The significant effects of fluid factors such as temperature-dependent viscosity, temperature-dependent thermal conductivity, nonlinear stretching parameter, magnetic field parameter, permeability parameter, Brownian motion parameter, thermophoresis parameter, Prandtl number, Lewis number on velocity, temperature and nano-particle concentration profiles are analysed numerically using the shooting method. Numerical values obtained were plotted into graphs varying the fluid parameters with basic at \(r=1\), \(n=2\), \({f}_{w}=0.2\), \(Re=1\), \(Ha=1\), \(La=1\), \(Le=5\), \(Nb=0.5\), \(Nt=0.5\), \(Pr=2\), \(E=1\), \(F=1\).

The influence of temperature-dependent viscosity on the \(f{\prime }\left(\eta \right)\), \(\theta \left(\eta \right)\), \(\varphi \left(\eta \right)\) are shown in Figs. 1, 2, 3. An increase in temperature-dependent viscosity led to an increase in the \(f{\prime }\left(\eta \right)\) of the fluid. As the temperature of the fluid increased, the viscosity decreased, which in turn led to a decrease in the internal friction of the fluid. As a result, the fluid flew more easily and at a higher velocity.

An increase in the values of temperature-dependent viscosity had significant impact on the temperature profiles. Viscosity of the fluid changes with temperature, which affect the flow behaviour of the fluid. A higher viscosity impedes the flow of heat through the fluid which result in a lower temperature gradient.

An increase in the temperature-dependent viscosity led to an increase in the concentration profiles of nano-particles. The increase in the viscosity resulted in a higher drag force on the Nano-particles, which caused them to accumulate near the stretching surface. This effect was enhanced when the temperature-dependent viscosity was considered since an increase in temperature led to a decrease in viscosity and subsequently lower drag force on the nano-particles and the concentration of the nano-particles in the fluid increased.

Figures 4, 5, 6 shows the effect of temperature-dependent thermal conductivity on the \(f{\prime }\left(\eta \right)\), \(\theta \left(\eta \right)\), \(\varphi \left(\eta \right)\)of the fluid. An increase in temperature-dependent thermal conductivity has an impact on the velocity profiles. As the thermal conductivity of the fluid increases with temperature, the heat transfer rate through the fluid also increases. This leads to a temperature gradient across the fluid, which in turn create thermal buoyancy forces that affect the fluid flow behaviour.

An increase in temperature-dependent thermal conductivity has a significant impact on the temperature profiles. The thermal conductivity of the fluid changes with temperature which affect the flow behaviour of the fluid. A higher thermal conductivity enhances the flow of heat through the fluid which result in a higher temperature gradient.

An increase in temperature-dependent thermal conductivity leads to a decrease in the concentration profiles of nano-particles. The decrease in the concentration of nano-particles is attributed to the phenomenon of Brownian motion. As the temperature of the fluid increases, the fluid molecules become more energetic leading to an increase in the rate of Brownian motion. The nano-particles in the fluid collide with the fluid molecules and become dispersed throughout the fluid leading to a decrease in their concentration near the stretching surface.

Figures 7, 8, 9 depict the effect of the nonlinear stretching parameter on the \(f{\prime }\left(\eta \right)\), \(\theta \left(\eta \right)\), \(\varphi \left(\eta \right)\)of the fluid. An increase in the values of the nonlinear stretching parameter also affect the velocity profiles. The stretching parameter describes the degree of nonlinearity in the stretching of the sheet and a larger stretching parameter implies a more significant stretching effect which in turn cause the fluid to accelerate and flow faster.

An increase in the values of the nonlinear stretching parameter have a significant impact on the temperature profiles. The stretching parameter describes the degree of stretching applied to the fluid which affect the flow behaviour of the fluid and also results in decrease of temperature distribution in the system.

