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Rotational motion of fractional Maxwell fluids in a circular duct due to a timedependent couple
 Naeem Sadiq^{1},
 Muhammd Imran^{1}Email authorView ORCID ID profile,
 Constantin Fetecau^{2} and
 Najma Ahmed^{3}
 Received: 10 August 2018
 Accepted: 15 January 2019
 Published: 25 January 2019
Abstract
The rotational motion of fractional Maxwell fluids in an infinite circular cylinder that applies a timedependent but not oscillating couple stress to the fluid is investigated using the integral transform technique. Such a flow model was not analyzed in the past both for ordinary and fractional rate type fluids. This is due to their constitutive equations which contain differential expressions acting on the shear stresses. The obtained solutions fulfill all the enforced initial and boundary conditions and are easily reduced to the solutions of Newtonian or ordinary Maxwell fluids having similar motion. At the end, the influence of pertinent parameters on velocity and shear stress variations is graphically underlined and discussed.
Keywords
 Fractional Maxwell fluid
 Cylindrical domain
 Shear stress on the boundary
1 Introduction
Each field of life depends on fluids such as water, milk, juices, blood, glycerin, grease, paints, oils, polymer solutions, etc. and their direct or indirect motions. For most of these fluids, a linear stressstrain relationship does not exist [1] and the classical Navier–Stokes equation cannot describe their behavior. There are lots of fluid models in literature depending upon their response under different circumstances. Among them, the model that has received more attention is the rate type fluid model. The first rate type model, which is viscoelastic and still utilized generally, was given by Maxwell [2]. Although Maxwell developed this model for air, not for polymeric liquids, his methodology can be summed up to provide a plethora of models. Rajagopal and Srinivasa [3] developed a comprehensive thermodynamic framework by using the concept of Maxwell’s work, which provides a base for making a class of rate type viscoelastic fluids. In the presence of transverse magnetic field, Nayak [4] studied both the heat and mass flow rate of viscous fluid through a medium which is porous considering both heat source and sink. Shateyi [5] used a numerical approach to study the MHD flow of a Maxwell fluid past a stretching plate in the presence of chemical reaction. Shah et al. [6] analyzed the unsteady flow of a magnetohydrodynamic (MHD) second grade fluid over a stretching sheet by using similarity transformations.
Many types of fluid motions, in different geometries, have important applications in chemical industry, bioengineering, mechanical engineering, plasma physics, geophysics, etc. The movement of fluids in circular cylinders has a lot of applications in biological analysis, food industry, petroleum industry, and oil exploitation [7–9]. A variety of Newtonian fluid motions was studied by Bathchelor [10] in circular pipes, but Ting [11] was the first author who found exact solutions about the movement of nonNewtonian fluids. Srivastava [12] was the first who studied the motions of Maxwell fluids through a circular cylinder and obtained analytical solutions. Other exact solutions for motions of Maxwell fluids in cylindrical domains have been obtained by Rahaman and Ramkisson [13], Fetecau and Corina Fetecau [14], Vieru et al. [15], Jamil and Fetecau [16], Jamil et al. [17], Zeb et al. [18], and Corina Fetecau et al. [19]. Recently, Nehad et al. [20] provided the first general solutions for rotational motions of rate type fluids between circular cylinders.
