Analytical Solution of Fractional Oldroyd-B Fluid via Fluctuating Duct

School of Science, Hunan City University, Yiyang 413000, China Department of Mathematics, Abbottabad University of Science and Technology, Havelian 22500, Pakistan Department of Mathematics, COMSATS University Islamabad, Wah Campus, Islamabad 47040, Pakistan Department of Pure and Applied Mathematics, University of Haripur, Haripur, KPK, Haripur, Pakistan Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan


Introduction
e liquids which change their viscosity under force to either more liquid or solid are famous. ese liquids are known as non-Newtonian fluids. e understanding can be improved by studying such types of fluids. A French physicist and engineer along with a mathematician named (Anglo-Irish, Claude-Louis Navier, and George Gabriel Stokes) described fluid flow through its environment. After it, these equations are known by their names like the Navier-Stokes equations. Navier-Stokes equations could describe the form and presentation of non-Newtonian fluids' flow quite well. ese were proved useful in various areas of science and engineering. Petroleum engineers reveal how oil flows from well or pipe using mathematical modeling exactly in the same way as biomedical investigators for blood flow Non-Newtonian fluids have gained prominence because of their many uses in commerce and architecture, as well as medicine.
e non-Newtonian behavior understanding is generally more complicated than the Newtonian one. According to Hartnett and Kostic [1], in non-Newtonian fluids, the theoretical predictions yield low estimates of the heat transfer under laminar flow conditions. e motion of non-Newtonian fluid in containers is a very functional issue in dynamics. First, Stokes [2] presented the precise result of oscillatory motion in a classical linearly viscous fluid. Rajagopal [3] answered the models of non-Newtonian fluids in regards to their motion. Rajagopal and Bhatnagar [4] studied the Oldroyd-B fluid solutions in Bessel function for torsional and the longitudinal oscillations of an enormously lengthy dowel. Mahmood et al. [5] used Laplace and finite Hankel transform and acquired the exact velocity's solutions and sinusoidal shear stress corresponding to second-grade fluid's flow. Hayat et al. [6] found out the five particular results used for the problems of an Oldroyd-B fluid, i.e., (i) Stokes problem, (ii) modified Stokes problem, (iii) the timeperiodic Poiseuille flow due to an oscillating pressure gradient, (iv) the nonperiodic flows between two boundaries, and (v) symmetric flow with an arbitrary initial velocity.
In the last decades, fractional calculus (FC) underwent intensive research and development [7]. e working and comprehension of artificial and characteristic frameworks require the conventional derivative and integral which are significant for innovation experts. e derivative operators and calculus integral can be characterized by fractional calculus which is the field of math wherein the fractional powers are utilized instead of integer powers. erefore, noninteger derivatives are portrayed by some memory impacts that are imparted to various materials such as polymers and viscoelastic materials and furthermore its uses in anomalous diffusions. By [8], we obtained exact solutions using an expansion theorem of Steklov for flows satisfying no-slip boundary conditions. Waters and King [9] evaluated the exact solution with the Laplace transform. ey investigated that the velocity sketches intensely subject to the flexible parameters and fluctuates approximately on their central position. Wood [10] studied start-up helical flows for Oldroyd-B in straight tubes of the annular and rounded cross-section. ey added that the fluid is originally at rest for completing the process of the solution.
Electromagnetic, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signal processing are just a few of the engineering and science fields where fractional calculus is used. Ray et al. [11] discussed fractional calculus as a modeling tool for engineering and physical advancements that are defined vigorously by fractional differential equations. Systems that require precise damping modeling used fractional derivative models to accurately model it. Various analytical and numerical techniques, as well as their presentations to new complications, have been projected in these fields [12,13].
Researches in fluid flow problems are present in terms of fractional derivatives. ey had observed the influence of fractional parameters on the flow profiles [14,15]. ey referred to the obtained governing equations as fractional partial differential equations. Moreover, through discrete Laplace transform and Fourier transform along with some well-known special functions, they got precise results [16]. Few scholars considered Oldroyd-B fluid for various models in terms of fractional derivatives. Fetecau et al. [17,18] solved Stokes's first problem of the velocity profile and the related tangential tension parallel to an Oldroyd-B fluid flow above abruptly stimulated smooth bowl analytically. Fetecau et al. [19] investigated the tangential pressures and velocity field between two perpendicular walls in the trembling flow of Oldroyd-B fluid by continuously accelerating a plate. ey obtained the exact solution with the utilization of the Fourier transform method. Different geometries exist in different types of solutions. In industry, ducts are normally used for managing different flows. erefore, as a prime process part in industrial units, it gained high importance. e rotational flow of fractional Oldroyd-B fluid in cylindrical domains was studied in [20,21]. Fetecau and Fetecau [22] used a rectangular cross-sectional channel and introduced precise results for two different kinds of trembling flows of an Oldroyd-B fluid. Nazar et al. [23] determined sine oscillations of the rectangular tube through studying the mandatory time-period to reach the steady-state. Riaz et al. [21] used fractional derivatives and investigated the rotating flow of an Oldroyd-B fluid caused by an infinite circular tube with a continuous couple. Ghada and Ahmed [24] find out the trembling flow of broad Oldroyd-B fluid by studying the analytic solution and the flow of fluid was in the oscillating rectangular tube.
Furthermore, Wang et al. [25] used an extended rectangular cross-sectional tube and investigated the vibratory flow of Maxwell fluid. e exact solution's singularities and appropriate expressions of velocity and phase variation were studied clearly. Sun et al. [26] used an isosceles right triangular cross-sectional lengthy tube and modeled the vibratory flow of the Maxwell fluid. ey obtained analytical terms for the flow compelled by the periodic pressure gradient. Farooq et al. [27] studied the generalized Maxwell fluid flow with magnetic and porous factors via quadrilateral duct. Sultan et al. [28] found out the trembling magnetohydrodynamic (MHD) flow of Oldroyd-B fluid through analytic solution in a permeable rectangular cross-section. Some studies related to time-fractional derivatives have obtained interesting results of such flow problems [29][30][31][32][33][34][35].
is paper aims to express the oscillatory motion of an Oldroyd-B fluid through a rectangular duct. e unsteady boundary layer equations of Oldroyd-B fluid are formulated. en, the exact solutions are derived for the comprehensive Oldroyd-B fluid through integral transform. More precisely, the researchers want to know the relation of the vibratory motion of the fractionalized Oldroyd-B fluid by discovering the shear stress and velocity motion, and the first "time" derivative of the velocity is taken "zero" as its extra condition to simplify the model at time t � 0. Furthermore, the effects and features are graphically represented that are relevant to velocity field's parameters. e remainder of this article is designed in such a way that, after the introduction, the statement of the problem is discussed in Section 2. In Section 3, we presented the exact solution of related velocity field and tangential stresses specific to Oldroyd-B fluid inside a vibratory rectangular tube with fractional derivatives. Section 4 discusses limited cases of the fractional Oldroyd-B fluid. e graphical results and the derived exact solutions compared with numerical results are investigated in Section 5. e conclusion of the paper is presented in Section 6. Also, see Table 1 for the dimension of the physical parameters.
Y � h. At time t � 0 + , the tube starts to vibrate along Z-axis. Due to these oscillations, an oscillatory motion in fluid gets started inside along the duct's boundary. e considered velocity field and tangential stress are as follows: where k is the unit vector aiming in Z-direction. Remember the first B kinematic tensor can be given from Rivlin-Ericksen as where † represents the operation transpose. e stress tensor T is is equation indicates p, I, and S which are the hydrostatic pressure, identity tensor, and extra stress tensor, respectively. e extra stress tensor [36] can be written as by the following relation: In (4) μ > 0, λ 1 and λ 2 are the dynamic viscosity, relaxation time, and retardation time, respectively. e operator denotes the Oldroyd derivative [36] D t and is given below Furthermore, the initial conditions for the considered fluid flow are e assumed governing equations for an incompressible fluid system are while ρ > o; for ease, body strength and pressure gradient are overlooked.

