**6. Closing remarks**

In this paper, the impacts of dependent viscosity parameter, magnetic field, and solid nanoparticle flow and the heat transfer of modified nanofluid flow at the exponential stretching surface have been analyzed numerically. The governing coupled partial differential equations are converted into ordinary coupled differential equations which are solved numerically by bvp4c method. The parametric analysis is executed to investigate the impacts of the governing physical parameters (magnetic field, variable viscosity (for both cases *θ<sup>e</sup>* <0 and *θe*>0), Biot number, and solid nanoparticle) on the flow and heat transfer properties. In particular, we focus on the effect of dependent viscosity when *θ<sup>e</sup>* < 0 and *θe*>0 of the *Al*2*O*<sup>3</sup> *Cu Ni=water* and *Al*2*O*<sup>3</sup> *Ni=water*. It is noted that the fluid viscosity and temperature are inverse function. The computational results are presented through graph and tables. The results of modified nanofluid flow and heat transfer properties show many exciting behaviors which deserve further study of modified nanofluid.
