COMPREHENSIVE EXAMINATION OF THE THREE-DIMENSIONAL ROTATING FLOW OF A UCM NANOLIQUID OVER AN EXPONENTIALLY STRETCHABLE CONVECTIVE SURFACE UTILIZING THE OPTIMAL HOMOTOPY ANALYSIS METHOD

This article explores the three-dimensional (3D) rotating flow of Upper Convected Maxwell (UCM) nanoliquid over an exponentially stretching sheet with a convective boundary condition and zero mass flux for the nanoparticles concentration. The impacts of velocity slip and hall current are being considered. The suitable similarity transformations are employed to reduce the governing partial differential equations into ordinary ones. These systems of equations are highly non-linear, coupled and in turn solved by an efficient semi-analytical scheme known as optimal homotopy analysis method (OHAM). The effects of various physical constraints on velocity, temperature, and concentration fields are analyzed graphically and discussed in detail. The impact of hall current is reduced the temperature field whereas increase to the velocity and the concentration fields. The present results are compared with the available results in the literature to check the legitimacy of the present semianalytical scheme and noted an excellent agreement for limiting cases.


INTRODUCTION
Boundary layer flow of a viscous fluid due to an impulsive motion over an elastic surface is involved in several areas of science and technology such as "drawing, annealing, and tinning of copper wires, rolling and manufacturing of plastic films, and artificial fibers," etc. In these application processes, the end product primarily depends on the rate of stretching of the surface, which is very significant. The pioneering work of Sakiadis (1961) considering stationary ambient fluid over a moving plate has brought new dimensions to the boundary layer theory. Crane (1970) obtained the closed-form of the exact solution for the velocity distribution and examined the work of Sakiadis (1961) by considering the velocity of the stretching sheet which is proportional to the distance from the slit. Rajagopal et al. (1984) made a comparative analysis between viscoelastic fluid and viscous fluid and examined the rate of cooling of viscoelastic fluid flow over a stretching sheet. Grubka and Bobba (1985) and Lawrence and Rao (1992) extended the work of Rajagopal et al. (1984) for heat transfer characteristics. Further, several researchers have examined the stretching sheet geometry problems for two-dimensional flows (Ganji et al. (2014); Parand et al. (2017); Rahimi et al. (2017) and three-dimensional flows (Hayat et al. (2012); Nadeem et al. (2013); Weidman and Ishak (2015)).
All the above studies confined their examinations to twodimensional/three-dimensional flows over linearly stretched sheets. On the other hand, the demands of the technological industries and previously mentioned applications are not only confined to a linearly stretched sheet but also to the non-linearly extruded sheet. The stretching of the sheet exponentially is one of the prominent methods to meet the nonlinear requirements of the industry. Given this, Magyari and Keller (1999) and Elbashbeshy (2001) analyzed the flow pattern due to an exponentially continuous stretching sheet. Sajid and Hayat (2008) and Bidin and Nazar (2009) employed HAM/Keller-box method to obtain the analytical/numerical solution to examine the impact of thermal radiation on the flow using exponentially stretching sheet. Further, Swati Mukhopadhyay (2013) investigated that the slip effects over a magnetic flow field with suction/blowing are prominent. Of late, Fazle et al. (2017) and Srinivasacharya and Jagadeeshwar (2017) extended the work of Magyari and Keller (1999) and Swati Mukhopadhyay (2013) by considering radiation effects.
In recent advancement, the nanotechnology is an attractive area of discussion owing to its enriching characteristics of controlling the thermal conductivity. A blend (Solid-Liquid) of very small-sized nanoparticle ( 100 nm) and base fluid is known as nanofluid. The colloids of the base fluids are usually made up of metal and oxides, which enhance both the conduction and convection coefficient and also improve significantly the heat transfer rates of the coolants. The nanofluids have emerged as special kinds of many applications in heat transfer such as nuclear reactor cooling, solar water heating, domestic refrigerators, drag reduction, and thermal energy storage, etc. Choi (1995) has initially experimented on the base fluid, which is the addition of the mixture of metal oxides to the base fluid and observed the enhancement of the thermal properties in the fluid. Buongiorno (2006) proposed a most conventional model to describe the convective transport based on the mechanism of Brownian motion with thermophoretic diffusion and remarked that the heat transfer performance has a vital role in the nanofluid. Further, Tiwari and Das (2007) examined two-sided lid-driven differentially heated square cavity filled with nanofluids model which different form conventional Buongiorno (2006) model and explained behavior of particle size, momentum/thermal diffusivity, and temperature. Some recent attempts to describe in this direction are Mustafa et al. (2016); Hayat et al. (2017); Animasaun et al. (2019); Prasad et al. (2018);
Inspired by subsequent developments in the available literature, our main objective of the present investigation is to analyze the threedimensional rotating flow of a UCM nanoliquid over an exponentially stretchable surface. Rotating flows usually involve in an anticyclone flow circulation, geological stretching of tectonic plate beneath the rotating ocean, centrifugal filtration process, in rotorstator systems, and cooling of skins of high-speed aircraft. The analysis is carried out in the presence of hall effect, velocity slip, convective boundary condition, and zero mass flux nanoparticle concentration. Here, the local similarity equations are derived and solved analytically for varying values of embedded parameters by the semi-analytical technique known as OHAM (see for details, Liao (2010); Marinca and Herisanu (2015); and Van Gorder (2019)). The impacts of different physical parameters on velocity, temperature, and concentration profiles are analyzed graphically. In addition to this, estimations of skin friction, local Nusselt number, and local Sherwood number are presented in the analysis, which is very important from the industrial application point of view.