However, in all the above discussed articles, the effects of longterm memory as one of viscoelastic properties of nonNewtonian fluids have been ignored. As far as we know, the memory formalism can be represented using fractional derivatives [21], and the fractional models have gained an increasing interest in many fields including viscoelasticity. The first authors who used fractional derivatives in viscoelasticity were Bagley and Torvik [22], while Caputo and Mainardi [23, 24] got a very good agreement with experimental data using fractional calculus. Recently, the applicability of fractional calculus in fluid mechanics has been continuously increasing because differential equations can describe some important physical phenomena’ more accurately with fractional derivatives instead of ordinary derivatives. Makris et al. [25] utilized exploratory information in order to calibrate a fractional derivative Maxwell model. All the more precisely, they found an estimation of the partial parameter for the relating material properties to be in superb concurrence with test results.
Based on the abovementioned remarks, in the last decade many researchers used the fractional derivatives as a remarkable tool to analyze the properties of viscoelastic fluids [26–34]. However, in all these works the motion of the fluid is generated by a cylinder that is rotating around its axis with a given velocity or applies to the fluid a shear stress that is given by a partial differential equation. Consequently, in the existing literature, there is no exact solution about the motions of fractional rate type fluid developed by an infinite cylinder that applies a constant, accelerated, or oscillating shear stress to the fluid. Such solutions for ordinary rate type fluids were recently obtained by Fetecau et al. [35] and Rauf et al. [36] for constant and oscillating stress, respectively, which is on the boundary, while the solutions from [28] and [37] do not examine the constant shear on the boundary as the researchers claimed there. On the other hand, as it was shown by Renardy [38, 39], the boundary conditions on tangential stresses are very significant and a wellposed boundary problem can be generated in this way.
Our objective in this note is to determine closed form solutions of rotational motion of fractional Maxwell fluid in an infinite circular pipe that applies a couple to the fluid which is time dependent. To do that, contrary to the usual rule from the literature, we use the constitutive equation for the tangential stress, which is the result of elimination of velocity field between the constitutive equations and relevant motion of fluid. The solutions for the current flow model that have been achieved fulfill all imposed initial and boundary conditions. Solutions for Newtonian and ordinary Maxwell fluids having similar motion are also obtained as limiting cases. Finally, the effect of fractional parameter and relaxation time on the velocity and shear stress fields as well as some comparisons with ordinary Maxwell and Newtonian fluids are graphically underlined and discussed.
2 Governing equations
3 Solution of the flow problem
3.1 Calculation of the stress field
3.2 Calculation of the velocity field
4 Limiting cases
4.1 Ordinary Maxwell fluid
4.2 Newtonian fluid
5 Numerical results and conclusions
In this note, the flow of a fractional Maxwell fluid through an infinite circular cylinder that applies a timedependent torque per unit length to the fluid is analytically studied using the integral transform technique. To do that, contrary to the usual line from the literature, the governing equation for the nontrivial shear stress is used and the first exact solutions for such motions of rate type fluids are obtained. These solutions, which are presented in series form in terms of some generalized functions, satisfy all imposed initial and boundary conditions and are easily reduced to similar solutions for Newtonian and ordinary Maxwell fluids. It is worth pointing out the fact that our limiting solution (28) for the shear stress corresponding to a Newtonian fluid is identical to that obtained in [45, Eq. (24)], while the adequate velocity field (29) corrects a similar result from the same reference.

