In order to get the physical insight into the flow problem, comprehensive numerical computations are conducted for various values of the parameters that describe the flow characteristics, and the results are illustrated graphically. The system of nonlinear ordinary differential Equations 10 and 12 with the boundary conditions (Equation 13) are integrated numerically by means of the efficient numerical shooting technique with a fourth-order Runge-Kutta scheme (MATLAB package). The step size *η *= 0.001 was used while obtaining the numerical solution with *η*_{max }= 6. The physical quantities of interest here are the skin friction coefficient *C*_{f }and the Nusselt number *Nu*_{x}, which are obtained and given in Equations 15 and 16. The distributions of the velocity *f'(η)*, the temperature *θ*(*η*) from Equations 10 and 12, the skin friction at the surface, and the Nusselt number for different types of nanofluids are shown in Figures ,,,,,,,,,,,,.

We consider three different types of nanoparticles, namely, copper (Cu), alumina (Al

_{2}O

_{3}), and titanium oxide (TiO

_{2}), with water as the base fluid. Table shows the thermo-physical properties of water and the elements Cu, Al

_{2}O

_{3}, and TiO

_{2}. The Prandtl number of the base fluid (water) is kept constant at 6.2. It is worth mentioning that this study reduces the governing Equations 10-12 to those of a viscous or regular fluid when

*ϕ *= 0. In order to verify the accuracy of the present method, we have compared our results with those of Cortell [

12,

13] for the rate of heat transfer -

*θ'*(0) in the absence of the nanoparticles (

*ϕ *= 0), without (

*N*_{R }→ ∞ (i.e., k

_{0 }= 1)) and with thermal radiation parameter. The comparisons in all the above cases are found to be in excellent agreement, as shown in Tables and . Table depicts the skin friction at the surface -

*f*"(0) for various values of nonlinear stretching sheet

*n*, with

*ϕ *= 0.1, Pr = 6.2,

*Ec *= 0.5, and

*N*_{R }= 5 for different types of nanoparticles when the base fluid is water. It can be seen from Table that |

*f*"(0)| increases with an increase in the nonlinear stretching parameter

*n*, and the Cu nanoparticles are the highest skin friction, followed by TiO

_{2 }and Al

_{2}O

_{3}.

| **Table 2**Comparison of - *θ*' (0) with *ϕ *= 0 and *N*_{R }→ ∞ (i.e., *k*_{0 }= 1). |

| **Table 3**Comparison of - *θ'*(0) for various values of thermal radiation parameter *N*_{R }with *ϕ *= 0 (regular fluid). |

| **Table 4**Values related to the skin friction for different values of n. |

Figures and illustrate the effect of nanoparticle volume fraction *ϕ *on the nanofluid velocity and temperature profile, respectively, in the case of Cu nanoparticles and water base fluid (Pr = 6.2) when *ϕ *= 0, 0.05, 0.1, and 0.2, with *Ec *= 0.1, *n *= 10, and *NR *= 1. It is clear that, as the nanoparticles volume fraction increases, the nanofluid velocity decreases, and the temperature increases. These figures illustrate this agreement with the physical behavior. When the volume of nanoparticles increases, the thermal conductivity increases, and then the thermal boundary layer thickness increases. Figures and depict the effect of nonlinearly stretching sheet parameter *n *on velocity distribution *f*'(*η*) and temperature profile *θ*(*η*), respectively. Figure illustrates that an increase of nonlinear stretching sheet parameter *n *tends to decrease the nanofluid velocity in the case of Cu-water when *n *= 0.75, 1.5, 3, 7, and 10, with *Ec *= 0.1, *N*_{R }= 1, and *ϕ *= 0.1. Furthermore, Figure shows that increasing the nonlinear stretching sheet parameter *n *tends to decrease the temperature distribution the same values, thus leading to higher heat transfer rate between the nanofluid and the surface. The effect of the viscous dissipation parameter *Ec *on the temperature profile in the case of Cu-water when the Eckert number *Ec *= 0, 0.5, 1, 1.5, 2, and 2.5 with *n *= 10, *N*_{R }= 1, and *ϕ *= 0.1 is shown in Figure . It is clear that the temperature distribution increases with an increase in the viscous dissipation parameter *Ec*. Figure shows the influence of thermal radiation parameter *N*_{R }on the temperature profile in the case of Cu-water. It is clear that the temperature decreases with an increase in the thermal radiation parameter *N*_{R}; this leads to an increase in the heat transfer rate. Moreover, Figure shows this effect of the thermal radiation parameter on the temperature distribution but for the different types of nanoparticles with water as the base fluid. It can be seen from Figure that *θ*(*η*) decreases with an increase in the thermal radiation parameter as shown in Figure , and the Cu nanoparticles have the highest value of temperature distribution than the nanoparticles Al_{2}O_{3 }and TiO_{2}. The influence of *Ec *and *n *on the temperature profiles for all types of nanoparticles is shown in Figures and , respectively. It is found that the temperature decreases with *n *and increases with *Ec *as shown in Figures and , respectively, and the TiO_{2 }nanoparticles proved to have the highest cooling performance for this problem.

The influence of nonlinear stretching sheet *n *on the skin friction at the surface -*f'' *(0) with *N*_{R }= 5, Pr = 6.2, *ϕ *= 0.1, and *Ec *= 0.5 is shown in Figure . It can be noticed that, from Table and Figure , the numerical values of |*f'' *(0)| for different kinds of nanofluids increase with an increase in the nonlinear stretching parameter *n*. This implies an increment of the skin friction at the surface where Cu nanoparticles have the highest skin friction than the other nanoparticles. Figures ,, display the behavior of the heat transfer rates under the effects of *N*_{R}, Ec, and *n*, respectively, using different nanofluids for Pr = 6.2 and *ϕ *= 0.1. These figures show that, when using different kinds of nanofluids, the heat transfer rates change, which means that the nanofluids will be important in the cooling and heating processes. It can be noticed from the results above that, as expected, the heat transfer rate increases with an increase in the thermal radiation parameter *N*_{R }and nonlinear stretching sheet parameter *n*, and decreases rapidly with an increase in the viscous dissipation parameter *Ec*.