# First Problem of Stokes for Generalized Burgers' Fluids

Authors: Muhammad Jamil
Journal: ISRN Mathematical Physics
 (2012)

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## 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.


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## Body

## 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 [4–19].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 [27–33] 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 parametersλ1, 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 tensorT for an incompressible generalized Burgers’ fluid is characterized by the following constitutive equations [30–33]:(2.1)T=-pI+S,S+λ1δSδt+λ2δ2Sδt2=μ[A+λ3δAδt+λ4δ2Aδt2],
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 [30–33].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 fieldV and an extra-stress tensor S of the form(2.2)V=V(y,t)=u(y,t)i,S=S(y,t),
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, then(2.3)V(y,0)=0,S(y,0)=∂S(y,0)∂t=0.
Equations (2.1) and (2.3) imply Syy=Syz=Szz=Sxz=0, and the meaningful equation(2.4)(1+λ1∂∂t+λ2∂2∂t2)τ(y,t)=μ(1+λ3∂∂t+λ4∂2∂t2)∂u(y,t)∂y,
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(2.5)∂τ(y,t)∂y-∂p∂x=ρ∂u(y,t)∂t,∂p∂y=∂p∂z=0.
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 form(2.6)(1+λ1∂∂t+λ2∂2∂t2)∂u(y,t)∂t=ν(1+λ3∂∂t+λ4∂2∂t2)∂2u(y,t)∂y2;y,t>0.Consider an incompressible generalized Burgers’ fluid occupying the space above a flat plate perpendicular to they-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 [34–36] is(2.7)u(y,0)=∂u(y,0)∂t=∂2u(y,0)∂t2=0,y>0,(2.8)u(0,t)=UH(t),t≥0,
where H(t) is the Heaviside function. Moreover, the natural conditions(2.9)u(y,t),∂u(y,t)∂y⟶0asy⟶∞,t>0
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 that(3.1)(1+λ1∂∂t+λ2∂2∂t2)∂us∂t+νξ2(1+λ3∂∂t+λ4∂2∂t2)us=νξU2π[H(t)+λ3δ(t)+λ4δ′(t)],
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 by(3.2)us(ξ,t)=2π∫0∞u(y,t)sin⁡(yξ)dy,
has to satisfy the initial conditions(3.3)us(ξ,0)=∂us(ξ,0)∂t=∂2us(ξ,0)∂t2=0,ξ>0.
By applying the Laplace transform to (3.1) and having in mind the initial conditions (2.7), we find that(3.4)u¯s(ξ,q)=νξU2πλ4q2+λ3q+1q[λ2q3+{λ1+λ4νξ2}q2+{1+λ3νξ2}q+νξ2].
Now, for a more suitable presentation of the final results, we rewrite (3.4) in the following equivalent form:(3.5)u¯s(ξ,q)=U2π1ξ[1q-λ2q2+λ1q+1λ2q3+{λ1+λ4νξ2}q2+{1+λ3νξ2}q+νξ2].
Inverting (3.5) by means of the Fourier sine formula, we can write u¯(y,q) as(3.6)u¯(y,q)=2Uπ∫0∞sin⁡(yξ)ξ[1q-λ2q2+λ1q+1λ2q3+{λ1+λ4νξ2}q2+{1+λ3νξ2}q+νξ2]dξ.
Finally, in order to obtain the velocity field u(y,t)=ℒ-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:(3.7)u(y,t)=UH(t)-2UH(t)πλ2∫0∞sin⁡(yξ)ξ×[(λ2q12+λ1q1+1)eq1t(q1-q2)(q1-q3)+(λ2q22+λ1q2+1)eq2t(q2-q1)(q2-q3)+(λ2q32+λ1q3+1)eq3t(q3-q1)(q3-q2)]dξ,
where(3.8)qi=si-λ1+λ4νξ23λ2,i=1,2,3,
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]),(3.9)s1=-β12+β124+α13273+-β12-β124+α13273,s2=Z-β12+β124+α13273+Z2-β12-β124+α13273,s3=Z2-β12+β124+α13273+Z-β12-β124+α13273
are the roots of the algebraic equation X3+α1X+β1=0, where(3.10)α1=1+λ3νξ2λ2-(λ1+λ4νξ2)23λ22,β1=νξ2λ2+2(λ1+λ4νξ2)327λ23-(λ1+λ4νξ2)(1+λ3νξ2)3λ22,Z=-1+i32.
From Routh-Hurwitz’s principle [39], we get Re(qi)<0 if λ1λ3-λ2+λ4>-2λ1λ3λ4, provided λ1,λ2,λ3,λ4>0. The corresponding shear stress (see also (A.2))(3.11)τ(y,t)=-2μUH(t)πλ2∫0∞cos⁡⁡(yξ)×[(λ4q12+λ3q1+1)eq1t(q1-q2)(q1-q3)+(λ4q22+λ3q2+1)eq2t(q2-q1)(q2-q3)+(λ4q32+λ3q3+1)eq3t(q3-q1)(q3-q2)]dξ,
is obtained in the same way from (2.4).


## 4. Special Cases

### 4.1. Burgers’ Fluid

Makingλ4→0 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

Makingλ2 and λ4=0 into (3.6) and following the same way as before, we get the velocity field (see also (A.3))(4.1)uOB(y,t)=UH(t)-2UH(t)πλ1∫0∞sin⁡(yξ)ξ[(λ1q8+1)eq8t-(λ1q7+1)eq7tq8-q7]dξ,
and the shear stress(4.2)τOB(y,t)=-2μUH(t)πλ1∫0∞cos⁡⁡(yξ)[(λ3q8+1)eq8t-(λ3q7+1)eq7tq8-q7]dξ,
corresponding to an Oldroyd-B fluid. In the above relations,(4.3)q7,q8=-{1+λ3νξ2}±{1+λ3νξ2}2-4νλ1ξ22λ1
and (4.1) is identical to (15) from [40].


