Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration

: In this article, the author has examined the unsteady flow over a rotating stretchable disk with deceleration. The highly nonlinear partial differential equations of viscous fluid are simplified by existing similarity transformation. Reduced nonlinear ordinary differential equations are solved by homotopy analysis method (HAM). The convergence of HAM solutions is also obtained. A comparison table between analytical solutions and numerical solutions is also presented. Finally, the results for useful parameters, i.e., disk stretching parameters and unsteadiness parameters, are found.


Introduction
Recently, due to the massive practical application in the scientific and technical field, the study of the rotating stretchable disk has become significant, such as thermal power generation system, medical equipment's, computer storage devices, rotating machinery, gas turbine routers, air cleaning machines, crystal growth process, and in aerodynamic applications [1]. Initially, von Kármán [2] conducted a study on rotating disk. Several researchers then illustrated the different aspects of this important analysis. Fang and Zhang [3] have highlighted the flow between two stretchable disks and found the exact solutions. The parameters analysis and optimization of entropy generation in unsteady magneto hydrodynamics flow over a rotating stretchable flow over a rotating disk using artificial neural network and practical swarm optimization algorithm was presented by Rashidi et al. [4]. Recently, Fang and Tao [5] wrote about the unsteady flow over a rotating stretchable disk with deceleration. After using the similarity analysis, they found the numerical solutions.
In many situations, exact solutions are very difficult and in most of the cases exact solutions are impossible. Therefore, series solutions are more useful if they satisfy the given initial and boundary value problems. Nevertheless, there are various analytical approaches, and each approach has certain limitations. However, homotopy analysis method (HAM) has many advantages over many analytical methods. Liao [6] introduced the idea of HAM, which is used by many researchers effectively. Some useful studies are cited in [7][8][9][10][11][12][13][14][15]. The purpose of this article is to illustrate the application of HAM for unsteady Newtonian fluid flow over a rotating stretchable disk with declaration. Tables provide a correlation between current HAM solution and Fang and Tao's [5] numerical solution.

Formulation of the Problem
Let us consider an incompressible, laminar, and unsteady flow of a viscous fluid or Newtonian fluid over a stretchable disk, which is rotating about the -axis with time dependent angular velocity Ω , where Ω is constant angular speed of the disk and 'b' is the measure of unsteadiness.
Flow is due to the rotation of the stretchable disk and is axisymmetric about the z-axis. Figure 1 describe the geometry of the proposed problem. The governing equations for an unsteady threedimensional flow of viscous fluid in cylindrical coordinates are shown below.
where is along the radial direction, is along the azimuthal direction, and is in normal direction to the axis. Here, Equation (1) is the continuity equation and Equations (2) and (4) represent the momentum equation for incompressible flow.
Where , , and are the velocities along , , and directions, is the density of fluid, is the pressure, is the kinematic viscosity, and , , , , are the stress which are defined as The proposed boundary conditions are specified in accordance with the geometry of the problem as where 'a' is the disk stretching parameter. Introducing, the similarity transformation used in [5] are Applying these similarities into the above equations, following non-dimensional equations along with boundary conditions can be obtained as where = Ω ⁄ is the unsteadiness parameter.

Homotopy Analysis Method
Homotopy Analysis Method (HAM) [6][7][8][9][10][11][12][13][14][15] is used to find an analytical solution to Equations (8) and (11). The velocity distribution ( ) and ( ) can be expressed by a set of base functions ( ) , ≥ 0 in the form in which , and , are the coefficients, the initial guesses and can be selected on the basis of the law of the solution expressions and of the boundary conditions: The auxiliary linear operators are which satisfy where ( = 1 − 5) are integral constants.

Zeroth-Order Deformation Equation
If ∈ 0,1 denote an embedding parameter, ℏ and ℏ indicate the non zero auxiliary parameters for ( ) and ( ), the zeroth-order deformations for the given problem are By Taylor's theorem and

Mth-Order Deformation
Differentiating the zeorth-order deformation Equations (21) and (23) with respect to , then setting = 0, and finally dividing them by !, the mth-order deformation equations can be obtained as where

Convergence of the HAM Solution
The homotopy analysis method includes the regulating parameter ℎ, which controls the region of convergence and HAM solution approximation. To ensure that the solutions converge within the admissible spectrum of auxiliary parameter values and ℎ and ℎ , ℎ − were sketched for 15th-order approximation. The ℎ − are plotted in Figures 2 and 3.

Results and Discussion
To solve Equations (8) and (9) homotopy analysis method (HAM) is applied as a subject to the boundary conditions (11). Homotopy analysis method is a strong analytical technique which is applied to obtain the convergent series solution of nonlinear differential equations.
where ( ) is the residual error of Equations (8) and (9) at the mth-order approximation. It is observed that 10th-order approximation is in good agreement with the numerical result. The convergence control parameter plays an important role. In Tables 5 and 6, the effect of ℏ on convergence is shown. Tables 5 and 6 show that the convergence of the solution depends strongly on ℏ. It can be seen easily that for one set of ℏ the convergence is faster than the other. Table 5. The Convergence analysis of ′′(0) for different ℏ when = 1, and = −1 2 ⁄ . Order Numerical result [5] ( ) = −0.8007. Table 6. The Convergence analysis of ′(0) for different ℏ when = 1, and = −1 2 ⁄ . In Figures 5 and 6, the comparison of 5th-order approximation with 10th-order approximation is shown, which again provide the facts for convergence. The Mathematica software is used to compute the results for higher-order approximation. As the given problem is highly nonlinear, the computation time increases if higher-order approximation is computed or increases the value of the parameters. For = 1 and = 0, the given problem becomes a special case as mentioned in the numerical paper [5]. Table 7 provides the convergence result for this special case as well. The results obtained in the present research for this special case are also in very good agreement with the numerical result. This shows the strength of homotopy analysis methods. It is found that for small , (0) decreases with the increase of 'a' as shown in Tables 2 and 3. Figure 7 represents the velocity distribution for different values of . It is observed that with the increase in disk stretching parameter the velocity decreases. Figures 8 and 9 show that with the decrease in the unsteadiness parameter, both tangential and radial velocities increase.

Conclusions
In this research, viscous axisymmetric flow is studied on a stretchable rotating disk with deceleration. It is found that Navier-Stokes equation admits a similarity solution, which depends on non-dimensionalized parameters and measuring unsteadiness and disk stretching, respectively. The resulting group of nonlinear ordinary differential equations is then solved analytically using homotopy analysis method (HAM). In numerical paper [5] it is mentioned that there are two solution branches. The upper solution branch is physically feasible, but the lower solution branch may not be practically possible. Here, the author has discussed and evaluated the outcome for a physical solution from the upper field.
The main results are summarized as