Abstract

Exact analytic solutions are obtained for the flow of a generalized second grade fluid in an annular region between two infinite coaxial cylinders. The fractional calculus approach in the governing equations of a second grade fluid is used. The exact analytic solutions are constructed by means of Laplace and finite Hankel transforms. The motion is produced by the inner cylinder which is rotating about its axis due to a constantly accelerating shear. The solutions that have been obtained satisfy both the governing equations and all imposed initial and boundary conditions. Moreover, they can be easily specialized to give similar solutions for second grade and Newtonian fluids. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between the three models, is underlined by graphical illustrations.

1. Introduction

The study of the non-Newtonian fluids has recently achieved much importance because of well-established applications in a number of processes that occur in industry. Such applications include the extrusion of polymer fluids, cooling of the metallic plate in a bath, animal bloods, foodstuffs, exotic lubricants and colloidal and suspension solutions. For these fluids, the classical Navier-Stokes theory is inadequate. Because of their complexity, there are several models of non-Newtonian fluids in the literature. One of the most popular models for non-Newtonian fluids is the model that is called second-grade fluid [1]. Although there are some criticisms regarding the applications of this model [2], it has been shown by Walters [3] that, for many types of problems in which the flow is slow enough in the viscoelastic sense, the results given using Oldroyd fluid will be substantially similar to those obtained for second grade fluid. Thus, if this is the manner of interpretation of the results, it is reasonable to use the second-grade fluid [4–6] to carry out the calculations. This is particularly so because of the fact that the calculations are generally simpler. This is true not only for exact analytic solutions but even for numerical solutions. The second-grade fluid is the simplest subclass of non-Newtonian fluids for which one can reasonably hope to obtain exact analytic solutions. Moreover, the exact analytic solutions are very important for several reasons. They provide a standard for checking the accuracies of many approximate solutions which can be numerical or empirical. These exact solutions can also be used as tests for verifying numerical schemes that are developed for studying more complex flow problems. Therefore, various researchers [7–9] are engaged in obtaining exact solutions.

Recently, the fractional calculus has encountered much success in the description of complex dynamics. In particular, it has been proved to be a valuable tool for handling viscoelastic properties. The starting point of the fractional derivative model of non-Newtonian fluids is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the so-called Riemann-Liouville fractional operator. This generalization allows one to define precisely noninteger order derivatives. The fractional calculus has been found to be quite flexible in describing viscoelastic behavior of fluids. In many different situations fractional calculus has been used to handle various rheological problems [10–21].

The aim of this note is to provide exact solutions for the flow of a generalized second grade fluid in the annular region between two infinite coaxial circular cylinders. The motion is produced by the inner cylinder that applies a time-dependent couple to the fluid. More exactly, we would like to extend the results from [7, Section 5] to a larger class of fluids and to a time-dependent couple on the boundary. The general solutions, obtained by means of the integral transforms, will be easily specialized to give the similar solutions for Newtonian and ordinary second grade fluids. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, will be underlined by graphical illustrations.

2. Governing Equations

The flows to be here considered have the velocity field of the form [22, 23] 𝐯=𝐯(𝑟,𝑡)=𝑤(𝑟,𝑡)𝐞𝜃,(2.1) where 𝐞𝜃 is the unit vector along the 𝜃-direction of the cylindrical coordinate system 𝑟, 𝜃, and 𝑧. For such flows the constraint of incompressibility is automatically satisfied. The nontrivial shear stress 𝜏(𝑟,𝑡)=𝑆𝑟𝜃(𝑟,𝑡) corresponding to such a motion of a second grade fluid is given by [24] 𝜏(𝑟,𝑡)=𝜇+𝛼1𝜕𝜕𝑡𝜕𝑤(𝑟,𝑡)−𝜕𝑟𝑤(𝑟,𝑡)𝑟,(2.2) where 𝜇 is the viscosity and 𝛼1 ia a material modulus. In the absence of a pressure gradient in the flow direction and neglecting the body forces, the balance of the linear momentum leads to the relevant equation [25, 26] 𝜌𝜕𝑤(𝑟,𝑡)=𝜕𝜕𝑡+2𝜕𝑟𝑟𝜏(𝑟,𝑡).(2.3)