### 4.3. Maxwell Fluid

Making the limit of (4.1) and (4.2) as λ3→0, we obtain the solutions(4.4)uM(y,t)=UH(t)-2UH(t)πλ1∫0∞sin⁡(yξ)ξ[(λ1q10+1)eq10t-(λ1q9+1)eq9tq10-q9]dξ,τM(y,t)=-2μUH(t)πλ1∫0∞cos⁡(yξ)[eq10t-eq9tq10-q9]dξ,
corresponding to a Maxwell fluid. The new roots q9 and q10 are given by(4.5)q9,q10=-1±1-4νλ1ξ22λ1.


### 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 restrictionλ≥λr and make λ1→0 into (4.1) and (4.2), we recover the expressions(4.6)uSG(y,t)=UH(t)[1-2π∫0∞sin⁡(yξ)ξ(1+λ3νξ2)exp⁡(-νξ2t1+λ3νξ2)dξ],(4.7)τSG(y,t)=-2μUH(t)π∫0∞cos⁡(yξ)1+λ3νξ2exp⁡(-νξ2t1+λ3νξ2)dξ,
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, makingλ1→0 into (4.4) or λ3→0 into (4.6) and (4.7), the solutions for a Newtonian fluid(4.8)uN(y,t)=UH(t)[1-2π∫0∞sin⁡(yξ)ξe-νξ2tdξ],(4.9)τN(y,t)=-2μUH(t)π∫0∞cos⁡(yξ)e-νξ2tdξ,
are achieved. The above equations for uN(y,t) and τN(y,t) can be written under classical forms(4.10)uN(y,t)=Uerfc(y2νt),τN(y,t)=-μUπνtexp⁡(-y24νt),
corresponding to the first problem of Stokes.


## 4.1. Burgers’ Fluid

Makingλ4→0 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

Makingλ2 and λ4=0 into (3.6) and following the same way as before, we get the velocity field (see also (A.3))(4.1)uOB(y,t)=UH(t)-2UH(t)πλ1∫0∞sin⁡(yξ)ξ[(λ1q8+1)eq8t-(λ1q7+1)eq7tq8-q7]dξ,
and the shear stress(4.2)τOB(y,t)=-2μUH(t)πλ1∫0∞cos⁡⁡(yξ)[(λ3q8+1)eq8t-(λ3q7+1)eq7tq8-q7]dξ,
corresponding to an Oldroyd-B fluid. In the above relations,(4.3)q7,q8=-{1+λ3νξ2}±{1+λ3νξ2}2-4νλ1ξ22λ1
and (4.1) is identical to (15) from [40].


## 4.3. Maxwell Fluid

Making the limit of (4.1) and (4.2) as λ3→0, we obtain the solutions(4.4)uM(y,t)=UH(t)-2UH(t)πλ1∫0∞sin⁡(yξ)ξ[(λ1q10+1)eq10t-(λ1q9+1)eq9tq10-q9]dξ,τM(y,t)=-2μUH(t)πλ1∫0∞cos⁡(yξ)[eq10t-eq9tq10-q9]dξ,
corresponding to a Maxwell fluid. The new roots q9 and q10 are given by(4.5)q9,q10=-1±1-4νλ1ξ22λ1.


## 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 restrictionλ≥λr and make λ1→0 into (4.1) and (4.2), we recover the expressions(4.6)uSG(y,t)=UH(t)[1-2π∫0∞sin⁡(yξ)ξ(1+λ3νξ2)exp⁡(-νξ2t1+λ3νξ2)dξ],(4.7)τSG(y,t)=-2μUH(t)π∫0∞cos⁡(yξ)1+λ3νξ2exp⁡(-νξ2t1+λ3νξ2)dξ,
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, makingλ1→0 into (4.4) or λ3→0 into (4.6) and (4.7), the solutions for a Newtonian fluid(4.8)uN(y,t)=UH(t)[1-2π∫0∞sin⁡(yξ)ξe-νξ2tdξ],(4.9)τN(y,t)=-2μUH(t)π∫0∞cos⁡(yξ)e-νξ2tdξ,
are achieved. The above equations for uN(y,t) and τN(y,t) can be written under classical forms(4.10)uN(y,t)=Uerfc(y2νt),τN(y,t)=-μUπνtexp⁡(-y24νt),
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 velocityu(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.Profiles of the velocity fieldu(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.
(a)(b)Profiles of the velocity fieldu(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.
(a)(b)The influence of the relaxation and retardation timesλ1 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.Profiles of the velocity fieldu(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.
(a)(b)Profiles of the velocity fieldu(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.
(a)(b)Profiles of the velocity fieldu(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.
(a)(b)Profiles of the velocity fieldu(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.
(a)(b)Finally, for comparison, the profiles of the velocityu(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.Profiles of the velocityu(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.
(a)(b)Profiles of the velocityu(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.
(a)(b)


## 6.  Concluding Remarks

In this paper, the velocity fieldu(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 solutions(6.1)uS(y)=uS(y,∞)=U,τS(y)=τ(y,∞)=0
are the same for both types of fluids if the conditions λ1λ3-λ2+λ4>-2λ1λ3λ4 and λ1λ3-λ2>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 constantsλ2 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 velocityu(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 constantsλ2 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.Profiles of the velocityu(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.
(a)(b)Profiles of the velocityu(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.
(a)(b)



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*Source: 831063-2012-03-04.xml*