Influence of Magnetic Field on Thermal Radiation and Particle Shapes of Copper-Water Nanofluid Considering Marangoni Boundary Layer

This problem aims to address hydrodynamic Marangoni boundary layer flow of incompressible nanofluid along different shapes of particle like sphere, tetrahedron, column and lamina with exponential temperature. Choosing appropriate transformations, the governing equations are reduced to non-linear ordinary differential equations and then solved by using a perturbation technique. Impacts in velocity and temperature profiles for the relevant considering parameters namely nanoparticle volume fraction, magnetic parameter, empirical shape factor and radiation parameter are evaluated and shown through graphs. Moreover, computational values for influences of physical parameters on local surface heat flux are presented in table. KeywordsMagnetic field, Thermal radiation, Nanofluid, Marangoni boundary layer.


Introduction
An analysis of electrically conducting fluids with magnetic field sources is named as Magnetohydrodynamic (MHD). Magnetic field can be applied to earn the liked nature of the industrial product via control the cooling rate. Therefore, the phenomenon of MHD flow has received great appreciation because of its applications in the branches of science and engineering. These applications involve bearings, plasma, crude oil purification, paper production, copper wire thinning, drawing and continuous strips cooling. The impact of magnetic field on fluid flow past a semi-infinite wall is developed initially by Rossow (1957). Although, Chaudhary and Kumar (2014) introduced the MHD boundary layer flow towards a flat plate and permeable surface. Latterly, some problems of fluid in the presence of magnetic field discussed by Chaudhary and Choudhary (2017), and Sharma et al. (2019).
Thermal radiation is a way of heat transmission from a system of higher temperature into a system of lower temperature with the condition that both systems are separated in space. The radiative heat transfer can be quite significant at high temperature and hence it can not be neglected. It has so many applications in the area of thermal engineering, chemical engineering, , copper oxide   CuO , copper   Cu and silver   Ag have higher thermal conductivity. So to enhance the heat transfer rate of liquids with a significant augmentation of thermal conductivity, suspended the nano solid particles into the base fluid, this type of mixture is known as nanofluid. Consequently, nano-solid particles are taken typically with the size of diameter as 1 to 100 nm. Examples of practical applications include heat exchangers, cancer therapy, thermal engineering and electronic chemical process. Notably, that applications are involved in heat sinks cooling, medical suspensions sterilization and powered engines. Choi and Eastman (1995) is pioneered the concept of nanofluid, in which he introduced an innovative way to increase advanced heat transfer by mixing nanoscaled particles into the ordinary fluid. However, some earlier published work on nanofluid illustrations are given by Bayat and Nikseresht (2012), and Sarafraz and Hormozi (2014). Further, Hajabdollahi and Hajabdollahi (2017), and Ahuja and Gupta (2019) are created the study of a few aspects like nanoparticle's size, shape and type in experimental or theoretical way.
Marangoni boundary layer can be created through liquid-gas or liquid-liquid interfaces. Marangoni convection has the appearance over to surface tension gradients, which may be both temperature and concentration gradient. Importance of Marangoni convection is carrying out by the feasibility of material processing in space craft, in this processing gravity force is taken as so small rather than the thermo-capillary one. Thus Marangoni effect have great attention in the applications of space processing and micro-gravity science like thin films spreading, processing of semi conductor, crystal growth melts, and nucleation vapor bubbles. Probably, Shivakumara et al. (1999) was the first one who pioneered the basic research work in the area of Marangoni boundary layer effect. Besides, some other studies have been carried out by Zhang and Zheng (2012), Gambaryan-Roisman (2015) and Aly and Ebaid (2016) with a Marangoni boundary layer under the different situations.
Keeping the discussed literature in view, the main goal of this article is to modulate the research problem of Lin et al. (2016) with the impact of a uniform transverse magnetic field on electrically conducting nanofluid. There are used solid nanoparticles as copper and base fluid as water containing different shapes of nanoparticles.