It is observed that with the passage of time velocity field and shear stress both increase for the above fractional fluid flow model.

It is noted that both shear stress and velocity field are decreasing functions of relaxation time λ and fractional parameter η.

As expected, velocity of the fluid decreases as fluid becomes more thick but tangential stress increases.

It can be seen from graphs that, for every physical parameter, shear stress and velocity field decrease smoothly from maximum (near the circular cylinder) to zero (at the center or axis of cylinder).

The effect of stress on Newtonian fluid is higher as compared to that on the fractional and ordinary Maxwell fluids. Due to quick response, the value of velocity field is greater than that of other fluid models.

From general solution, we recover the solution for shear stress for Newtonian fluid [45, Eq. (24)].

In all figures we use SI units, and roots are approximated by \({r_{n}} = \frac{{(4n1)\pi } }{{({4R})}}\).
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analysed during the current study.
Funding
This research is supported by the Government College University, Faisalabad, Pakistan and the Higher Education Commission Pakistan.
Authors’ contributions
NS and NA made the mathematical model and mathematical calculation of the paper. MI and NS made the numerical results and graphs of the paper. CF was a major contributor in writing the manuscript. NS and MI checked the calculation and revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Shifang, H.: Constitutive Equation and Computational Analytical Theory of NonNewtonian Fluids. Science Press, Beijing (2000) (in Chinese) Google Scholar
 Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. A 157, 26–78 (1866) Google Scholar
 Rajagopal, K.R., Srinivasa, A.R.: A thermodynamical framework for rate type fluid models. J. NonNewton. Fluid Mech. 88, 207–227 (2000) View ArticleGoogle Scholar
 Nayak, M.K.: Chemical reaction effect on MHD viscoelastic fluid over a stretching sheet through porous medium. Meccanica 51, 1699–1711 (2016) MathSciNetView ArticleGoogle Scholar
 Shateyi, S.: A new numerical approach to MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction. Bound. Value Probl. 2013, 196 (2013) MathSciNetView ArticleGoogle Scholar
 Shah, R.A., Rehman, S., Idrees, M., Ullah, M., Abbas, T.: Similarity analysis of MHD flow field and heat transfer of a second grade convection flow over an unsteady stretching sheet. Bound. Value Probl. 2017, 162 (2017) MathSciNetView ArticleGoogle Scholar
 Miyazaki, H.: Combined free and forced convective heat transfer and fluid flow in a rotating curved circular tube. Int. J. Heat Mass Transf. 14, 1295–1309 (1971) View ArticleGoogle Scholar
 Ishigaki, H.: Laminar flow in rotating curved pipes. J. Fluid Mech. 329, 373–388 (1996) View ArticleGoogle Scholar
 Mukunda, P.G., Shailesh, R.A., Kiran, A.S., Shrikantha, S.R.: Experimental studies of flow patterns of different fluids in a partially filled rotating cylinder. J. Appl. Fluid Mech. 2, 39–43 (2009) Google Scholar
 Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000) View ArticleGoogle Scholar
 Ting, T.W.: Certain nonsteady flows of secondorder fluids. Arch. Ration. Mech. Anal. 14(1), 1–26 (1963) MathSciNetView ArticleGoogle Scholar
 Srivastava, P.N.: Nonsteady helical flow of a viscoelastic liquid. Arch. Mech. Stosow. 18(2), 145–150 (1966) Google Scholar
 Rahaman, K.D., Ramkissoon, H.: Unsteady axial viscoelastic pipe flows. J. NonNewton. Fluid Mech. 57, 27–38 (1995) View ArticleGoogle Scholar
 Fetecau, C., Fetecau, C.: Unsteady helical flows of a Maxwell fluid. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 5(1), 13–19 (2004) MathSciNetMATHGoogle Scholar
 Vieru, D., Akhtar, W., Fetecau, C., Fetecau, C.: Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42, 573–583 (2007) View ArticleGoogle Scholar
 Jamil, M., Fetecau, C.: Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary. Nonlinear Anal., Real World Appl. 11(5), 4302–4311 (2010) MathSciNetView ArticleGoogle Scholar
 Jamil, M., Fetecau, C., Khan, N.A., Mahmood, A.: Some exact solutions for helical flows of Maxwell fluid in an annular pipe due to accelerated shear stresses. Int. J. Chem. React. Eng. 9, Article ID A20 (2011) Google Scholar
 Zeb, M., Haroon, T., Siddiqui, A.M.: Study of corotational Maxwell fluid in helical screw rheometer. Bound. Value Probl. 2015, 146 (2015) MathSciNetView ArticleGoogle Scholar
 Fetecau, C., Imran, M., Fetecau, C.: Taylor–Couette flow of an OldroydB fluid in an annulus due to a timedependent couple. Z. Naturforsch. 66a, 40–46 (2011) View ArticleGoogle Scholar
