Mathematical analysis of Heat and mass transfer for Williamson nanofluid flow over an exponentially stretching surface subject to the exponential order surface temperature and heat flux

This article investigates the features of heat and mass transfer for the steady two-dimensional Williamson nanofluid flow across an exponentially stretched surface depending on suction/injection. The boundary conditions incorporate the impacts of the Brownian motion and thermophoresis boundary. The analysis of heat transfer is carried out for the two cases of prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). The ongoing flow problem is mathematically modeled under the basic laws of motion and heat transfer. The similarity variables are allowed to transmute the governing equations of the problem into a similarity ordinary differential equation (ODEs). The solution of this reduced non-linear system of ODEs is supported by the Homotopy analysis method (HAM). The combination of HAM arrangements is acquired by plotting the h-curve. In order to evaluate the influence of several emergent parameters, the outcomes are presented numerically and are plotted diagrammatically as a consequence of velocity, temperature and concentration profiles.


Introduction
Crane [1] was one of the early researchers who addressed thepsteadywtwo-dimensionalpflow in terms of the Newtonian0fluid, which was conducted by means of an elastic stretching sheet with a linearly varying velocity. The additional introductory judgment of the boundary layer flow across a stretched surface was donated by Sakiadis [2]. Wang [3] studied the Navier-Stokes equations as a function of time that were initially whiteuced to ODEs assisted by a dimensionless transformationpand later solved through the shooting method. In accordance of the above concept, this modern research was further developed by a large number of researchers with the aim of considering different characteristics of flowpandkheat transfer that occurs in an infinite range of stretching surface. In these modern years, moderate importance has been established in the research of stretching boundary in view of its considerable and growing industrial technology applications containing sheet extraction, paper production, cooling of microchipping or metal film, hot rolling, bundle wrapping, and several others. As a result of above cases the ultimate compound of desired properties relies onpthepratepofpcooling and the manner of stretching. In the context of these effective applications, significant work is carried out in multiple directions regarding the flow of the boundaryplayer andpheat transfer by providing the surface stretching. The excess of research is provided in the publications about the steady flows of stretching phenomenon particularly in [4,5]. Although some effort has been conducted in terms of unsteady flows of stretching phenomenon [6,7]. Subsequently, the notion of the stretched surface was formulated by several other researchers [8][9][10][11][12][13] in various fluid models.
Initially, the study of magneto-hydrodynamics (MHD) was reported in geophysical and astrophysical problems. During the last several years, this topic has come to the special focus based on their variety of applications in the medical, engineering, and petroleum-refining sectors. The existence of MHD in the nanofluid flow of threedimensional coordinate was planned by Sheikholeslami and Ellahi [14]. It was detected that the presence of MHD raises the resistive (drag) force and minimize the convection current. Additionally, thepratepofpheatptransfer is visible to be developed. The thermo-physical properties of carbon nanotubes in MHD flow across a moving sheet have been addressed by Haq et al. [15]. It seems that the strength of the magnetic field escalates the fluid temperature. The channel flow of the rotating fluid describing the effect of the transversal magnetic field was recently studied by Mehmood et al. [16], which declares that the force of the magnetic field decays the wall flux. Thepobliquepstagnationppointpflow with steady MHD forces was addressed by Borrelli et al. [17]. It was highlighted that if the strength of the electric field disappears, at that time the magnetic field occupies the flat plane of the stream rather than in the parallel directive to the flow. Additionally, the flowpof the obliquepstagnation point occurs solely when the applicable magneticpfield is inpthepdirection of dividing streamlines. Nadeem et al. [18] investigated the two-dimensional viscous flow of a nanofluid relating to the effect of the magneticpfield across a curved surface. Some other studies regarding MHD flows are [19][20][21][22].
The existing research focuses on the two-dimensional steady Williamson nanofluid flow across an exponentially stretched surface depending on both suction as well as injection. The boundary conditions incorporate the impacts of the thermophoresis boundary and Brownian motion. The ongoing flow problem is mathematically modelled under the basic laws of motion and heat transfer. The similarity variables are allowed to transmute the governing equations of the problem into a non-linear ODEs. The solution of this reduced non-linear system of ODEs is supported by HAM. In order to assess the importance of several emergent parameters and variables, the outcomes are presented numerically and are plotted graphically.