MATHEMATICAL FORMULATION AND PHYSICAL DESCRIPTION OF THE STUDIED FLOW PROBLEM
Let's consider a steady three-dimensional (3D) rotating flow of a viscous incompressible Upper Convected Maxwell (UCM) nanoliquid by an exponentially stretchable surface subjected to the slip velocity, convective boundary condition and zero mass flux concentration. The Cartesian coordinate system is adopted in such a way that the surface is aligned with and x y-axes and the fluid is taken in the space 0 z (see Fig. 1).

Fig. 1
Geometry of the Maxwell nanoliquid flow model.
The fluid is rotating about Z -axis with constant angular velocity . The fluid is considered electrically conducting, and a transverse magnetic field 0 B is applied in the Z -direction. Further, the hall current effect is taken into account. In general, the hall current and electrically conducting fluid affect the flow in the presence of a strong magnetic field. The effect of hall current gives rise to a force in the Z -direction, and hence the flow becomes three dimensional. The generalized Ohm's law with hall current is defined as Here, the following assumptions are considered. a) Joule heating is neglected. b) The wall is impermeable w i.e.,v 0 .
c) The sheet is stretchable with a variable velocity and slip velocity is given by where o U is the reference velocity, L is the characteristic length, 1 k is the slip constant. The physical problem under consideration includes the connections of momentum, energy, and mass. These relations can be condensed as with the following realistic boundary conditions (BCs) as .
x L w Here ,vand u ware the fluid velocity components along the , and x y z -direction, respectively. Further, v is the kinematic viscosity, 1 is the relaxation time, is the density of the fluid, m is the hall effect parameter, T is the temperature, is the thermal diffusivity, is the ratio of the effective heat capacity of the nanoparticle material and heat capacity of the fluid, B D is the Brownian diffusion coefficient, C is the concentration of nanoparticles, T D is the thermophoresis diffusion coefficient, T is the ambient fluid temperature, k is the thermal conductivity, h is the heat transport coefficient, f T is the hot fluid temperature. The magnetic field 0 B is considered to be uniform.
Stretchable surface Nano size particles

Similarity variables and dimensionless governing equations
To simplify the mathematical analysis of the model by the following similarity variables (see details Hayat et al. (2017)) are evoked, (9) where the prime superscripts represent the differentiation concerning . Using the above transformations, the continuity equation given by Eq.
(3) is automatically verified, while Eqs. (4) -(7) are reduced to In the above expression, is the rotation parameter, is the Deborah number, Mn is the magnetic parameter, Pr is Prandtl number, A is the temperature exponent, Nb is the Brownian motion parameter, Nt is the thermophoresis motion parameter, Sc is the Schmidt number, B is concentration exponent, 1 K is velocity slip parameter, Bi is the Biots number and defined as follow 2 2 1 0

Skin-friction, heat and mass transfer coefficients
The local skin-friction coefficients , , a n d where wx and wy are the skin-friction (at wall) along and x yaxis, and q is the heat flux from the plate are defined as in terms of non-dimensional quantities are obtained as Particularly, for zero nanoparticles mass flux condition, the dimensionless mass flux denoted by the local Sherwood number x Sh is identically zero, where Re / x w u L v represents the local Reynolds number.

