The First Solution for the Helical Flow of a Generalized Maxwell Fluid within Annulus of Cylinders by New Definition of Transcendental Function

Abstract

Most articles choose the transcendental function to define the finite Hankel transform, and very few articles choose . The derivations of and are also considered the same. In this paper, we find that the derivative formulas for the transcend function are different and prove the derivative formulas for and . Based on the exact formulas of and , we keep on studying the helical flow of a generalized Maxwell fluid between two boundless coaxial cylinders. In this case, the inner and outer cylinders start to rotate around their axis of symmetry at different angular frequencies and slide at different linear velocities at time . We deduced the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms. According to generalized G and R functions, the solutions we obtained are given in the form of integrals and series. The solution of ordinary Maxwell fluid has been also obtained by solving the limit of the general solution of fractional Maxwell fluid.

1. Introduction

Most industry and academic workers are interested in the motion of fluids on plates or cylinders. Regarding the flow of Newtonian fluid in cylinders, we can find the transient velocity distribution in [1]. With regard to the flow of non-Newtonian fluid in the cylindrical domain, the first exact solutions are those of Srivastava [2] with Maxwell fluids and Ting [3] with second-grade fluids. Casarella and Laura [4] obtained an analytical expression on a smooth circular cylindrical rod. They studied rods with longitudinal and torsional motion. For flow in the same case, Rajagopal [5] found two solutions. Rajagopal and Bhatnagar [6] obtained these solutions for Oldroyd-B fluid. Khan et al. [7] gave an exact analytical solution for the flow of a Burger’s fluid via Fourier transform. The oscillating pressure gradient they considered was the main driving force for the motion. With regard to second-order fluids, Hayat et al. [8, 9] gave an exact analytical solution for unsteady unidirectional flows. They also considered pressure gradients and the flows induced by the motion of one or two plates or the pressure gradient. In terms of unsteady unidirectional flows, Rajagopal [10] also established two types of exact solutions. Among these two types of flows, one is owing to the rigid plate that oscillates in its own direction and the other is the time-periodic Poiseuille flow on account of the oscillating pressure gradient. Recently, many papers of this type have been obtained by Yang and Wang [11–13].

Fractional calculus has made great progress in describing the motion of viscoelastic fluids in recent years. Tong et al. [14, 15] discussed an exact analytic solution for the unsteady transient rotational flow of an Oldroyd-B fluid in terms of the fractional calculus definition. Many researchers have studied the rotational flow of a generalized Maxwell fluid between two boundless coaxial cylinders. For example, Fetecau and Fetecau [16] established analytic solutions for this motion. The longitudinal vibration shear stresses that were imposed by the inner cylinder were the main driving force for the motion. Shaowei and Mingyu [17] considered the unsteady Couette flow with fractional derivative. When the outer cylinder remained immobile, the inner cylinder started to move around their axis of symmetry at a fixed angular velocity. They used the integral transform and the generalized Mittag-Leffler function to get the first solution. Zheng et al. [18] discussed the unsteady rotating flow with a fractional derivative model, where the flow is induced by the rotation of the inner cylinder and the oscillation of the pressure gradient. The first solution is given by integral and series via Laplace transform and Hankel transform. Mahmood et al. [19] also discussed the torsional oscillatory motion. The movement of the fluid was produced when the inner and outer cylinders began to oscillate around their symmetry axis. Chaudry et al. [20] established the first analytic solution for the oscillatory flow, when the outer cylinder oscillated along the axis of symmetry and the inner cylinder was stationary.

In the problem of using Hankel transform to solve the unsteady flow in a coaxial cylinder, we need to derive the transcendental function in the Hankel transform. Most articles [17–19] select the transcendental function when defining the finite Hankel transform, and very few articles [21] choose . However, these documents use the same derivation conclusions when calculating the derivatives of and ; it is proved that the derivative formula is dependent on N; in fact, their derivation conclusions are different. Motivated by this problem, in Section 2 of this paper, we study the derivative of the transcendental function , when N takes different values, and use the mathematical induction method to prove the derivative formula of the transcendental function . Then, we introduce the basic governing equation in Section 3. Based on the exact formula of and , we calculate the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms in Sections 4 and 5. According to generalized G and R functions, the solutions we obtained are given in the form of integrals and series. In Section 6, we obtained the solution of Maxwell fluid by solving the limit of the general solution.