Mathematical Model
Assume theptwo-dimensionalpsteady Williamson nanofluid flowpover a stretched exponential surface. It was recognized that the sheet is exponentially stretchedpwithpthe varying velocity U w in the x-direction as well as the fluid which is occupied in y-direction is governed by the velocity U w . Moreover, an exterior magneticpfield of intensity B 0 is orthogonally placed topthe direction ofpthe stretched surface and the suction/injection phenomenon is signified by v w . With these preconditions, the main boundaryplayer MHD equationspforpthepcontinuity, momentum,3energy, as well as concentration are respectively defined as [23][24][25][26][27][28][29].

Similarity solution of the governing equations
The governing equations (1-4) are non-linear PDE's. We use the similarity transformation given below to convert the non-linear PDE's into a non-linear ODE's The boundary conditions for the two cases of PEHF and PEST associated with the above equations (1-4) are given as PEST Case PEHF Case In view of the similarity transformation defined above, equation (1) fulfills in identical manner as well as equations (2)(3)(4) are reduced to the subsequent set of non-linear ODE's Using the similarity transformation into boundary conditions (5), we obtain Boundary conditions for PEST case Boundary conditions for PEHF The similarity parameters appeared in above Eqs. (9)(10)(11)(12)(13)(14)(15)(16) are N t , N b , L e , P r , M , λ andR e which respectively represents the thermophoresis and Brownian motion parameter, Lewispnumber,pPrandtl number,pHartmann number,pWilliamson parameter and the Reynolds number. These parameters are defined as

Solution by Homotopy Analysis Method
We will use HAM to work out the equations (9-13) connected to the boundary conditions (14)(15)(16). This method requires an initial guess which is taken as and the linear operators are These linear operator satisfy, where k 1 to k 7 are constants. Now we define the deformation of order zero as follows.

UsingpTaylorpseriespwe have
PEST Casef The mth-order problem will be like follow.
Employing the mathematica we get the solution  Figure 2: h-curvespforpvelocity, temperature and concentration profilepforpPEHF.
Order of approximation f (0) g (0) Θ (0) where b 0 m,0 , b k m,0 , B k m,n , G k m,n are constants. PHEF case is solved in a similar way.

Convergence of solution
The region for which the solution of momentum, energy and concentration equations converge, is obtained by plotting h curves. The range for auxiliary parameter are.
h-curves described the rang of parameter for which the solution f , g, and Θ are converge.  The behavior of concentration profile forpdifferentpvalues of physical parameters P r is respectively shown in figures 13 & 14. These figures signify the change in nano-particles volume fraction which is caused by making the change in P r number. One can see that, for the joint conditions of PEST and PEHF, there is a negligible change in g for different values of P r as well as the same behavior is detected for the boundary layer. Similarly, the concentration profile for several values of N b is shown in figures 15 & 16. We have noted that the escalation of N b , induces a decrease in concentration profile g for PEST conditions, although g increases for PEHF conditions. The effect of L e number on the nano-particles volume fractions is shown in figures 17 & 18. We see the same behavior for the joint conditions of PEST and PEHF, that is with the increase of L e number the concentration g individually shoots up for PEST as well as PEHF conditions. Figures 19 & 20. represent the effect of N t on g. It appears that with the escalation of N t , there is an increase in g for PEHF case whereas reduction is detected in case of PEST case.

Conclusion
• The velocity profile modifies for the enhancement of suction/injection parameter in both PEST and PEHF case.
• The temperaturepprofilepmodifies for the enhancement of P r only for PEST case, whereas it declines in the case of PEHF.
• The escalation of N b values causes to boost the temperaturepprofilepfor both PEST as well as PEHF case while.
• The escalation of N t values causes to improve both the temperature profile and concentration profile only in case of PEST whereas a reduction is detected in case of PEST for both profiles.
• The escalation of N b causes a decrease in concentration profile g for PEST case, although g increases for PEHF case.
• The escalation of L e number causes to increase the nanoparticlespvolumepfractionpfor both PEST andpPEHF case.