**3. Imposing Condition and Geometry**

Initially, (when *t* = 0) the FBF is at rest in an infinite circular pipe with radius *R*( > 0) as shown in Figure 1. After time *t* = 0+, the pipe suddenly starts to rotate about its Z-axis having the angular velocity Ω*tp*. According to mathematical situation, appropriate initial and boundary conditions are

$$F(\mathbf{r},0) = \frac{\partial F(\mathbf{r},0)}{\partial t} = 0, \ \mathbf{r}(\mathbf{r},0) = 0; \ \mathbf{r} \in [0,R]. \tag{8}$$

$$F(\mathbb{R}, t) = R\Omega t^p; \qquad t > 0, \quad p \in \mathbb{N}, \ p > 0,\tag{9}$$

where Ω denotes the angular constant and *N* is the set of natural numbers.

**Figure 1.** Geometry of the problem.

#### *3.1. Velocity Field Due to Rotating Circular Pipe*

By applying the Laplace transform to Equations (5) and (9),

$$\frac{\partial^2 \overline{F}(\mathbf{r}, s)}{\partial \mathbf{r}^2} + \frac{1}{\mathbf{r}} \frac{\partial \overline{F}(\mathbf{r}, s)}{\partial \mathbf{r}} - \left[\frac{1}{\mathbf{r}^2} + a(s)\right] \overline{F}(\mathbf{r}, s) = 0,\tag{10}$$

and

$$\mathbb{F}(\mathbb{R}, s) = \frac{R \Omega p \, ! \,}{\text{s}^{p+1}} \, , \tag{11}$$

where

$$a(\mathbf{s}) = \frac{s(1 + \lambda\_1^{\alpha}\mathbf{s}^{\alpha} + \lambda\_2^{2\alpha}\mathbf{s}^{2\alpha})}{\nu(1 + \lambda\_3^{\beta}\mathbf{s}^{\beta})}.$$

Also *<sup>F</sup>*(r,*s*) and *<sup>F</sup>*(*<sup>R</sup>*,*<sup>s</sup>*) are the Laplace transforms of the functions *<sup>F</sup>*(r, *t*) and *<sup>F</sup>*(*<sup>R</sup>*, *t*) respectively. By applying the variable transformation *z* = r*a*(*s*) in Equation (10)

$$z^2 \frac{\partial^2 \overline{F}}{\partial \mathbf{r}^2} + z \frac{\partial \overline{F}}{\partial \mathbf{r}} - (\mathbf{1}^2 + z^2)\overline{F} = 0. \tag{12}$$

The above Equation (12) is the modified Bessel equation of order 1. So, the general solution of this equation is given by the successive equation

$$\overline{F}(z,s) = \mathbb{C}\_1 I\_1(z) + \mathbb{C}\_2 K\_1(z),\tag{13}$$

where *C*1 and *C*2 are constants and *I*1, *K*1 are the modified Bessel functions of first and second kind of order 1. For a finite solution at r = <sup>0</sup>(*z* = <sup>0</sup>), the constant *C*2 should be zero i.e., *C*2 = 0. So, Equation (13) implies

$$\mathsf{F}(z,s) = \mathsf{C}\_1 I\_1(z). \tag{14}$$

The value of *C*1 is calculated from Equation (14) by taking Equation (11) into account

$$\mathcal{C}\_1 = \frac{\Omega R p!}{s^{p+1} I\_1(\mathcal{R}\sqrt{a(s)})}.\tag{15}$$

Therefore the Equation (14) implies

$$\overline{F}(\mathbf{r}, \mathbf{s}) = \frac{\Omega R \, p!}{\mathbf{s}^{p+1}} \times \frac{I\_1(\mathbf{r}\sqrt{a})}{I\_1(R\sqrt{a})}.\tag{16}$$