Eliminating 𝜏(𝑟,𝑡) between (2.2) and (2.3), we get the governing equation 𝜕𝑤(𝑟,𝑡)=𝜕𝜕𝑡𝜈+𝛼𝜕𝜕𝑡2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑡),(2.4) where 𝜈=𝜇/𝜌 is the kinematic viscosity of the fluid, 𝜌 is its constant density and 𝛼=𝛼1/𝜌.

Generally, governing equations for generalized fluids with fractional derivatives are derived from those of the ordinary fluids by replacing the inner time derivatives of an integer order with the so-called Riemann-Liouville operator [11, 27] 𝐷𝛽𝑡1𝑓(𝑡)=𝑑Γ(1−𝛽)𝑑𝑡𝑡0𝑓(𝜏)(𝑡−𝜏)𝛽𝑑𝜏,0≤𝛽<1,(2.5) where Γ(⋅) is the Gamma function.

Consequently, the governing equations corresponding to the motion (2.1) of a generalized second grade fluid are (cf. [22, Equations (2) and (4)]) 𝜕𝑤(𝑟,𝑡)=𝜕𝑡𝜈+𝛼𝐷𝛽𝑡𝜕2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑡);𝜏(𝑟,𝑡)=𝜇+𝛼1𝐷𝛽𝑡𝜕−1𝜕𝑟𝑟𝑤(𝑟,𝑡),(2.6) where the new material constant 𝛼1 (for simplicity, we are keeping the same notation) goes to the initial 𝛼1 for 𝛽→1.

In this paper, we are interested into the motion of a generalized second grade fluid whose governing equations are given by (2.6). The fractional partial differential equations (2.6), with adequate initial and boundary conditions, can be solved in principle by several methods, the integral transforms technique representing a systematic, efficient, and powerful tool. The Laplace transform will be used to eliminate the time variable and the finite Hankel transform to remove the spatial variable. However, in order to avoid the lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform will be used.

3. Rotational Flow between Two Infinite Cylinders

Consider an incompressible generalized second grade fluid at rest in the annular region between two infinitely long coaxial cylinders. At time 𝑡=0+, let the inner cylinder of radius 𝑅1 be set in rotation about its axis by a time-dependent torque per unit length 2𝜋𝑅1𝑓𝑡 and let the outer cylinder of radius 𝑅2 be held stationary. Owing to the shear, the fluid between cylinders is gradually moved, its velocity being of the form (2.1). The governing equations are given by (2.6) and the appropriate initial and boundary conditions are (see also [7, Equations (5.2) and (5.3)]) 𝑤𝑅(𝑟,0)=0;𝑟∈1,𝑅2𝜏𝑅,(3.1)1=,𝑡𝜇+𝛼1𝐷𝛽𝑡𝜕𝑤(𝑟,𝑡)−𝜕𝑟𝑤(𝑟,𝑡)𝑟|𝑟=𝑅1𝑅=𝑓𝑡,𝑤2,𝑡=0;𝑡≥0,(3.2) where f is a constant.

3.1. Calculation of the Velocity Field

Applying the Laplace transform to (2.6)1 and (3.2), we get 𝑞𝑤(𝑟,𝑞)=𝜈+𝛼𝑞𝛽𝜕2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑞),(3.3)𝜏𝑅1=,𝑞𝜇+𝛼1𝑞𝛽𝜕−1𝜕𝑟𝑟𝑤(𝑟,𝑞)|𝑟=𝑅1=𝑓𝑞2;𝑤𝑅2,𝑞=0,(3.4) where 𝑤(𝑟,𝑞) and 𝜏(𝑅1,𝑞) are the Laplace transforms of the functions 𝑤(𝑟,𝑡) and 𝜏(𝑅1,𝑡), respectively.

