Rotational motion of fractional Maxwell fluids in a circular duct due to a timedependent couple
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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 boundary1 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, 8, 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, 27, 28, 29, 30, 31, 32, 33, 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 \({r_{n}} = \frac{{(4n1)\pi } }{{({4R})}}\).
Notes
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.
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.
Funding
This research is supported by the Government College University, Faisalabad, Pakistan and the Higher Education Commission Pakistan.
Competing interests
The authors declare that they have no competing interests.
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