Research Article | Open Access
Critical Study on Rotational Flow of a Fractional Oldroyd-B Fluid Induced by a Circular Cylinder
We considered the unsteady flow of a fractional Oldroyd-B fluid through an infinite circular cylinder with the help of infinite Hankel and Laplace transforms. The motion of the fluid is produced by the cylinder that, at time is subject to a time-dependent angular velocity. The established solutions have been presented under series form in terms of the generalized functions satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, ordinary and fractional Maxwell, ordinary and fractional second-grade, and Newtonian fluids, performing the same motion, are acquired as limiting cases of general solutions. The keynote points regarding this work to mention are that (1) we extracted the expressions for velocity field and shear stress corresponding to the motion of fractional second-grade fluid as a limiting case of general solutions; (2) the expressions for velocity field and shear stress are in the most simplified form in contrast with the studies of Siddique and Sajid (2011), in which the expression for the velocity field involves the convolution product as well as the integral of the product of generalized functions. Finally, numerical results are presented graphically and discussed in order to reveal some physical aspects of obtained results.
The motion of a fluid in cylindrical domains has applications in the food industry, oil exploitation, chemistry, and bioengineering , it being one of the most important problems of motion near translating or rotating bodies. The non-Newtonian fluids are now considered to play a more important and appropriate role in technological applications in comparison with Newtonian fluids. The first exact solutions corresponding to motions of non-Newtonian fluids in cylindrical domains seem to be those of Ting  for second-grade fluids, Srivastava  for Maxwell fluids, and Waters and King  for Oldroyd-B fluids. In the meantime a lot of papers regarding these motions have been published. However, the most of them deal with motion problems in which the velocity field is given on the boundary. To the best of our knowledge, the first exact solutions for motions of non-Newtonian fluids due to a shear stress on the boundary are those of Bandelli and Rajagopal  and Bandelli et al.  for second-grade fluids. Other similar solutions have been also obtained in [7–15].
Fractional calculus is a field of applied mathematics that deals with derivatives and integral of arbitrary orders [16, 17]. During the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics. A lot of applications of fractional calculus can be found in turbulence and fluid dynamics, plasma physics and controlled thermonuclear fusion, nonlinear control theory, nonlinear biological systems, and astrophysics [18–21] while its applications to polymer physics, biophysics, and thermodynamics can be found in . The use of the method of fractional derivatives in the theory of viscoelasticity has the advantages that it affords possibilities for obtaining constitutive equations for elastic complex modulus of viscoelastic materials with only few experimentally determined parameters .
Our intention here is to establish exact solutions for the velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives through an infinite circular cylinder. The motion of the fluid is produced by the cylinder, which, at time , begins to rotate about its axis with a time-dependent angular velocity. The solutions that have been obtained, presented under series form in terms of the generalized functions, are established by means of the finite Hankel and Laplace transforms. The similar solutions for an ordinary Oldroyd-B, Maxwell with fractional derivatives, ordinary Maxwell fluids as well as those for the fractional and ordinary second-grade fluids, can be obtained as limiting cases of general solutions for ; and ; , and ; and ; , and , respectively. The solutions for a Newtonian fluid performing the same motion are obtained as limiting cases of the solutions for an Oldroyd-B fluid with fractional derivatives when , , and .
At the end with the help of Computer Algebra System (CAS) MATHCAD 2.0, we presented diagrams of velocity as well as of shear stress against different values of radius, time, and of the pertinent parameters. Also in one of the diagrams, we compared the velocity profiles and shear stresses corresponding to the motion of different fluid models, for the same value of time and of the common material parameters.
2. Governing Equations
The flows to be here considered have the velocity and the extrastress of the form  where is the unit vector in the -direction of the cylindrical coordinates system , , and . For such flows, the constraint of incompressibility is automatically satisfied. Furthermore, if initially the fluid is at rest, then The governing equations corresponding to such motions of Oldroyd-B fluid are given by  where is the dynamic viscosity, is the kinematic viscosity, with being the constant density of the fluid, is the relaxation time, is the retardation time, and is the nontrivial shear stress.
The governing equations corresponding to an incompressible fractional Oldroyd-B fluid, performing the same motion, are obtained by replacing the inner time derivatives with respect to from (2.3), by the fractional differential operator given as [22–24] where is the Gamma function defined by Consequently, the governing equations to be used here are We can easily observe that when , (2.6) and (2.7) reduce to (2.3), because . Furthermore, the new material constants and (although, for the simplicity, we keep the same notation) reduce to the previous ones for .
3. Starting Flow through an Infinite Circular Cylinder
Suppose that an incompressible fractional Oldroyd-B fluid is lying at rest in an infinite circular cylinder of radius >. At time , the cylinder suddenly begins to rotate about its axis with an angular velocity . Owing to the shear the inner fluid is gradually moved, with its velocity being of the form (2.1). The governing equations are given by (2.6) and (2.7), while the appropriate initial and boundary conditions are where is a constant.
