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

Most articles choose the transcendental function B1(rrn) to define the finite Hankel transform, and very few articles choose B0(rrn). -e derivations of B0(rrn) and B1(rrn) are also considered the same. In this paper, we find that the derivative formulas for the transcend function BN(rrn) are different and prove the derivative formulas for B0(rrn) and B1(rrn). Based on the exact formulas of B0(rrn) and B1(rrn), 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 t � 0. We deduced the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms. According to generalizedG and R functions, the solutions we obtained are given in the form of integrals and series.-e solution of ordinary Maxwell fluid has been also obtained by solving the limit of the general solution of fractional Maxwell fluid.


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. ey 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. e 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. ey 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.
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. ey 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. e first solution is given by integral and series via Laplace transform and Hankel transform. Mahmood et al. [19] also discussed the torsional oscillatory motion. e 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 B N (rr n ) in the Hankel transform. Most articles [17][18][19] select the transcendental function B 1 (rr n ) when defining the finite Hankel transform, and very few articles [21] choose B 0 (rr n ). However, these documents use the same derivation conclusions when calculating the derivatives of B 0 (rr n ) and B 1 (rr n ); 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 B N (rr n ), when N takes different values, and use the mathematical induction method to prove the derivative formula of the transcendental function B N (rr n ). en, we introduce the basic governing equation in Section 3. Based on the exact formula of B 0 (rr n ) and B 1 (rr n ), 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.

Derivative of the Transcendental Function B N (rr n )
According to the Bessel functions, Eldabe et al. [22] defined the finite Hankel transformation of f(r) in (a, b) as follows: where the transcendental function is B N rr n � J N rr n Y N br n − J N br n Y N rr n , r n are the positive roots of the transcendental equation: J N br n Y N ar n − J N ar n Y N br n � 0, and J N is the first-type Bessel function of the N-order and Y N is the second-type Bessel function of the N-order [23]. In this part, we will correct the derivative of the transcendental function B N (rr n ) and obtain the following conclusions.
where B N rr n � J N rr n Y N br n − J N br n Y N rr n , Proof. When N � 1, To prove z zr B 1 rr n � r n B 1 rr n − 1 r B 1 rr n � r n J 0 rr n − 1 r J 1 rr n Y 1 br n − J 1 br n r n Y 0 rr n − 1 r Y 1 rr n , we need to obtain z zr J 1 rr n � r n J 0 rr n − 1 r J 1 rr n , Since we get z zr J 1 rr n � r n J 0 rr n − 1 r J 1 rr n .

Mathematical Problems in Engineering
Similarly, z zr Y 1 rr n � 2 π r n J 0 rr n − 1 r J 1 rr n ln rr n 2 where so we establish the following: Assume that when N � m, the conclusion is true. Now, we prove that when N � m + 1, B m+1 rr n � J m+1 rr n Y m+1 br n − J m+1 br n Y m+1 rr n , B m+1 rr n � J m rr n Y m+1 br n − J m+1 br n Y m rr n .
To prove z zr B m+1 rr n � r n B m+1 rr n − m + 1 r B m+1 rr n � r n J m rr n − m + 1 r J m+1 rr n Y m+1 br n − J m+1 br n r n Y m rr n − m + 1 r Y m+1 rr n , Mathematical Problems in Engineering we need to prove z zr J m+1 rr n � r n J m rr n − m + 1 r J m+1 rr n , where Since

Mathematical Problems in Engineering
Similarly, So we obtain z zr Hence, the conclusion is true.

Corollary 1
(1) Derivative formula of transcendental function B 0 (rr n ) is as follows: (2) Derivative formula of transcendental function B 1 (rr n ) is as follows:

Proof
(1) Since we get z zr B 0 rr n � z zr J 0 rr n Y 0 br n − J 0 br n Y 0 rr n � − r n J 1 rr n Y 0 br n − J 0 br n − r n Y 1 rr n � − r n J 1 rr n Y 0 br n − J 0 br n Y 1 rr n .
(2) It can be directly obtained from eorem 1 conclusion.

Basic Governing Equations
In this paper, based on the exact formula of B 0 (rr n ) and B 1 (rr n ), 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 ω(r, t) is the rotating velocity and v(r, t) is the sliding velocity. e inner and outer cylinders start to rotate around their axis of symmetry at different angular frequencies ω 1 and ω 2 and slide along the same axis of symmetry at different linear velocities A 1 t and A 2 t(A 1 ≠ A 2 ) at time t � 0 + . e 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 S rθ and S rz are the shear stresses. Without considering the body forces and pressure gradient, the momentum conservation equation is where ρ is the constant density of the fluid.

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 ω(r, s) and v(r, s) are the Laplace transformations of ω(r, t) and v(r, t). Similarly, applying the Laplace transforms (28) and (29), we can obtain According to Section 2, we know that are the finite Hankel transforms of ω(r, s) and v(r, s), where r n and r m are the positive roots of equations B 1 (R 1 r) � 0 and B 0 (R 1 r) � 0, and Multiplying both sides of (38) and (39) by rB 1 (rr n ) and rB 0 (rr m ), integrating with respect to r from R 1 to R 2 , and considering conditions (40), we can obtain [28] (44)

Mathematical Problems in Engineering
Since we can get According to (43)-(47), or equivalently 8 Mathematical Problems in Engineering (50) 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 a n � then Mathematical Problems in Engineering (57) 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 G a,b,c (d, t): where is the generalized G function and (c) j is the Pochhammer polynomial [33]. If f(t) � L − 1 (f(s)) and g(t) � L − 1 (g(s)), then We attain the following velocity field: We let t � 0, r � R 1 , and r � R 2 in (63) and (64) and found that v(r, 0) � ω(r, 0) � 0, So the velocity field satisfies our initial and boundary conditions. Mathematical Problems in Engineering 11 When α � 1, the fractional Maxwell fluid model is an ordinary Maxwell fluid model. erefore, (79)-(82) are the analytical solutions for the ordinary Maxwell fluid.

Conclusion
As we know, most articles choose the transcendental function B 1 (rr n ) and few articles choose B 0 (rr n ) to define the finite Hankel transform; moreover, the derivations of B 0 (rr n ) and B 1 (rr n ) 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 B N (rr n ).
en, according to the momentum conservation law, the relevant and meaningful equation and constitutive equation are given. Based on the exact formula of B 0 (rr n ) and B 1 (rr n ), 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. e 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 α � 1, 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
e authors declare that they have no conflicts of interest.