Flow Problem.
Firstly, the researchers carefully weighed the flow of Oldroyd-B fluid and the possible constitutive equations for it. en, after the exact modification, the researchers found out the desired results for the fluids' flow.
(1) satisfies the equation of continuity, i.e., (6). While (1), (2) and (4), (3), together with the initial conditions, i.e., (5), we can get for all t > 0, Using the same procedure [28] and with the help of (1) and (8)-(10), we can reduce (7) to e appropriate conditions are where U 0 is the amplitude, H(t) is a unit step function, and ω is the velocity frequency of edge. While forgetting the fractional Oldroyd-B fluid flow equations, the researchers need to change the inner time derivatives (11) with left-sided Caputo fractional time derivatives z α t and z β t for 0 < α ≤ β < 1, and it can be diverted into the model with the same original boundary conditions accurately, where the Caputo's fractional derivative [13] is and Γ(.) is the gamma function.

Symbols
Quantity To balance the dimension of (13), we bring into the power α and β on λ 1 and λ 2 , respectively. Moreover, we will use the following dimensionless quantities to normalize the (13).
With the help of the above dimensionless quantities and dropping the hats sign, the (13) will become or , (16), then integrating concerned X and Y over [0, 1] × [0, 1], and utilizing the transmuted original and boundary conditions (18), we will get

