**2. Mathematical Modeling**

We examine the 3D flow of Prandtl nanofluid with a convectively heated surface. Thermal convection and zero nanoparticles mass flux are the two boundary conditions discussed in this model. Thermophoretic and Brownian motion impacts are also examined. The fluid is electro conductive with the magnetic field applied in the direction of *z*-axis, shown by the geometrical interpretation represented through Figure 1. The figure shows that the sheet is extended in *x* and *y*- direction, and the fluid is moving at that extended sheet. The model is discussed in Cartesian coordinates system. The velocity *Uw* is along the *x*-axis and the velocity *Vw* is along the *y*-axis at *z* = 0. Along z-axis, magnetic field *B0* is applied. The current hall and magnetic field impacts are ignored for a small value of Reynold number. Then the resulting ODEs along with the boundary conditions after solving the system of PDEs [49] are:

$$
\beta\_1 f'''' + (f+g)f'' - f'^2 + \beta\_2 f''^2 f''' - (Ha)^2 f' = 0,\tag{1}
$$

$$
\beta\_1 \mathbf{g''} + (f+\mathbf{g})\mathbf{g''} - \mathbf{g'}^2 + \beta\_2 \mathbf{g''}^2 \mathbf{g'''} - (Ha)^2 \mathbf{g'} = 0,\tag{2}
$$

$$\theta'' + \Pr\left( (f+g)\theta' + N\_b \theta' \phi' + N\_l \theta'^2 \right) = 0,\tag{3}$$

$$
\phi'' + \mathcal{Sc}(f+\mathfrak{g})\phi' + \frac{N\_{\mathfrak{k}}}{N\_{\mathfrak{b}}}\theta'' = 0,\tag{4}
$$

And the BCs are;

$$\begin{aligned} f(0) = \mathbf{g}(0) &= 0, \mathbf{f}'(0) = 1, \mathbf{g}'(0) = a, \\ \theta'(0) = -\gamma (1 - \theta(0)), N\_b \theta'(0) + N\_t \theta'(0) &= 0, \end{aligned} \tag{5}$$

$$f'(\infty) \to 0,\\ g'(\infty) \to 0,\\ \theta(\infty) \to 0,\\ \phi(\infty) \to 0,\tag{6}$$

**Figure 1.** Flow Diagram.