The solution in Equation (16) is in the complex form of the first and second kind of Bessel functions. Normally, it is difficult to find out the solution of such complex expressions with an ordinary inverse Laplace transformation. However, an alternately inverse Laplace of Equation (16) was numerically calculated by using the "Stehfest's algorithm" [27]

$$f\_n(\mathbf{x}) = \ln(2)\mathbf{x}^{-1} \sum\_{i=1}^{2n} a\_i(n) F(i\ln(2)\mathbf{x}^{-1}), \qquad n \ge 1, \mathbf{x} > 0,\tag{17}$$

where *ai*(*n*) are coefficients defined as follows

$$a\_i(n) = \frac{(-1)^{n+i}}{n!} \sum\_{s=\lfloor \frac{(i+1)}{2} \rfloor}^{\min(i,n)} s^{n+1} \, ^\text{\textquotedblleft C}\_s \, ^\text{\textquotedblleft C}\_{i-s} \, ^\text{\textquotedblright C}\_{i-s} \, \qquad n \ge 1, 1 \le i \le 2n,\tag{18}$$

and "MATHCAD" software.

#### *3.2. Shear Stress Due to Rotating Circular Cylinder*

By taking the Laplace transform of Equation (6)

$$\overline{\mathbb{Q}}(\mathbf{r},s) = \sqrt{b(s)} \left[ \frac{\partial \overline{\mathbb{F}}(\mathbf{r},s)}{\partial \mathbf{r}} - \frac{1}{\mathbf{r}} \mathbb{F}(\mathbf{r},s) \right],\tag{19}$$

where

$$b(\mathbf{s}) = \frac{\mu(1 + \lambda\_3^{\beta}\mathbf{s}^{\beta})}{(1 + \lambda\_1^{\alpha}\mathbf{s}^{\alpha} + \lambda\_2^{2\alpha}\mathbf{s}^{2\alpha})}.$$

Putting Equation (16) into (17) and after taking the few steps of simplification,

$$\overline{Q}(\mathbf{r},s) = \frac{\sqrt{b}}{s^{p+1}} \times \frac{p!\Omega R}{I\_1(R\sqrt{a})} \left[ \sqrt{a}I\_0(\mathbf{r}\sqrt{a}) - \frac{2}{r}I\_1(\mathbf{r}\sqrt{a}) \right]. \tag{20}$$

Again it is in a complicated form. So "Gaver Stehfest's algorithm" and "MATHCAD" software were used for the inverse Laplace transformation of the result given in Equation (20) .

#### **4. Results and Discussions**

In this article, the main goal is to establish the numerical technique to develop the solutions of the Burgers' fluid for the rotational flow of Burgers' fluid within cylindrical domain. The expressions of velocity and shear stress are found for an incompressible non-integer order model of the Burgers' fluid in a circular pipe. The Laplace transformation is used to establish the solutions. As there are complicated expressions in Equations (16) and (20) with respect to the Laplace transformation, it was not easy to find out final results by using direct inverse Laplace on Equations (16) and (20). Although, "MATHCAD" is used along with the "Stehfest's algorithm" instead of the inverse Laplace transformation to find out the numerical results. The numerical solutions for Oldroyd-B, Maxwell, second grade and Newtonian fluids are also derived with generalization of main results i.e., Equations (16) and (20). To determine the impact of physical parameters, the graphical illustrations are made. The behaviour of time on the velocity field and stress is shown in Figures 2 and 3. These figures show that the influence of velocity and stress depending upon the parameters r, *λ*1, *λ*2, *λ*3, *α*, *β*, *ν* and *μ*. Figures 2 and 3 show that the velocity and shear stress increase with respect to the increase in the dependent variable *t* with fixed values of other dependent parameters. Similarly, a clear increase in velocity and stress can also be seen in Figures 4 and 5 for r with respect to the time variable *t* while fixing the other dependent parameters. It is also observed from Figure 4 that velocity has linear relation for r with respect to *t*.