We denote by [22, Equation (34)] 𝑤𝐻𝑟𝑛=,𝑞𝑅2𝑅1𝑟𝑤(𝑟,𝑞)𝐵𝑟𝑟𝑛𝑑𝑟,(3.5) the finite Hankel transform of the function 𝑤(𝑟,𝑞), where 𝐵𝑟𝑟𝑛=𝐽1𝑟𝑟𝑛𝑌2𝑅1𝑟𝑛−𝐽2𝑅1𝑟𝑛𝑌1𝑟𝑟𝑛,(3.6) where 𝑟𝑛 are the positive roots of the equation 𝐵(𝑅2𝑟)=0, and 𝐽𝑝(⋅), 𝑌𝑝(⋅) are the Bessel functions of the first and second kind of order 𝑝.

The inverse Hankel transform of 𝑤𝐻(𝑟𝑛,𝑞) is given by [22, Equation (35)] 𝜋𝑤(𝑟,𝑞)=22∞𝑛=1𝑟2𝑛𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛𝑤𝐻𝑟𝑛.,𝑞(3.7)

By means of (3.4)2 and of the identity 𝐽1(𝑧)𝑌2(𝑧)−𝐽2(𝑧)𝑌12(𝑧)=−,𝜋𝑧(3.8) we can easily prove that 𝑅2𝑅1𝑟𝜕2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑞)𝐵𝑟𝑟𝑛𝑑𝑟=−𝑟2𝑛𝑤𝐻𝑟𝑛+2,𝑞𝜋𝑟𝑛𝜕−1𝜕𝑟𝑟𝑤(𝑟,𝑞)|𝑟=𝑅1.(3.9)

Combining (3.3), (3.4), and (3.9), we find that 𝑤𝐻𝑟𝑛=,𝑞2𝑓𝜋𝑟𝑛1𝑞21𝜌𝑞+𝛼1𝑞𝛽𝑟2𝑛+𝜇𝑟2𝑛.(3.10) Writing 𝑤𝐻(𝑟𝑛,𝑞) under the equivalent form 𝑤𝐻𝑟𝑛=,𝑞2𝑓𝜇𝜋𝑟3𝑛1𝑞2−1+𝛼𝑟2𝑛𝑞𝛽−1𝑞𝑞+𝜈+𝛼𝑞𝛽𝑟2𝑛=2𝑓𝜇𝜋𝑟3𝑛1𝑞2−𝑞−𝛽−1+𝛼𝑟2𝑛𝑞−2𝑞1−𝛽+𝛼𝑟2𝑛+𝜈𝑟2𝑛𝑞−𝛽,(3.11) and applying the inverse Hankel transform and using the identities 1𝑞1−𝛽+𝛼𝑟2𝑛+𝜈𝑟2𝑛𝑞−𝛽=∞𝑘=0−𝜈𝑟2𝑛𝑘𝑞−𝛽𝑘𝑞1−𝛽+𝛼𝑟2𝑛𝑘+1,𝜋(3.12)∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛=12𝑅1𝑅22𝑅𝑟−22𝑟,(3.13) we find that 𝑓𝑤(𝑟,𝑞)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟1𝑞2−𝜋𝑓𝜇∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛×∞𝑘=0−𝜈𝑟2𝑛𝑘𝑞1−𝛽+𝛼𝑟2𝑛𝑘+1𝑞−𝛽𝑘−𝛽−1+𝛼𝑟2𝑛𝑞−𝛽𝑘−2.(3.14)Now applying the inverse Laplace transform to (3.14), we find for the velocity field the expression 𝑓𝑤(𝑟,𝑡)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟𝑡−𝜋𝑓𝜇∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛×∞𝑘=𝑜−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−𝛽−1,𝑘+1−𝛼𝑟2𝑛,𝑡+𝛼𝑟2𝑛𝐺1−𝛽,−𝛽𝑘−2,𝑘+1−𝛼𝑟2𝑛,,𝑡(3.15) where the generalized function 𝐺𝑎,𝑏,𝑐(𝑑,𝑡) is defined by [28, Equations (97) and (101)] 𝐺𝑎,𝑏,𝑐(𝑑,𝑡)=𝐿−1𝑞𝑏(𝑞𝑎−𝑑)𝑐=∞𝑗=0𝑑𝑗Γ(𝑐+𝑗)𝑡Γ(𝑐)Γ(𝑗+1)(𝑐+𝑗)𝑎−𝑏−1Γ[]||||𝑑(𝑐+𝑗)𝑎−𝑏;Re(𝑎𝑐−𝑏)>0,𝑞𝑎||||<1.(3.16)