SOLUTION METHODOLOGY BY MEANS OF OHAM
In this section, we solve Eqs. (10) -(13) with BCs. (14) by a productive semi-analytical algorithm known as Optimal Homotopy Analysis Method (see for details, Liao (2010); Marinca and Herisanu (2015); and Van Gorder (2019)). Consequently, a nonlinear problem is changed into an infinite number of linear sub-problems. In the frame of OHAM, we have an incredible opportunity to pick initial approximation and auxiliary linear operators and are picke as Bi Nb It is worth noting here that the auxiliary linear operators in Eq. (20) satisfy the properties In the present procedure, we construct the following zero th -order deformation equations are given by , with the following BCs where 0,1 q is an embedding parameter, ( , , , ) Frontiers in Heat and Mass Transfer (FHMT), 14, 11 (2020) DOI: 10.5098/hmt.14.11 Global Digital Central ISSN: 2151-8629 where the convergence of the above series strongly depends upon , , and The p th -order deformation equations and their corresponding BCs are   , , In practice the evaluation of the residual errors are ˆˆˆ, , , consumed a large amount of time.
So, instead of computing the exact residual errors, it is feasible to handle the accuracy of the problem using the average residual errors defined by  which have been analyzed by minimizing the averaged residual error and total residual error at 12 th -order approximation. It can be observed that the averaged squared residual error, and total residual errors are consistently reduced as increases the order of approximations. Further, the average squared residual error and total residual error of each governing equations are found in the diminishing function of the order of approximation, as shown in Fig.  2(a-b).

MODEL VALIDATION
Here, we present the exact solutions in certain special cases. These solutions have much importance due to they provide as a baseline for the comparison with the obtained results in the literature through the numerical solutions. In the absence of somenon-dimensional parameters, with boundary conditions are 0 (0) 0, ' 0 1, '(0) (1 (0)), To authenticate and validate the exactness of the proposed OHAM procedure, the present outcomes are compared with those reported by CPU time 2 6.09 10 -4 2.86 10 -3 2.57 10 -5 3.53 10 -6 3.51s 4 3.59 10 -5 3.89 10 -4 8.28 10 -7 6.19 10 -7 34.59 s 6 4.96 10 -6 9.74 10 -5 4.40 10 -9 8.93 10 -8 263.18s 8 1.17 10 -6 3.55 10 -5 1.31 10 -9 1.69 10 -8 1261.57s 10 4.16 10 -7 1.67 10 -5 1.11 10 -9 5.52 10 -9 4898.92s 12 1.97 10 -7 9.29 10 -6 3.21 0 -10 2.27 10 -9 24101.19s  Hayat et al. (2017) for some special cases. The correctness found to be in superior agreement (see Table 3).   Table 4. Further, the smaller Deborah number gives a viscous effect compared to the elastic effect, whereas the larger exhibit in the elastically solid material in nature. With reference to ( ), both and increases the temperature field, which is recorded in Fig.   3(c). The concentration profile exhibits the decreasing trend for and (see Fig. 3(d)). Figure   Bi on ( ) is sketched in Fig. 6(b). The terms / Bi increase. It is noticed that the fluid temperature is zero when 0 Bi and it is prescribed temperature at the wall when it tends to infinity. noted that the nanoparticle volume fraction increases with the increase in Nt (increase in thermophoresis force) and thus, an enhancement in the thickness of the concentration boundary layer is observed. In this case, the nanoparticles move away from the hot stretching sheet towards the cold ambient fluid under the influence of the temperature gradient. But in the case of Nb (smaller nanoparticles), the result is the opposite. However, Nb will stifle the diffusion of nanoparticles away from the surface, which results in a decrease in nanoparticle concentration values in the boundary layer. Finally, Figure 8(a-c) displays the 3D plot of velocities and these plots exhibit similar results as that of velocity profiles. Table 4 is tabulated to exhibit the influence of embedding parameters on the skin-friction coefficient, the local Nusselt number, and the local Sherwood number. It is seen that the rising values of

CONCLUSIONS
Some of the interesting findings of the present work are summarized below.
The rotation parameter decreases the '( ), f g( ), ( ) and ( ) whereas the hall current parameter exhibits reverse trend. A substantial variation in Deborah number reduces '( ) f and ( ) while ( ) g and ( ) rises.
The enhanced magnetic parameter and velocity slip parameter decreases '( ) f . Increased Prandtl number and temperature exponent and samller magnetic parameter, reduces ( ) .
An increase in ( ) is due to the increase in the Schmidt number, temperature exponent, thermophoresis parameter and the Brownian motion parameter. ' differentiation with respect to .