The partial differential equations (2.6) and (2.7), also containing fractional derivatives, can be solved in principle by several methods; the integral transforms technique representing a systematic, efficient, and powerful tool. In the following we will use the Laplace transform to eliminate the time variable and the finite Hankel transform for the spatial variable. However, in order to avoid the burdensome calculations of residues and contour integrals, we will apply the discrete inverse Laplace transform method.
3.1. Calculation of the Velocity Field
Applying the Laplace transform to (2.7) and (3.2), we get where and are the Laplace transforms of the functions and , respectively. We will denote by  the finite Hankel transform of the function , and the inverse Hankel transform of is given by  where is the positive roots of the equation and is the Bessel function of the first kind of order .
Multiplying now both sides of (3.3) by , integrating with respect to from 0 to , and taking into account (3.4), (3.5), and the result which we can easily prove we find that It can be also written in the suitable form where Applying the inverse Hankel transform to (3.10) and using the known formula we getUsing the identity (3.12b) can be written as After taking the inverse Hankel transform of (3.9) and with the help of (3.12a) and (3.14), it leads to Now taking the inverse Laplace transform of (3.15), the velocity field is given by where the generalized function is defined by [26, Equations (97) and (101)]
3.2. Calculation of the Shear Stress
Applying the Laplace transform to (2.6), we find that where, in view of (3.12a) and (3.12b), we have Now differentiating the last expression for partially with respect to , we get and introducing (3.20) into (3.18), we have In view of identity (3.13), (3.21) can be equivalently written as Now taking the inverse Laplace transform of both sides of (3.22) and using (3.17), we find that
4. The Special Cases
4.1. Ordinary Oldroyd-B Fluid
4.2. Maxwell Fluid with Fractional Derivatives
4.3. Ordinary Maxwell Fluid
4.4. Second-Grade Fluid with Fractional Derivatives
Making , into (3.16) and (3.23), and using (A1) and (A2), we obtain velocity field and associated shear stress corresponding to a second-grade fluid with fractional derivatives performing the same motion. By taking and (the material constants for second grade fluid) into (4.7) and (4.8), we will get the similar solutions for fractional second-grade fluid as we get directly from the governing equations of fractional second-garde fluid as under
4.5. Ordinary Second-Grade Fluid
4.6. Newtonian Fluid
Making and into (3.16) and (3.23) and using (A5) we get velocity field as and the associated shear stress corresponding to a Newtonian fluid performing the same motion. Of course, the Newtonian solutions (4.13) and (4.14) can be also obtained by making into (4.5) and (4.6) and using (A5) or making into (4.12).
The underlying idea of this research paper is to calculate exact solutions for velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives through an infinite circular cylinder. The motion of the fluid is produced by the cylinder, which, at time , begins to rotate about its axis subject to a time-dependent angular velocity. The solutions that have been obtained, presented under series form in terms of the generalized functions, are established by means of the finite Hankel and Laplace transforms. The similar solutions for an ordinary Oldroyd-B, Maxwell with fractional derivatives, ordinary Maxwell fluids as well as those for fractional and ordinary second-grade fluids, can be obtained as limiting cases of general solutions for ; and ; , and ; and ; , and , respectively. The solutions for a Newtonian fluid performing the same motion are obtained as limiting cases of the solutions for an Oldroyd-B fluid with fractional derivatives when , , and .
6. Numerical Results
Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity as well as those of the shear stress , are depicted against for different values of time and of the pertinent parameters. Figures 1(a) and 1(b) clearly show that both velocity and the shear stress are increasing functions of . They are also increasing functions of , excepting on a small interval near the boundary. The influence of the kinematic viscosity on the fluid motion is shown in Figures 2(a) and 2(b), velocity is an increasing function of , while shear stress is a decreasing function of .
The influences of the relaxation time and of the retardation time on the velocity and shear stress are shown in Figures 3 and 4. The two parameters, as expected, have opposite effects on the fluid motion. More exactly, velocity and the shear stress are decreasing functions with respect to and increasing ones with regards to . The influences of the fractional parameters and on the fluid motion are presented in Figures 5 and 6. Their effects are also opposite, but they are qualitatively the same as those of and , respectively. More exactly velocity and the shear stress are increasing functions with regards to and decreasing with regards to .
At the end, for comparison, the profiles of corresponding to the motion of the seven models (Newtonian, ordinary second grade, fractional second grade, ordinary Maxwell, fractional Maxwell, ordinary Oldroyd-B, fractional Oldroyd-B) are together depicted in Figure 7, for the same values of and of the common material parameters. The fractional Maxwell fluid, as it results from figure, is the slowest while the fractional second-grade fluid is the swiftest on the whole flow domain. The units of the material constants are SI units within all figures, and the roots have been approximated by .
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