3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.6)2, we find that 𝜏(𝑟,𝑞)=𝜇+𝛼1𝑞𝛽𝜕−1𝜕𝑟𝑟𝑤(𝑟,𝑞).(3.17) In order to get a suitable form for 𝜏(𝑟,𝑡), we rewrite (3.10) under the equivalent form 𝑤𝐻𝑟𝑛=,𝑞2𝑓𝜋𝑟3𝑛1𝑞2𝜇+𝛼1𝑞𝛽−2𝑓𝜋𝑟3𝑛1𝑞𝜇+𝛼1𝑞𝛽𝑞+𝛼𝑞𝛽𝑟2𝑛+𝜈𝑟2𝑛.(3.18)

Applying the inverse Hankel transform to (3.18) and using (3.7) and the identity (3.13), we find that 𝑓𝑤(𝑟,𝑞)=2𝑅1𝑅22𝑅𝑟−22𝑟1𝑞2𝜇+𝛼1𝑞𝛽−𝜋𝑓∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛1𝑞𝜇+𝛼1𝑞𝛽𝑞+𝛼𝑞𝛽𝑟2𝑛+𝜈𝑟2𝑛.(3.19) Introducing (3.19) into (3.17), it results that 𝑅𝜏(𝑟,𝑞)=1𝑟2𝑓1𝑞2+𝜋𝑓∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛1𝑞𝑞+𝛼𝑞𝛽𝑟2𝑛+𝜈𝑟2𝑛,(3.20) or equivalently (see also (3.12)) 𝑅𝜏(𝑟,𝑞)=1𝑟2𝑓1𝑞2+𝜋𝑓∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛∞𝑘=0−𝜈𝑟2𝑛𝑘𝑞−𝛽𝑘−𝛽−1𝑞1−𝛽+𝛼𝑟2𝑛𝑘+1,(3.21) where 𝐵1(𝑟𝑟𝑛)=𝐽2(𝑟𝑟𝑛)𝑌2(𝑅1𝑟𝑛)−𝐽2(𝑅1𝑟𝑛)𝑌2(𝑟𝑟𝑛).

Now taking the inverse Laplace transform of both sides of (3.21), we get 𝑅𝜏(𝑟,𝑡)=1𝑟2𝑓𝑡+𝜋𝑓∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛∞𝑘=0−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−𝛽−1,𝑘+1−𝛼𝑟2𝑛.,𝑡(3.22)

4. The Special Case 𝛽→1

Making 𝛽→1 into (3.15) and (3.22), we obtain the similar solutions 𝑤SG𝑓(𝑟,𝑡)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟𝑡−𝜋𝑓𝜇∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛×1+𝛼𝑟2𝑛∞𝑘=0−𝜈𝑟2𝑛𝑘𝐺0,−(𝑘+2),𝑘+1−𝛼𝑟2𝑛,𝜏,𝑡SG𝑅(𝑟,𝑡)=1𝑟2𝑓𝑡+𝜋𝑓∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛∞𝑘=0−𝜈𝑟2𝑛𝑘𝐺0,−(𝑘+2),𝑘+1−𝛼𝑟2𝑛,,𝑡(4.1) corresponding to a second grade fluid performing the same motion.

