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 x as depicted in Figure 1. The interface condition and velocity components can be visualized in Figure 1.

Governing equations are therefore, as follows:

……. (2)

……. (3)

……. (4)

with following boundary conditions,

….. (5)

The surface tension σ, being a function of T and C can be defined as follows:

……… (6)

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 B₀ 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 and heat capacity of base fluid (ρc) _fl. D_Br is Brownian diffusion, D_Th is Thermophoresis.

Define,

……… (8)

Using (8) in equations 1234, we get

…………. (9)

……… (10)

………….. (11)

subject to the following boundary conditions:

……… (12)

where r=(C₀ 𝜸_C)/(T₀ γ_T ), M=(L² σ_l B₀)/μ, Pr=ν_fl /α_fl is the Prandtl number N_b=(ρc)_np D_B𝜸 C₀ x²) / (ρc)_lf L² a), N_t=(ρc)_np D_Th x²) / (ρc)_lf L² a), L= μν fl)/(σ₀ T₀ 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 α₁/(ρ_fl L²).

Homotopy analysis method 1819202122 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,

  …. (13)

The auxiliary parameters can be defined as follows:

……. (14)

such that,

……. (15)

where M_i are constants for i=1-7. Subsequently, the zeroth order problems of deformation are written as follows:

with following boundary conditions:

Therefore,

where p∈ (0,1) is a typical embedding parameter and h ̂_f , h ̂_θ, h ̂_ϕ are so-called auxiliary parameters with N_f, N_θ, N_ϕ are the non-linear operators. For p =0,1 , we have:

……. (19)

The m^th problems of deformation are

where g_m=1 for m>1 otherwise 0.

Finally,

Thus,

…… (22)

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 (f), temperature distribution (θ) and concentration distribution (ϕ) are termed as convergence control parameters. These parameters are significant to speed-up the convergence. Convergence intervals are sketched in Figure 2. The interval of interest for convergence of the aforementioned profiles is (-0.60, 0.30).

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.