**Figure 2.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *t* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 3.** The aspect of stress *Q*(r, *t*) for various values of *t* and fixed values for *μ* = 0.010, Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 4.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of r and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 5.** The aspect of stress *Q*(r, *t*) for various values of r and fixed values for *μ* = 0.09, Ω = 1.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.7, *ν* = 0.053, *n* = 12, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

 

Also, to uncover the aspects of other physical and fraction parameters on velocity and stress the graphs drawn and presented in Figures 6–17. All the graphs for velocity and stress in Figures 6–17 are plotted for concerned parameter against the change factor "time". One can say that the fractional parameter *α*, relaxation parameters *λ*1 and *λ*2 have decreasing behaviour for velocity and stress with respect to *t*. This fact can be observed in Figures 6–9, 12 and 13 respectively. Whereas, the velocity and stress are also increasing functions for the retardation parameter *λ*3, fractional parameter *β*, viscosity *μ* and *ν* with respect to time *t* as shown in Figures 10, 11, 14–17. Figures 2–17 make it possible to check the point to point variations, increment or decrement in two parameters (among which the graphs are made) for velocity and stress profile.

Finally, the comparison among the different generalized cases of Burgers' fluid are made in Figure 18 for velocity function against time *t*. The results of these generalized cases are derived by the implementation of the "Stehfest's algorithm" and "MATHCAD" to Equation (16). It is observed that the second grade fluids have more velocity as compared to other Newtonian and non-Newtonian fluid

cases of this model while taking the same values of different dependent parameters and variation in time *t*. The Oldroyd-B fluid behaves as like the second grade fluid but have less velocity as compared to the second grade fluid's velocity. Similarly, Figure 18 also shows the behaviour of velocity for Newtonian and Maxwell fluids. The Maxwell fluids have minimum velocity for this flow model with prescribed conditions among all the Newtonian and non-Newtonian cases. As a description it is clearing that in all the Figures 2–18, the units of the material constants are SI units. The comparison of solutions that obtained from two different methods analytically [29,30] and numerically (derived from Equations (16) and (20)) are given in Tables 1 and 2 for two different cases Maxwell and Newtonian fluids. These tables showing clearly that two different methods for the same problem have the same results. Which shows the consistency of our numerical technique and results for fractional model of Burgers' fluid with already published literature [29,30].

**Figure 6.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *λ*1 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*2 = 20 and *λ*3 = 55.

**Figure 7.** The aspect of stress *Q*(r, *t*) for various values of *λ*1 and fixed values for *μ* = 0.09, Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*2 = 20 and *λ*3 = 55.

**Figure 8.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *λ*2 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*3 = 55.

**Figure 9.** The aspect of stress *Q*(r, *t*) for various values of *λ*2 and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*3 = 55.

**Figure 10.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *λ*3 and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*2 = 20.

**Figure 11.** The aspect of stress *Q*(r, *t*) for various values of *λ*3 and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.2, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5 and *λ*2 = 20.

**Figure 12.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *α* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 13.** The aspect of stress *Q*(r, *t*) for various values of *α* and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *β* = 0.7, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 14.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *β* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 15.** The aspect of stress *Q*(r, *t*) for various values of *β* and fixed values for *μ* = 0.09, Ω = 1.2, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 16.** The aspect of velocity *<sup>F</sup>*(r, *t*) for various values of *ν* and fixed values for Ω = 1, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 17.** The aspect of stress *Q*(r, *t*) for various values of *μ* and fixed values for Ω = 2.5, *R* = 1, *p* = 1,*s* = 1, *α* = 0.3, *β* = 0.8, *ν* = 0.013, *n* = 12, r = 0.8, *λ*1 = 5, *λ*2 = 20 and *λ*3 = 55.

**Figure 18.** Comparisons of fluid velocity for different fluid models for same value of parameters.

**Table 1.** Comparison of exact solutions [30] and numerical solutions (obtained from "Stehfest's algorithm") for the fractional order derivative model of the Maxwell fluid.



**Table 2.** Comparison of exact solutions [29] and numerical solutions (obtained from "Stehfest's algorithm") for the fractional order derivative model of the Newtonian fluid.
