**Author details**

**Figure 6** reveals the variation of dimensionless quantity of Biot number on the temperature profile. The relative transport of internal and external resistances is called the Biot number. The thermal boundary layer increases with increasing in the

**Figure 7** shows the impact of the porosity parameter on the velocity profile. It is noted that velocity profiles decreases for the higher values of the porosity parameter. The boundary layer thickness decreases for large values of porousity

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

*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

focus on the effect of dependent viscosity when *θ<sup>e</sup>* < 0 and *θe*>0 of the

biot number.

parameter.

nanofluid.

**Nomenclature**

Pr Prandtl number

*Bi* Biot number *θ* temperature profile *R* permeability

*Tw* wall temperature *T*<sup>∞</sup> ambient temperature *ν<sup>f</sup>* fluid kinematic viscosity *νnf* nanofluid kinematic viscosity

*ρ* density *f* fluid

*ρCp* 

*ρCp* 

**164**

*Φ*<sup>1</sup> nanoparticle volume fraction of Al2O3 *Φ*<sup>2</sup> nanoparticle volume fraction of Cu *Φ*<sup>3</sup> nanoparticle volume fraction of Ni

*f* velocity profile along x-direction

*νhnf* hybrid nanofluid kinematic viscosity *νmnf* modified nanofluid kinematic viscosity

*κ<sup>f</sup>* thermal conductivity of fluid *κnf* thermal conductivity of nanofluid

*hnf* heat capacity of hybrid nanofluid

*mnf* heat capacity of modified nanofluid

*κhnf* thermal conductivity of hybrid nanofluid

**6. Closing remarks**

*Nanofluid Flow in Porous Media*

Sohail Nadeem and Nadeem Abbas\* Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan

\*Address all correspondence to: nabbas@math.qau.edu.pk

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
