Heat and Mass Transfer of Rotational Flow of Unsteady Third-Grade Fluid over a Rotating Cone with Buoyancy Effects

Objective of this paper is a study of the impact of heat and mass transfer on time-dependent flow of a third-grade convective fluid due to an infinitely rotating upright cone. An interesting fact is observed that the similarity solutions only exist, if we take the angular velocities of the cone and far away from the fluid as an inverse function of time. (e analytical solutions of the reduced ordinary differential equations of third-grade fluids are offered by the optimal homotopy analysis method (OHAM). (e results for important parameters are illustrated graphically as well as in tabular form. (e precision of the present results is also checked by comparison with the numerical outcomes published earlier. (e impact of non-Newtonian fluid parameters is found to decrease the primary skin-friction coefficient. (ere is an aggregate in Nusselt and Sherwood numbers for increasing the ratio of the buoyancy forces.


Introduction
In the modern era, non-Newtonian fluid flows gain appreciable attention due to its huge applications in massive engineering and industrial processes.
is phenomenon achieves great importance in theoretical as well as practical viewpoint. Heat and mass transfer in non-Newtonian fluids are of significance in various manufacturing, industrial, and engineering applications such as food processing, nanomaterial production, and lubrication.
Nazir et al. [1] studied transport phenomenon in Carreau fluid with variable diffusion. Makinde and Chinyoka [2] reported numerical study of unsteady hydromagnetic generalized Couette flow of a reactive third-grade fluid with asymmetric convective cooling. Generalized Couette flow of a third-grade fluid with slip: the exact solutions were analyzed by Ellahi et al. [3]. Salahuddin et al. [4] found the solutions of MHD flow of Williamson fluid over a stretching sheet. Attia et al. [5] explored unsteady Couette flow of a thermally conducting viscoelastic fluid with porosity. Krishna et al. [6] offered multiple solutions for non-Newtonian fluids along an elongating surface. Hayat et al. [7] discussed impressions of radiations and chemical reactions in stretched flows of Jeffrey fluids. Properties of hall and ion slip with Jeffrey peristaltic fluid were deliberated by Ellahi et al. [8].
e process of mixed convection exists when induced and normal convection variations are compatible. It has a marvelous role heat exchangers, solar collectors and nuclear reactors. In the current work, a vertically aligned cone is fixed in a non-Newtonian fluid with the outward flow is inspected. Problems related to heat transfer with convective flow about conic shaped objects are superbly applied in automobile and organic productions. In numerous realworld situations, the unsteady combined convective flows do not essentially give similarity solution and for the past era, several problems have been examined, where the nonsimilarity has to be measured. e body arc is responsible for fluid flow for free stream velocity. e essential scientific complications intricate in finding nonsimilar solutions for such studies have limited many scientific minds to confine their findings either to the nonsimilar steady flows or to the unsteady semisimilar flows. In self-similar solutions, a set of partial differential equations can be converted to a scheme of ordinary differential equations. In modern studies, Anilkumar and Roy [9] and Saleem et al. [10] did an important work for mixed convection flow on a rotating cone in a rotating fluid. Significant features of Dufour and Soret effect on viscous fluid by a rotating cone with entropy generation were inspected by Khan et al. [11]. e variable surface temperature and heat flow on a vertical permeable circular cone was inspected on a free convection by Hassainetal et al. [12]. Shamkha et al. [13] examined the analysis of radiation impacts on a cone embedded in a porous nanofluid-filled medium in a mixed convection. ere are numerous developments in flows depending on time and with surface mass transfer (suction/injection), where the thermal and mass diffusion of temperature and concentration gradient results in the buoyancy effects. erefore, it is worthwhile studying the boundary layer fluid over a cone with the thermal diffusion and mass diffusion if the freely flow rate changes randomly with time [14][15][16][17][18] in order to improve the examination on mutual convection. e main task of the existing paper is to observe the variation of heat and mass transfer on third-grade fluid flow induced by the rotating cone. e highly nonlinear ordinary differential equations of the third-grade fluid with the prescribed wall temperature conditions are solved by the optimal homotopy analysis method (OHAM) [19][20][21][22][23][24][25][26][27]. Usually, the nonlinear system of ODE is difficult to tackle with the analytical method. Mostly, some well-known techniques such as perturbation methods are applied for this purpose. ese techniques have limitations as they rely on small/large parameters. Moreover, such algorithms fail to control the convergence region of series solutions. Liao [19] was the first who developed a semianalytical method to handle highly nonlinear equations with the specialty to regulate and control the region of convergence.
Furthermore, graphs and numerical tables show the effects of interesting parameters on the velocities, stress tensor, temperature, and concentration fields. In addition, comparisons of the findings with the literature previously available have verified the exactness of the analytical methodology.