 Shah, N.A., Fetecau, C., Vieru, D.: First general solutions for unsteady unidirectional motions of rate type fluids in cylindrical domains. Alex. Eng. J. 57, 1185–1196 (2018) View ArticleGoogle Scholar
 Povstenko, Y.Z.: Linear Fractional DiffusionWave Equation for Scientists and Engineers. Springer, Cham (2015) View ArticleGoogle Scholar
 Bagley, R.L., Torvik, P.J.: A theoretical basis for the applications of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983) View ArticleGoogle Scholar
 Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971) View ArticleGoogle Scholar
 Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971) View ArticleGoogle Scholar
 Makris, N., Dargush, G.F., Constantinou, M.C.: Dynamic analysis of generalized viscelastic fluids. J. Eng. Mech. 119, 1663–1679 (1993) View ArticleGoogle Scholar
 Imran, M., Athar, M., Kamran, M.: On the unsteady rotational flow of a generalized Maxwell fluid through a circular cylinder. Arch. Appl. Mech. 81(11), 1659–1666 (2011) View ArticleGoogle Scholar
 Kamran, M., Imran, M., Athar, M., Imran, M.A.: On the unsteady rotational flow of fractional OldroydB fluid in cylindrical domains. Meccanica 47(3), 573–584 (2012) MathSciNetView ArticleGoogle Scholar
 Mathur, V., Khandewal, K.: Exact solution for the flow of OldroydB fluid due to constant shear stress and time dependent velocity. IOSR J. Math. 10, 38–45 (2014) View ArticleGoogle Scholar
 Mathur, V., Khandelwal, K.: Exact solution for the flow of OldroydB fluid between coaxial cylinders. Int. J. Eng. Res. Technol. 3, 949–954 (2017) Google Scholar
 Raza, N., Haque, E.U., Rashidi, M.M., Awan, A.U., Abdullah, M.: Oscillating motion of an OldroydB fluid with fractional derivatives in a circular cylinder. J. Appl. Fluid Mech. 10, 1421–1426 (2017) View ArticleGoogle Scholar
 Kamran, M., Imran, M., Athar, M.: Exact solutions for the unsteady rotational flow of a generalized OldroydB fluid induced by a circular cylinder. Meccanica 48, 1215–1226 (2013) MathSciNetView ArticleGoogle Scholar
 Kamran, M., Imran, M., Athar, M.: Exact solutions for the unsteady rotational flow of a generalized second grade fluid through a circular cylinder. Nonlinear Anal., Model. Control 15, 437–444 (2010) MathSciNetMATHGoogle Scholar
 Sultan, Q., Nazar, M., Imran, M., Ali, U.: Flow of generalized Burgers fluid between parallel walls induced by rectified sine pulses stress. Bound. Value Probl. 2014, 152 (2014) MathSciNetView ArticleGoogle Scholar
 Qureshi, M.I., Imran, M., Athar, M., Kamran, M.: Analytic solutions for the unsteady rotational flow of an OldroydB fluid with fractional derivatives induced by a quadratic timedependent shear stress. Int. J. Appl. Comput. Math. 10, 484–497 (2011) MathSciNetMATHGoogle Scholar
 Fetecau, C., Rana, M., Nigar, N., Fetecau, C.: First exact solutions for flows of rate type fluids in a circular duct that applies a constant couple to the fluid. Z. Naturforsch. 69a, 232–238 (2014) Google Scholar
 Rauf, A., Zafar, A.A., Mirza, I.A.: Unsteady rotational flows of an OldroydB fluid due to tension on the boundary. Alex. Eng. J. 54(4), 973–979 (2015) View ArticleGoogle Scholar
 Tong, D., Liu, Y.: Exact solutions for the unsteady rotational flow of nonNewtonian fluid in an annular pipe. Int. J. Eng. Sci. 43(3–4), 281–289 (2005) MathSciNetView ArticleGoogle Scholar
 Renardy, M.: Inflow boundary conditions for steady flows of viscoelastic fluids with differential constitutive laws. Rocky Mt. J. Math. 18, 445–453 (1988) MathSciNetView ArticleGoogle Scholar
 Renardy, M.: An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions. J. NonNewton. Fluid Mech. 36, 419–425 (1990) View ArticleGoogle Scholar
 Bandelli, R., Rajagopal, K.R.: Startup flows of second grade fluids in domains with one finite dimension. Int. J. NonLinear Mech. 30(6), 817–839 (1995) MathSciNetView ArticleGoogle Scholar
 Fetecau, C., Rubbab, Q., Akhter, S., Fetecau, C.: New methods to provide exact solutions for some unidirectional motions of rate type fluids. Therm. Sci. 20(1), 7–20 (2016) View ArticleGoogle Scholar
 Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
 Debnath, L., Bhatta, D.: Integral Transforms and Their Applications, 2nd edn. Chapman & Hall, Boca Raton (2007) MATHGoogle Scholar
 Lorenzo, C.F., Hartley, T.T.: Generalized functions for the fractional calculus. NASA/TP (1999) Google Scholar
 Fetecau, C., Awan, A.U., Fetecau, C.: TaylorCouette flow of an OldroydB fluid in a circular cylinder subject to a timedependent rotation. Bull. Math. Soc. Sci. Math. Roum. 2, 117–128 (2009) MathSciNetMATHGoogle Scholar