The authors have declared that no competing interests exist.

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.

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

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.

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.

We consider a second grade nanofluid forced convection due to the Lorentz forces instigated into the model by induction of an applied Magnetic field. The effect of magnetic field develops normal to the surface upon which the fluid is flowing. The Marangoni effect is utilized to apprehend the fluid flow in forward direction. The problem is considered in two dimensions such that the fluid flows along x-axis and the y-axis extends normal to the surface. There is no fluid motion along y-axis therefore, v=0 is taken. Along x-axis the fluid undergoes the Lorentz forces that are induced into the model due to the applied magnetic field neglecting the Hall effects. The interface temperature is taken as function of

Governing equations are therefore, as follows:

with following boundary conditions,

The surface tension σ, being a function of

where

Here u , ν are the horizontal and vertical velocity components, respectively _{fl}. is the kinematic viscosity, _{fl } is the density of fluid B_{0} is the magnetic effect involved in the model for MHD, _{fl} is is thermal diffusivity of the fluid, the ratio between heat capacity of the nanoparticles (_{np} and heat capacity of base fluid (_{fl}. _{Br }is Brownian diffusion, _{Th} is Thermophoresis.

Define,

……… (8)

Using (8) in equations

…………. (9)

subject to the following boundary conditions:

where r=(C_{0} 𝜸_{C})/(T_{0} γ_{T} ), M=(L^{2} σ_{l} B_{0})/μ, Pr=ν_{fl} /α_{fl} is the Prandtl number N_{b}=(ρc)_{np} D_{B}_{𝜸} C_{0} x^{2}) / (ρc)_{lf} L^{2} a), N_{t}=(ρc)_{np} D_{Th} x^{2}) / (ρc)_{lf} L^{2} a), L= μν fl)/(σ_{0} T_{0} γ_{T}) are the Marangoni factor (r), modified Hartman number (M), Prandtl factor (Pr), Brownian motion parameter (Nb), Schmidt number (Sc), Thermophoresis parameter (Nt) and reference length (L), respectively. The second grade fluid parameter (α) is defined as α_{1}/(ρ_{fl} L^{2}).

Homotopy analysis method

The auxiliary parameters can be defined as follows:

such that,

where _{i} are constants for

with following boundary conditions:

Therefore,

where p_{f }, h ̂_{θ}, h ̂_{ϕ} are so-called auxiliary parameters with N_{f}, N_{θ}, N_{ϕ} are the non-linear operators. For

The m^{th} problems of deformation are

where g_{m}=1 for m>1 otherwise 0.

Finally,

Thus,

are the general solutions where Mi are the arbitrary constants for i=1-7 and f_{m}^{*} (η) ,θ_{m}^{*} (η), ϕ_{m}^{*} (η) are special solutions.

The auxiliary parameters introduced in (series solution) for the velocity profile (

This section is concerned with the discussion on graphical results that are obtained through Mathematica based HAM code for specific values of pertinent fluid parameters including the second grade fluid parameter, the Magnetic number, the Marangoni ratio, the Prandtl and Schmidt numbers as well as the Brownian and Thermophoretic parameters.

This study concludes with the impact of Marangoni effect and Lorentz force generated by MHD on second grade nanofluid flow. Flow model is formulated mathematically in PDEs which are transformed into ODEs using transformation and HAM is applied to get the convergent series solutions. We conclude that velocity profile reduces for stronger Marangoni effect in second grade nanofluid. The Brownian motion and Thermophoresis have significant impact on the flow profiles. Furthermore, the temperature and concentration profile shows augmented behavior with augmented Marangoni ratio. One can see that due to strong impact of Lorentz forces, a reduction in hydraulic boundary layer is noticed however, an opposite behavior is shown the other two profiles.