Now, in view of the identity ∞𝑘=0−𝜈𝑟2𝑛𝑘𝐺0,−(𝑘+2),𝑘+1−𝛼𝑟2𝑛=1,𝑡𝜈𝑟2𝑛−1−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛,(4.2) equation (4.1) can be written under the simplified forms 𝑤SG𝑓(𝑟,𝑡)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟𝑡−𝜋𝑓𝜇𝜈∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟3𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛×1+𝛼𝑟2𝑛−1−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛,𝜏(4.3)SG𝑅(𝑟,𝑡)=1𝑟2𝑓𝑡+𝜋𝑓𝜈∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝑟2𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛−1−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛.(4.4) The velocity field can be also processed to give the equivalent form 𝑤SG𝑓(𝑟,𝑡)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟𝛼𝑡−1𝜇−𝜋𝑓𝜇𝜈∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟3𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛×1−1+𝛼𝑟2𝑛−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛.(4.5)

Making 𝛼1 and then 𝛼→0 into (4.3) and (4.4), the velocity field 𝑤𝑁𝑓(𝑟,𝑡)=𝑅2𝜇1𝑅22𝑅𝑟−22𝑟𝑡−𝜋𝑓𝜇𝜈∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵𝑟𝑟𝑛𝑟3𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛1−exp−𝜈𝑟2𝑛𝑡(4.6) and the associated shear stress 𝜏𝑁(𝑅𝑟,𝑡)=1𝑟2𝑓𝑡+𝜋𝑓𝜈∞𝑛=1𝐽21𝑅2𝑟𝑛𝐵1𝑟𝑟𝑛𝑟2𝑛𝐽22𝑅1𝑟𝑛−𝐽21𝑅2𝑟𝑛1−exp−𝜈𝑟2𝑛𝑡,(4.7) corresponding to a Newtonian fluid, are obtained.

5. Conclusions

The purpose of this note is to provide exact analytic solutions for the velocity field 𝑤(𝑟,𝑡) and the shear stress 𝜏(𝑟,𝑡) corresponding to the unsteady rotational flow of a generalized second grade fluid between two infinite coaxial cylinders, the inner cylinder being set in rotation about its axis by a constantly accelerating shear. The solutions that have been obtained, presented under series form in terms of usual Bessel (𝐽1(⋅), 𝐽2(⋅), 𝑌1(⋅), and 𝑌2(⋅)) and generalized 𝐺𝑎,𝑏,𝑐(⋅,𝑡) functions, satisfy all imposed initial and boundary conditions. They can be easily specialized to give the similar solutions for ordinary second grade and Newtonian fluids. Furthermore, in view of some recent results [29, Equation (3.15)], our velocity field (4.3) is in accordance with that obtained in [7, Equation (5.17)] by a different technique.

Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity 𝑤(𝑟,𝑡) and of the shear stress 𝜏(𝑟,𝑡) are depicted against 𝑟 for different values of the time 𝑡 and of the pertinent parameters. From Figures 1(a) and 1(b), containing the diagrams of the velocity and the shear stress at several times, it clearly results in the influence of the rigid boundary on the fluid motion. The velocity is an increasing function of 𝑡. For the same values of the parameters, the shear stress, in absolute value, is also an increasing function of 𝑡. The influence of the kinematic viscosity 𝜈 on the fluid motion is shown in Figures 2(a) and 2(b). The velocity is a decreasing function of 𝜈. The shear stress, in absolute value, on the first part of the flow domain, near the moving cylinder, is a decreasing function of 𝜈. It is an increasing function of 𝜈 in the neighborhood of the stationary cylinder. Figure 3 shows the influence of the parameter 𝛼 on the flow motion. Both the velocity and the shear stress, in absolute value, on the first part of the flow domain, near the moving cylinder, are increasing functions of 𝛼. They are decreasing functions of 𝛼 in the neighborhood of the stationary cylinder. The influence of the fractional parameter 𝛽 on the fluid motion is shown in Figure 4. Its effect on the fluid motion is qualitatively opposite to that of parameter 𝛼.

