Marangoni Effect in Second Grade Forced Convective Flow of Water Based Nanofluid

Advances in nanotechnology especially in nanofluids comprising of a typical base fluid saturated with nano-size metallic particles with enhanced thermophysical properties is one of the hot topics in industrial as well as engineering applications of applied mathematics. This article explores the impact of Lorentz forces and Maragoni effect on second grade nanofluid forced convective flow. A two phase model is chosen to validate the nanofluid. Series solutions are achieved through HAM after transformation of PDEs into ODEs. The Brownian motion effect, Thermophoresis and the Marangoni effect are the main influencing factors for present flow model. In addition, the influence of pertinent fluid parameters such as Schmidt number, Magnetic number and Prandtl number on the velocity, temperature and concentration profiles is discussed with the help of graphs. With an enhanced Marangoni factor the hydraulic boundary layer thickness shows enhancement. DOI : Coming soon Corresponding author: Ghulam Rasool, School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR-China, Email: grasool@zju.edu.cn


Introduction
The concept of nanofluid is very simple but it has broadly improved the efficiency of fluids especially from industrial point of view. The suspension of nanoparticles in typical base fluid results in a very highly conductive fluid that certainly reduces the human effort as well as the machine heating. These suspensions can meet the cooling requirement for any type of thermal system. The foundation of nanofluid was laid by Choi [1] and later on, the concept became famous in all the related fields of fluid dynamics due to its industrial applications. Ibanez et al. [2] discussed an analytic investigation for MHD nanoliquid flow with suction/injection and radiation effects.
Surface tension as well as the gradients of temperature and concentration results in Marangoni convection appearing in fluid flow analysis. The topic is interesting due to its applications in crystal growth mechanism, beam melting and welding etc. Lin et al. [12] reported the Marangoni convection in nanoliquid flow via thermal gradient varying the magnetic effect. Surface tension is taken as a nonlinear temperature and concentration function. Aly and Ebaid [13] found exact solutions considering Marangoni effect in nanofluid flow using Laplace transformation method. Most recent articles can be seen in [14][15][16][17] and reference cited therein. Present inspiration is covered by the following novel aspects. Firstly, to model a second grade nanofluid forced convective flow due to the Lorentz forces instigated into the fluid by induction of a variable magnetic field.
Secondly, to achieve series solutions through HAM. The PDEs are transformed into ODEs using usual similarity transformations. Finally, to discuss the results through graphs with sufficient concluding remarks.

Problem Statement
We consider a second grade nanofluid forced convection due to the Lorentz forces instigated into the model The surface tension, σ being a function of T and C can be defined as follows: where …….. (7) Here u , ν are the horizontal and vertical velocity components, respectively v fl . is the kinematic viscosity, ρ fl is the density of fluid B 0 is the magnetic effect involved in the model for MHD, σ is the surface tension, α fl is is thermal diffusivity of the fluid, the ratio between heat capacity of the nanoparticles (ρc) np

Series Solutions
Homotopy analysis method [18][19][20][21][22] for convergent series solutions is very convenient to obtain the approximated solutions for a given nonlinear system. The method is independent of small/large physical parameters.
Thus it is an efficient method as compared to other conventional methods for solving nonlinear systems. Assuming the following initial guesses,

Results and Discussion
This section is concerned with the discussion on graphical results that are obtained through Mathematica  4. Similar is the case noticed for Magnetic number as displayed in Fig. 5. A significant drop in velocity profile appears at first sight however, the variation slows down with the stronger effect of magnetic number. The temperature profile, as displayed in Fig. 6, shows augmented behavior with augmented values of Marangoni ratio. However, after certain limitation, the effect can be seen in opposite nature. The Prandtl number shows enhancement in temperature profile due to an enhanced thermal diffiusivity as one can see in Fig.7. Fig. 8 and Fig. 9 are the display of variation in Temperature profile varying the values of Brownian motion parameter and Thermophoretic parameter, respectively.
The rapid and in-predictive movement of nanoparticles as well as the more stronger Thermophoretic force results in incremental behavior of temperature profile. The effect of Marangoni ratio on concentration profile is displayed in Fig. 10. The results persist with those appearing in Temperature profile for the same parameter. An opposite behavior is seen upon variation of Thermophoretic parameter and Schmidt number for concentration profile as displayed in Fig. 11 and Fig. 12, respectively.