Solution of Velocity Profile
where is the binary Fourier alteration of Ω(X, Y, t). By applying Laplace transformation and appropriate transform conditions on equation (15) we will get Ω lp (q) as or where which can be written as So it is denoted by and the inverse LT of the F(q) is Now, we consider the function , the expression for A lp (q) can be written as 4 Complexity e inverse LT of the expression (29) is where G e,f,g (h, t) is the generalized G-function as defined in [37].
e transformed velocity can be rewritten as e inverse LT of the (32) is where dq denotes the convolution product. Taking the inverse Fourier transform to (33) and using the formula [38], the velocity field for sin oscillation is It can be rewritten as e dimensionless tangential stresses T 1 and T 2 associated with the fractional Oldroyd-B fluid in such motions are given by where T 1s � dS xz /μu 0 and T 2s � hS yz /μu 0 . Taking the Laplace transform of (36), we can get Rewrite (38) as where Invoking z X Ω(X, Y, t) in (40), we will get e inverse LT of the above relation is where Similarly, we can calculate T 2s from (37).  (17), we obtain Now, taking the LT of (43) with an appropriate transform condition, we will obtain the expression for Ω lp (q) as Putting F lp (q) in (45), where K(q) � q(1 + λ α 1 q α )/q 2 + ω 2 , the inverse LT of the K(q) is Rewrite Ω lp (q), Ω lp (q) � a lp q q 2 + ω 2 − a lp K(q)A lp (q).
(48) e inverse LT (48) is where k(t) * a lp (t) � t o f(t − s)a lp (t)ds denotes the convolution product of k(t) and a lp (t). Taking the inverse Fourier alteration of (49) and utilizing the formula [38], the velocity profile is

(50)
Rewrite (50), Using the same technique as of the above section, we can find the tangential stresses under the forms: Similarly, we can calculate T 2c .

Classical Oldroyd-B Fluid.
Creating α ⟶ 1 and β ⟶ 1 into (35), (42), (51), and (52), we can acquire a similar solution of the velocity distribution of both cases for trembling flows of an ordinary Oldroyd-B fluid. us, the velocity field and shear stresses decrease to ,

Fractional Maxwell Fluid.
Making λ 2 ⟶ 0 into (35), (42), (51), and (52), then we can acquire both cases of identical solution of velocity dispersion and shear stress for generalized Maxwell fluid's flows [38]. us, the velocity field and shear stresses decrease to where  [23]. us, the velocity field and shear stresses decrease to Complexity 7    Complexity

Numerical Results
In this section, we will give graphically results for the velocity and shear stresses profiles for the various parameters. Also, we will show a comparison between the analytical and numerical results in a tabular form. e numerical results were obtained by Stehfest's and Tzou's numerical inverse Laplace algorithms. e access of the various physical parameters for time is graphically presented in Figures 2 to 13.
In Figure 2, the researchers strategized the absolute values of the velocity field versus time. e given diagrams are strategized for four values of the fractional coefficient α. e velocity of the fluid decreases (absolute values) is observed as α ⟶ 1 for the back and forth moment of both sine and cosine vibrations. In Figure 3, the researchers drew the consequences of the second fractional parameter β on the velocity field versus time. It was expected an opposite behavior concerning the first fractional parameter α and seen it comes true in the plot of Figure 3. In Figures 4 and 5, the effect of the relaxation parameter and the delay time on fluid motion is seen. From this figure, it is observed that its behavior is identical to that of a fractional parameter α, and when the fractional derivative parameter extends to 1 for the back and forth moment of both sine and cosine, the velocity  In Figure 8, the dimensionless shear stress versus t drew for various values of fractional coefficient α. e given diagrams are strategized for three values of the α. e stress on the fluid decreases for half interval of time, and the next half, it increases for sine oscillation because the standards of fractional parameter α increase. Figure 9 is sketched to show the dimensionless shear stress versus t directed for various values of fractional coefficient β. It can be observed that the effect of parameter β on the fluid motion has an opposite behavior compared to parameter α. Figures 10 and 11 are showing the effects of relaxation and retardation time on fluid motion. ey have the opposite effect on stress as expected due to the relation between the return of a perturbed system into equilibrium and delayed response to an applied force or stress. Figure 12 gives the impact of frequency factor w on the shear stress. From Figure 12, it is clearly sighted that shear stress of the fluid goes to decay due to the increase of w. e comparison of dimensionless shear stresses for fractional Oldroyd-B, ordinary Oldroyd-B,   e Stehfest's algorithm shows quite good agreement with our analytical solutions as compared to the numerical results obtained by Tzou's algorithm.

Conclusion
is communication aims to provide exact solutions for the fractionalized Oldroyd-B fluid in a fluctuating quadrilateral duct by applying the discrete Laplace and double finite Fourier transforms. Also, a comparison is shown in tables for the analytical and numerical results. e corresponding results were not studied before and have important remarks concerning the prevailing equations for the nontrivial shear stress. ese results fulfill all the executed initial and boundary conditions and were easily converted into parallel solutions. e parallel solutions for fractional Maxwell fluid, classical Oldroyd-B, and Maxwell fluid were regained as regulating case of the conventional solution.
e following conclusions were drawn: (i) Both sine and cosine oscillations of the velocity field decrease with the increase of fractional parameter α and vice versa for β (ii) An increase in the values of λ 1 and λ 2 decreases the velocity profile of both sine and cosine oscillations (iii) Dimensionless velocities comparison figured out fractional Oldroyd-B fluid is swiftest than fractional Maxwell fluid (iv) Dimensionless shear stress changes behavior after half interval, and the opposite effect was seen between α and β fractional parameters (v) Also, the analytical solutions show good agreement with the numerical results In future, we will study, what will be the effects on fractional Oldroyd-B fluid via a fluctuating duct by adding the magnetic and porous factors.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.