Mathematical Model
Let consider the laminar incompressible flow of a thirdgrade fluid on a rotational vertical cone in a rotating fluid. e motion is unsteady due to the rotation of the cone and the fluid along the cone axis. e geometry of flow field is described in Figure 1.
A rectangular coordinate system is considered where u, v, and w are the velocity components in the x-tangential, y-azimuthal, and z-normal directions, respectively. Axisymmetric flow is considered while wall temperature T w and the wall concentration C w vary linearly in x. e momentum, energy, and diffusion equations for a third-grade fluid are given as [9] e appropriate physical boundary conditions are defined as where α i (i � 1, 2) and β i (i � 1, 2, 3) are the material parameters of fluid considered. Note that the viscous dissipation effects in the energy equation are assumed to be negligible. Introducing the following nondimensional quantities [9], where flow field accelerates for positive s (unsteady parameter) and vice versa. λ 1 is the buoyancy force parameter, N is the ratio of the buoyancy forces, and ε i (i � 1, 2, 3) are the third-grade fluid parameters. e governing equation along with boundary conditions (1-6) is systematically satisfied and written as Mathematical Problems in Engineering e skin-friction coefficients in the primary and secondary directions are, respectively, given by where where Re x � (x 2 Ω sin α * (1 − st * ) − 1 /υ). e rate of heat and mass transfer in nondimensional form is written as

Optimal Convergence-Control
where E t m is the residual error, δη � 0.5, k � 20. By using Mathematical package BVPh2.0, we have reduced the average residual error. Different sets of global optimal convergence-control parameters are obtained at various approximations to achieve minimum values of related total averaged-squared residual error (see Table 1). e average and total squared residual errors are defined in equations (30) to (33), respectively, and presented in Table 2 at diverse order of approximations. Increasing the order of approximations error can be reduced.

Results and Discussion
e variations of the profiles of the flow velocities, temperature, and concentration, as well as the skin-friction coefficients and the Nusselt and Sherwood numbers against the ratio of angular velocities c, ratio of the buoyancy force N, ε 1 , ε 2 , and ε 3 are third-grade parameter, and the unsteadiness parameter s are discussed through graphs and numerical tables.
Figures 2(a) to 2(d) describe the behavior of the tangential velocity − f ′ (η) for, ε 1 , ε 2 , and ε 3 . It is found from Figure 2(a) that for c � 0.5 the fluid and the cone are rotating in the same direction with equal angular velocity and only because of favorable gradient pressure, i.e., λ 1 � 1. When c > 0.5, the velocity − f ′ (η) increases its magnitude while it decreases for c < 0.5.
e velocity − f ′ (η) at the edge of the boundary layer can be seen asymptotically, when c < 0 is reached in oscillatory fashion. In the physical area of the boundary layer, such oscillations which arise from the excess convection of the angular momentum. Figures 2(b) and 2(c) elucidate that the magnitude of the velocity − f ′ (η) reduces for a second-grade parameter ε 1 , while there is an increase in the magnitude for ε 2 .
e velocity − f ′ (η) is a decreasing function of ε 3 (see Figure 2(d)). e influences of ε 1 , ε 2 , and ε 3 on the velocity g(η) is presented in Figures 3(a) to 3(d). Figure 3(a) displays that the velocity g(η) decreases for c > 0.5 , where as it is reversed when c < 0.5. It is evident in Figures 3(b) and 3(c) that the velocity g(η) enhances with an increase in ε 1 and ε 2 , respectively. It is seen from Figure 3(d) that ε 3 causes a reduction in the magnitude of the velocity g(η). ere is a rising behavior in the Nusselt and Sherwood numbers for increasing N and s (see Figures 4(a) and 4(b)). Table 3 is reported in order to confirm the serious solutions obtained. Table 3 indicates that the analysis and numerical findings follow the appropriate criteria.

Mathematical Problems in Engineering
e numerical values of the skin-friction coefficients in both directions for different emerging parameters are displayed in Table 4. e influence of ε 1 , ε 2 , ε 3 , and s is to decrease the primary skin-friction coefficient. On the other hand, the secondary skin-friction coefficient varies directly with ε 1 , ε 3 , and s but inversely proportional to ε 2 .  Table 2: Residual errors using optimal values at m � 8 from Table 1.  Mathematical Problems in Engineering

Conclusions
e problem of a third-grade fluid's combined convection flow is analyzed in the current work on a vertical rotating cone. e reduced nondimensional differential equations of third-grade fluid, temperature, and concentration are unraveled by the optimal homotopy analysis method (OHAM). e existing calculated outcomes are acknowledged in a conventional agreement with the previously published results available in the literature: (i) It is found that the tangential velocity has a reducing manner for ε 1 and ε 3 , however increases for ε 2 (ii) e skin-friction coefficients are increasing functions of the ratio of the buoyancy forces N (iii) e impact of third-grade fluid parameters ε 1 , ε 2 , ε 3 , and s is found to decrease the primary skin-friction coefficient (iv) ere is increasing activity with Nusselt and Sherwood numbers for increasing N values

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that they have no conflicts of interest.