(a)
(a)
(b)
(b)
Figure 1 
Profiles of the velocity 𝜔 ( 𝑟 , 𝑡 ) and the shear stress 𝜏 ( 𝑟 , 𝑡 ) given by (3.15) and (3.22), for 𝑓 = − 1 ,   𝑅 1 = 0 . 3 , 𝑅 2 = 0 . 5 , 𝜈 = 0 . 0 0 0 2 5 9 , 𝜇 = 0 . 2 4 6 , 𝛼 = 0 . 0 0 1 5 3 , 𝛽 = 0 . 3 , and different values of 𝑡 .
(a)
(a)
(b)
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Figure 2 
Profiles of the velocity 𝜔 ( 𝑟 , 𝑡 ) and the shear stress 𝜏 ( 𝑟 , 𝑡 ) given by (3.15) and (3.22), for 𝑓 = − 1 , 𝑅 1 = 0 . 3 , 𝑅 2 = 0 . 5 , 𝛼 = 0 . 0 0 1 5 3 , 𝛽 = 0 . 3    𝑡 = 1 0  s and different values of 𝜈 .
(a)
(a)
(b)
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Figure 3 
Profiles of the velocity 𝜔 ( 𝑟 , 𝑡 ) and the shear stress 𝜏 ( 𝑟 , 𝑡 ) given by (3.15) and (3.22), for 𝑓 = − 1 , 𝑅 1 = 0 . 3 , 𝑅 2 = 0 . 5 , 𝜈 = 0 . 0 0 0 2 5 9 , 𝜇 = 0 . 2 4 6 , 𝛽 = 0 . 3 and different values of 𝛼 .
(a)
(a)
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Figure 4 
Profiles of the velocity 𝜔 ( 𝑟 , 𝑡 ) and the shear stress 𝜏 ( 𝑟 , 𝑡 ) given by (3.15) and (3.22), for 𝑓 = − 1 , 𝑅 1 = 0 . 3 , 𝑅 2 = 0 . 5 , 𝜈 = 0 . 0 0 0 2 5 9 , 𝜇 = 0 . 2 4 6 , 𝛼 = 0 . 0 0 1 5 3 and different values of 𝛽 .

Finally, for comparison, the profiles of 𝑤(𝑟,𝑡) and 𝜏(𝑟,𝑡) corresponding to the motion of the three models (Newtonian, ordinary second grade, and generalized second grade) are together depicted in Figure 5, for the same values of 𝑡 and of the common material parameters. In all the cases the velocity of the fluid is a decreasing function with respect to 𝑟, and the Newtonian fluid is the swiftest, while the generalized second grade fluid is the slowest in the region near the moving cylinder. The units of the material constants are SI units within all figures, and the roots 𝑟𝑛 have been approximated by 2(𝑛−1)𝜋/[2(𝑅2−𝑅1)].

(a)
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Figure 5 
Profiles of the velocity 𝜔 ( 𝑟 , 𝑡 ) and the shear stress 𝜏 ( 𝑟 , 𝑡 ) corresponding to Newtonian, second grade and generalized second grade fluids, for 𝑓 = − 1 , 𝑅 1 = 0 . 3 , 𝑅 2 = 0 . 5 , 𝜈 = 0 . 0 0 0 2 5 9 , 𝜇 = 0 . 2 4 6 , 𝛼 = 0 . 0 0 1 5 3    𝛽 = 0 . 3 and 𝑡 = 1 0  s.

Acknowledgments

The authors would like to express their gratitude to the referees for their careful assessment, constructive comments, and suggestions regarding the initial form of the manuscript. Specifically, M. Kamran is thankful to COMSATS Institute of Information Technology Wah Cantt Pakistan; M. Imran to Government College University Faisalabad Pakistan; M. Athar to University of Education Lahore Pakistan; and last but not the least the Higher Education Commission of Pakistan for supporting and facilitating this research work.