Analytical solution of Velocities and Temperature fields using Homotopy Analysis Method

1 Department of Mathematics,Kongu Engineering College, Perundurai, Erode, Tamil Nadu, India. e-mail : kvtamilselvi@gmail.com Ramanujan Research Centre in Mathematics, Saraswathi Narayanan College (Autonomous), Perungudi, Madurai, Tamil Nadu, India. e-mail : rajaram23292@gmail.com 3 Ramanujan Research Centre in Mathematics, Saraswathi Narayanan College (Autonomous), Madurai, Tamil Nadu, India. e-mail : meensphd@gmail.com Article History:Received:11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: In this paper, analysis of nonlinear partial differential equations on velocities and temperature with convective boundary conditions are investigated. The governing partial differential equations are transformed into ordinary differential equations by applying similarity transformations. The system of nonlinear differential equations are solved using Homotopy Analysis Method (HAM). An analytical solution is obtained for the values of Magnetic parameter M, Prandtl number Pr, Porosity parameter λ, Radiation parameter R, Biot number Bi, Stretching parameter c. The present results are compared with the numerical result [1], showing good agreement with each other and with different parameters. The influence of the skin friction and local Nusselt number for different parameters are discussed. The effects of various parameters on velocity and temperature profiles are presented graphically.


INTRODUCTION
S. Shaw et.al [1] explained the study of Magnetohydrodynamic (MHD) flow 3D casson fluid. It has many industrial applications, such as purification of crude oil, Magnetohydrodynamic electrical power generation [1]. Many authors [2][3][4][5] have been investigated experimentally and numerically for the MHD flow of Casson fluid with convective boundary conditions..
The main objective of this paper is to investigate the analytical solution of boundary layer flow of electrically conducting Casson fluid past. The system of nonlinear partial differential equations has been diminished into the system of nonlinear ordinary differential equation [6][7][8]. S. Shaw et.al [1] presented the numerical solution of 3D Casson fluid flow using Spectral relaxation method. Still now there is no analytical results for 3D Casson fluid flow. In this manuscript, the coupled nonlinear equations are solved using Homotopy analysis method [9][10][11]. Analytical results are compared with Previous result [1]. Moreover analytical solution of Skin friction Co-efficient and local Nusselt number are given. In the HAM, the auxiliary parameter h converge the obtained analytical solution. Differential sensitivity analysis are graphically represented for different parameters. It shows that the behaviors of each of the parameter. Fig.1: Physical model and coordinate [1] The conservation equation of Continuity, momentum and energy equation is as follows [1]:

USING HOMOTOPY ANALYSIS METHOD (HAM) ANALYTICAL EXPRESSION OF VELOCITIES AND TEMPERATURE
The homotopy analysis method is a semi-analytical technique to solve nonlinear problems. This method was introduced by Liao [12]. The basic concept of this method [13,14]. Using the HAM, the approximate analytical expressions are given as follows:

ANALYTICAL EXPRESSION FOR SKIN FRICTION CO-EFFICIENT AND LOCAL NUSSELT NUMBER
The expressions for skin friction coefficient is given by [1]: is the skin friction, and are skin friction along the -and -directions and are defined in [1] The analytical expressions for skin friction co-efficient along the and direction are represented as follows: 3) The dimensionless local Nusselt number is [1]: . The analytical expression for local Nusselt number is

RESULT AND DISCUSSION
Eqs. The velocity of the fluid for different parameters are presented in the graphs Figs.2 to 9. Fig.2 shows the influence of the velocity ′ ( ) decreases with increases of casson parameter . From Fig.3, it is inferred that the velocity decreases with increases of parameter. Fig.4 and 5 have been plotted to demonstrate the effects of parameters M and C on the velocity profiles. It is depicted that as M and C increases the velocity ′ ( ) of the fluid decreases. In Fig.6 and 7, depict to analyze the velocity profile for various values and . It is noticed that the ′ ( ) increases with decreases of and parameter. Fig.8 and 9 are displayed to show the influence of M and C on the velocity profiles. It is observed that as M increases the velocity ′ ( ) of the fluid decreases. It is clear that the velocity ′ ( ) increases with increases of the parameter C. The temperature of the different parameters are presented through Figs.10 to 12. From Fig.10, It is depicted to examine the effects of with temperature. It is clear that the temperature ( ) decreases with increases of the parameter . In Fig.11, It is revealed that the influence of temperature profile for various values . This is because the temperature ( ) increases with also increases of the parameter . From Fig.12 It is clear that the parameter R increases with temperature ( ) also increases.
The skin friction co-efficient and local Nusselt number are presented in Figs.13 to 15. From Fig.13, There is prominent that the skin friction co-efficient in direction increases when M also increases. It is noticed that when C increases the skin friction co-efficient direction decreases. In Fig.14, It illustrate that the skin friction co-efficient direction increases when the parameters M and C also increases. From Fig.15, Indicated the local Nusselt number decreases when M increases. It is evident that the parameter C increases with local Nusselt number also increases.

Determining the validity region of h
Eqs.(3.1) and (3.2) contains so called convergence control auxiliary parameter h. It plays a vital role for converging the solution series. In order to obtain the h region, the analytical solution is independent of h. To study the influence of h on the convergence of solution, the h curves of f and g are plotted in Fig.16. This figure clearly indicates that the valid region of h is about ( -0.1 ).

Differential Sensitivity Analysis
Eq.(3.1) represent the new approximate analytical expression for the fluid in terms of the parameters , , , differentiating the fluid partially with respect to determined. The percentage of change in fluid with respect to , , , are 5.28%, 0%, 87.69%, 8.83% respectively. From this, it is evident that parameter have more impact on fluid. These parameter are highly sensitive parameter. The parameter is called as moderately sensitive parameter as it has 8.83% of influence over fluid . The remaining two parameters M and C are less sensitive and zero sensitives. The spread sheet analysis of these result is described in Fig.17.
Eq.(3.2) represent the approximate analytical expression for the fluid in terms of the parameters , , , differentiating the fluid partially with respect to determined. The percentage of change in fluid with respect to , , , are 5.06%, 18.18%, 69.99%, 7.54% respectively. From this, it is inferred that parameter have high impact on fluid. These parameter are greatest sensitive parameter. The parameter C is called as moderately sensitive parameter as it has 18.18% of influence over fluid . The remaining two parameters M and are lowest sensitive. The fluid analysis of these result is described in Fig.18.
Eq.(3.3) represent the approximate analytical expression for the temperature in terms of the parameters , , , , , , differentiating the temperature partially with respect to determined. The percentage of change in temperature with respect to , , , , , , are 82.92%, 0%, 124.37%, 20.73%, 0%, 1.95%, 141.82% respectively. From this, it is revealed that parameter R have sorely impact on temperature. These parameter are biggest sensitive parameter. The parameter is called as nearest sensitive parameter as it has 124.37% of influence over temperature. The parameter M is said to center sensitive parameter as it has 82.92% of examine over temperature. The remaining four parameters , and , are smallest and zero sensitives. The temperature analysis of these result is described in Fig.18.