Problem Description
Let assume the Marangoni boundary layer flow of electrically conducting copper-water nanofluid along with the impact of radiation heat transfer and various particle shapes towards the temperature on surface , where  T is the ambient fluid temperature, 0 T is the reference temperature and 0 L is the reference length. The flow is assumed steady, twodimensional and viscous. Although, X and Y are the Cartesian coordinates with the origin along and perpendicular to the surface, respectively and flow is bound at half plane 0  Y . The solids and the liquid are taken in thermal equilibrium, and there is no slip condition between them. As presented in Figure 1, a variable magnetic field is utilized in perpendicular direction of the flow, where 0 B is the magnitude of magnetic field strength. If the magnetic Reynolds number is surmised sufficiently small then caused magnetic field appears to be negligible. Until, the effect of Marangoni surface imitates as a boundary condition on the flow field governing equations. Further, thermophysical resources of water and copper are presented in Table 1, which is followed by Hayat et al. (2016).
the boundary conditions are expressed as , 0, at 0 where subscript nf denotes the nanofluid's thermophysical properties, U and V are the velocity is the radiative heat flux, *  is the Stefan- is the tension at the surface, 0  is the positive constant and T  is the temperature coefficient of the surface tension. It is considered that temperature difference inside the flow is very small like that 4 T can be signified as a linear function of  T . Thus, expanding 4 T into the Taylor series about ,  T which after leaving higher order terms given the form . Ali et al. (2016), viscosity, density, electrical conductivity, thermal conductivity and heat capacity of the copper-water nanofluid are simplified as where, subscript s and f indicate for copper particles and base fluid water, respectively,  is the is the empirical shape factor and  is the particles' sphericity. The proportion of the surface area of sphere to the surface area of particle along equal volumes is known as sphericity of the particle. Different shapes of the copper particles like sphere, tetrahedron, column and lamina are presented in Figure 2. Consequently, values of  and m for surmised shapes are shown in Table 2 given by Lin et al. (2016).  Introduce the following non-dimensional variables pursued by Lin et al. (2016) to convert the dimensional governing equations (1) to (3) into dimensionless form where x and y are dimensionless coordinates along the plate and perpendicular to it respectively, u and v are non-dimensional velocity along the  x and  y axes respectively,  (1) to (4) . where,

Transformation of Model
Substituting the following dimensionless similarity variables (Lin et al. (2016)). is the non-dimensional temperature. Therefore, by using equation (15) in equations (12) to (14), obtain the following set of equations.
with the corresponding boundary conditions where prime denotes differentiation with respect to  and is the magnetic parameter.
The physical quantity is the local Nusselt number Nu , which is described in dimensional form as Using the above non-dimensional variables equation (15), the equation (19)

Solution by Perturbation Technique
The solution set of equations (16) and (17) with the relevant boundary conditions equation (18) is found by using the perturbation technique. Utilize the power series in the term of the small magnetic parameter M is considered as follows.
Substituting equation (21) and its derivative in the equations (16) to (18) and then equating the coefficients of like power of M  Lin et al. (2016) were found out the equation (22) for the non-magnetic case. Applied the shooting technique with fourth order Runge-Kutta method for the computational results of the remaining ordinary differential equation, which is employed with step size 001 . 0 to attain the solution. The above procedure is replicated until the results correct up to the admissible accuracy of

Discussion of the Results
The purpose of this section is to illustrate the velocity     International Journal of Mathematical, Engineering andManagement Sciences Vol. 5, No. 5, 957-970, 2020 https://doi.org/10.33889/IJMEMS.2020.5.5.073 966 Influence of  , M , m and Nr on     are sketched in Figures 5 to 8 respectively. These figures display that enhancement in  , M , Nr implies to accretion in the thickness of thermal boundary layer. Apparently, It is also notable that for different shapes of nanoparticles particularly sphere, tetrahedron, column and lamina, fluid temperature step-up in the increasing order respectively.

Conclusions
In the present study, a numerical model for the effect of radiative heat transfer and exponential temperature on the MHD Marangoni convection flow of incompressible copper-water nanofluid along the different four types shape of nanoparticles is explored. By using the non-dimensional transformation, the governing partial differential equations for continuity, momentum and energy with the appropriate boundary conditions are remodeled into the ordinary differential equations. Numerical results of the appeared equations are established through the perturbation technique.
According to the observation of present study, the fluid velocity decreases with the enhancement in the value of the nanoparticle volume fraction and the magnetic parameter, while opposite circumstance exists in temperature of the fluid and heat transfer rate for the enhancing value of the nanoparticle volume fraction, the magnetic parameter and the radiation parameter. Consequently, lamina shaped nanoparticles have high temperature and surface heat flux as compared to the other nanoparticle shapes.