journal of Porous Media, vol. 11, no. 3, pp. 277–287, 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. M. Khan, R. Malik, C. Fetecau, and C. Fetecau, “Exact solutions for the unsteady flow of a Burgers' fluid between two sidewalls perpendicular to the plate,” Chemical Engineering Communications, vol. 197, no. 11, pp. 1367–1386, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. D. Tong, “Starting solutions for oscillating motions of a generalized Burgers' fluid in cylindrical domains,” Acta Mechanica, vol. 214, no. 3-4, pp. 395–407, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. M. Jamil and C. Fetecau, “Some exact solutions for rotating flows of a generalized Burgers' fluid in cylindrical domains,” Journal of Non-Newtonian Fluid Mechanics, vol. 165, no. 23-24, pp. 1700–1712, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. M. Jamil, A. A. Zafar, C. Fetecau, and N. A. Khan, “Exact analytic solutions for the flow of a generalized burgers fluid induced by an accelerated shear stress,” Chemical Engineering Communications, vol. 199, no. 1, pp. 17–39, 2012. View at Publisher · View at Google Scholar
  34. R. I. Tanner, “Note on the Rayleigh problem for a visco-elastic fluid,” Zeitschrift für Angewandte Mathematik und Physik, vol. 13, pp. 573–580, 1962.
  35. P. N. Srivastava, “Non-steady helical flow of a visco-elastic liquid,” Archives of Mechanics, vol. 18, pp. 145–150, 1966.
  36. I. C. Christov, “Stokes' first problem for some non-Newtonian fluids: results and mistakes,” Mechanics Research Communications, vol. 37, no. 8, pp. 717–723, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, NY, USA, 1951.
  38. A. Kuros, Cours d’Algebre Superieure, Edition Mir Moscow, 1973.
  39. F. R. Gantmacher, Applications of the Theory of Matreices, Wiley, New York, NY, USA, 1959.
  40. C. Fetecau, T. Hayat, M. Khan, and C. Fetecau, “Erratum: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate (Acta Mechanica, vol. 198 no. 21–33, 2008,” Acta Mechanica, vol. 216, no. 1–4, pp. 359–361, 2011. View at Publisher · View at Google Scholar