**2. Governing Equations**

Here, the formulation of velocity **V** and the extra-stress **S** for the fluid under consideration are as [28]

$$\mathbf{V} = \mathbf{V}(\mathbf{r}, t) = F(\mathbf{r}, t)\mathbf{e}\_{\theta}, \quad \mathbf{S} = \mathbf{S}(\mathbf{r}, t) \,, \tag{1}$$

where **e***θ* is the unit vector of the cylindrical coordinates system and *<sup>F</sup>*(r, *t*) is the component of velocity along **e***θ*. The initial conditions for the fluid at rest position are, as in [28];

$$\mathbf{V}(\mathbf{r},0)=\mathbf{0},\qquad\mathbf{S}(\mathbf{r},0)=\mathbf{0}.\tag{2}$$

The governing equations for the motion of Burgers' fluid are [29]

$$\left(1+\lambda\_1\frac{\partial}{\partial t}+\lambda\_2\frac{\partial^2}{\partial t^2}\right)\frac{\partial F}{\partial t}=\nu\left(1+\lambda\_3\frac{\partial}{\partial t}\right)\left(\frac{\partial^2}{\partial \mathbf{r}^2}+\frac{1}{\mathbf{r}}\frac{\partial}{\partial \mathbf{r}}-\frac{1}{\mathbf{r}^2}\right)F(\mathbf{r},t)\,\tag{3}$$

$$
\left(1 + \lambda\_1 \frac{\partial}{\partial t} + \lambda\_2 \frac{\partial^2}{\partial t^2}\right) Q(\mathbf{r}, t) = \mu \left(1 + \lambda\_3 \frac{\partial}{\partial t}\right) \left(\frac{\partial}{\partial \mathbf{r}} - \frac{1}{\mathbf{r}}\right) F(\mathbf{r}, t) \,, \tag{4}
$$

where *μ* is the coefficient of viscosity, *ν* = *μ*/*ρ* is kinematic viscosity, *ρ* is constant density of the fluid, *λi* (*i* = 1, 2, 3) new material constants and *Q*(r, *t*) = *S*r*<sup>θ</sup>* (r, *t*) = 0 shear stress. By altering the inner time derivatives with the fractional derivatives in Equations (3) and (4), the governing equations for the fractional derivative of Burgers' fluid (FBF) can be obtained as

$$\left(1+\lambda\_1^a D\_t^a + \lambda\_2^{2a} D\_t^{2a}\right) \frac{\partial F(\mathbf{r},t)}{\partial t} = \nu \left(1+\lambda\_3^\beta D\_t^\beta\right) \left(\frac{\partial^2}{\partial \mathbf{r}^2} + \frac{1}{\mathbf{r}} \frac{\partial}{\partial \mathbf{r}} - \frac{1}{\mathbf{r}^2}\right) F(\mathbf{r},t) \,, \tag{5}$$

$$\left(1+\lambda\_1^a D\_t^a + \lambda\_2^{2a} D\_t^{2a}\right) Q(\mathbf{r}, t) = \mu \left(1+\lambda\_3^\beta D\_t^\beta\right) \left(\frac{\partial}{\partial \mathbf{r}} - \frac{1}{\mathbf{r}}\right) F(\mathbf{r}, t) \,, \tag{6}$$

where *α* and *β* are the fractional parameters such as 0 ≤ *α* ≤ *β* ≤ 1. The Caputo fractional derivative is defined as [16,18]

$$D\_t^\kappa f(t) = \begin{cases} \frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int\_0^t \frac{f(\tau)}{(t-\tau)^\alpha} d\tau, & 0 \le \alpha < 1; \\\\ \frac{d}{dt} f(t), & \alpha = 1, \end{cases} \tag{7}$$

where <sup>Γ</sup>(·) is the gamma function. When *α* , *β* → 1, Equations (5) and (6) reduce to Equations (3) and (4), because *D*1*t f* = *d f dt*.