2. Derivative of the Transcendental Function

According to the Bessel functions, Eldabe et al. [22] defined the finite Hankel transformation of in as follows:where the transcendental function is

are the positive roots of the transcendental equation:and is the first-type Bessel function of the N-order and is the second-type Bessel function of the N-order [23]. In this part, we will correct the derivative of the transcendental function and obtain the following conclusions.

Theorem 1. Derivative formula of transcendental function :where

Proof. When ,To provewe need to obtainSincewe getSimilarly,whereso we establish the following:Assume that when , the conclusion is true. Now, we prove that when ,To provewe need to provewhereSincethenSimilarly,So we obtainHence, the conclusion is true.

Corollary 1. (1)Derivative formula of transcendental function is as follows:(2)Derivative formula of transcendental function is as follows:

Proof. (1)Since we get(2)It can be directly obtained from Theorem 1 conclusion.

3. Basic Governing Equations

In this paper, based on the exact formula of and , we keep on studying the helical flow of a generalized Maxwell fluid between two boundless coaxial cylinders. We research the velocity of the helical flow of the following form:where is the rotating velocity and is the sliding velocity. The inner and outer cylinders start to rotate around their axis of symmetry at different angular frequencies and and slide along the same axis of symmetry at different linear velocities and at time .

The initial and boundary conditions we gave are

By introducing the time derivative with a fractional differential operator to the conservation and constitutive equations of an incompressible Maxwell fluid, the governing equation is given by [15, 24, 25]where λ is the material constant, μ is the dynamic viscosity of the fluid, and and are the shear stresses. Without considering the body forces and pressure gradient, the momentum conservation equation iswhere ρ is the constant density of the fluid.

4. Calculation of the Velocity Field

Applying the Laplace transform [26, 27] to (30)–(33), we obtain

According to (34)–(37), we obtain the following ordinary differential equation:where the functions and are the Laplace transformations of and . Similarly, applying the Laplace transforms (28) and (29), we can obtain

According to Section 2, we know thatare the finite Hankel transforms of and , where and are the positive roots of equations and , and

Multiplying both sides of (38) and (39) by and , integrating with respect to r from to , and considering conditions (40), we can obtain [28]

Sincewe can get

According to (43)–(47),or equivalently

We first write (49) and (50) under the suitable form as follows:and we use the inverse Hankel transform formula [29, 30]:

As we know if then

Ifthenso

So we can obtain

To make the calculation of calculus more convenient, we can write [31, 32] the following:and we use the Laplace transform of the functions :whereis the generalized G function and is the Pochhammer polynomial [33].

If and , then

We attain the following velocity field:

We let , and in (63) and (64) and found that

So the velocity field satisfies our initial and boundary conditions.

5. Calculation of the Shear Stress

According to (34) and (35), we can obtain that

By using Corollary 1,whereso

Introducing (70) and (71) into (66) and (67), we can obtain

To make the calculation of calculus more comfortable, we writeand using the Laplace transform of the generalized function and function [33], we obtainwhere

We apply the inversion Laplace transform to (72) and (73) and obtain the shear stress as follows:

6. Limiting Case

(1)When , (63) and (64) can be reduced to(2)When , (77) and (78) can be reduced to

When , the fractional Maxwell fluid model is an ordinary Maxwell fluid model. Therefore, (79)–(82) are the analytical solutions for the ordinary Maxwell fluid.

7. Conclusion

As we know, most articles choose the transcendental function and few articles choose to define the finite Hankel transform; moreover, the derivations of and are often considered the same. Actually, in this paper, by use of the mathematical induction method, we prove the derivative formula for the transcend function . Then, according to the momentum conservation law, the relevant and meaningful equation and constitutive equation are given. Based on the exact formula of and , we calculate the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms in Sections 4 and 5. In order to present our solution in a simpler and more comfortable form, we combine the generalized G function and R function. The solutions we obtained are given in the form of integrals and series containing the generalized G function and R function. At the end of Section 4, we explain that these solutions satisfy initial and boundary conditions that we gave. When , the fractional Maxwell fluid model is an ordinary Maxwell fluid model. As before, we obtained the solution of Maxwell fluid by solving the limit of the general solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

F. Wang was supported by the Natural Science Foundation of Hunan Province under grant no. 2019JJ50659, the Double First-class International Cooperation Expansion Project of Changsha University of Science and Technology under grant no. 2019IC39, and the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering under grant no. 2017TP1017 (Changsha University of Science and Technology).