An increase in the nonlinear stretching leads to a decrease in the concentration profiles of nano-particles. The nonlinear stretching parameter describes the rate at which the stretching surface increases in size along a particular direction. An increase in the nonlinear stretching parameter results in a faster increase in the size of the stretching surface, which in turn lead to a decrease in the concentration of nano-particles in the fluid near the stretching surface.

As the stretching surface expands rapidly, the fluid molecules near the surface experience a rapid increase in temperature, leading to a significant temperature gradient. This temperature gradient causes the nano-particles to move away from the stretching surface in response to thermophoresis. However, the random motion of the particles due to Brownian motion causes them to diffuse back towards the stretching surface. The time available for the particles to diffuse back towards the surface decreases, leading to a decrease in the concentration of nano-particles near the surface.

Figures 10 and 11 show how the \(f{\prime }\left(\eta \right)\) and \(\theta \left(\eta \right)\) of the fluid are affected by the magnetic field parameter. An increase in the values of magnetic field parameter affect the velocity profiles. The magnetic field parameter describes the strength of the applied magnetic field, and a larger magnetic field parameter implies a stronger magnetic field affect. A stronger magnetic field generate a magnetic Lorentz force that oppose the fluid motion leading to a reduction in velocity profiles.

An increase in the values of the magnetic field has a significant impact on the temperature profiles. The magnetic field parameter describes the strength of the applied magnetic field which affect the flow behaviour of the fluid and the distribution of temperature in the system. A higher magnetic field enhance the conduction of heat through the fluid which results in a higher temperature gradient.

Figures 12 and 13 illustrate how the permeability parameter affects the \(f{\prime }\left(\eta \right)\) and \(\theta \left(\eta \right)\) of the fluid. An increase in the values of the permeability parameter can also affect the velocity profiles. The permeability parameter describes the degree of permeability of the stretching sheet, and a larger permeability implies a higher degree of permeability which increases the drag force on the fluid and in turn reduce the fluid velocity.

An increase in the values of permeability parameter has a significant impact on the temperature profiles. The permeability parameter describes the degree of permeability of the stretching sheet, which affect the flow behaviour of the fluid and the distribution of temperature in the system. A higher permeability results in a higher rate of heat transfer from the stretching sheet to the fluid which lead to a higher temperature gradient in the system.

Figures 14 and 15 depicts how the \(f{\prime }\left(\eta \right)\) and \(\theta \left(\eta \right)\) is affected by the Brownian motion parameter of the fluid. An increase in the values of the Brownian motion parameter affect the velocity profiles. The Brownian motion parameter describes the degree of Brownian motion of the nano-particles suspended in the fluid and a higher degree of Brownian motion leads to an increased collision frequency between the nano-particles and the fluid, which in turn enhance the fluid velocity.

The Brownian motion parameter represents the effect of random thermal motion on the nano-particles suspended in the base fluid. An increase in the Brownian motion parameters result in an increase in the random motion of the nano-particles which leads to an increase in their collision frequency with the fluid molecules. This increased frequency result in enhanced heat transfer between the nano-particles and the fluid, leading to an increase in the temperature profiles. An increase in the Brownian motion parameters result in an increase in the effective thermal conductivity of the nano-particles fluid mixture, as the enhanced nano-particle motion leads to better mixing of the fluid and the nano-particles. This increased effective thermal conductivity further enhances the heat transfer between the nano-particles and the fluid leading to an increase in the temperature profiles.

Figures 16, 17 and 18 highlight the effect of thermophoresis parameter on the \(f{\prime }\left(\eta \right)\), \(\theta \left(\eta \right)\), \(\varphi \left(\eta \right)\) of the fluid. An increase in the values of the thermophoresis parameter affect the velocity profiles. The thermophoresis parameter describes the degree of thermophoresis of the nano-particles suspended in the fluid and a higher degree of thermophoresis leads to an increased mass flux of the nano-particles towards the stretching sheet which in turn enhance the fluid velocity.

It is observed that the temperature increases with increasing values of thermophoresis parameters. Thermophoresis parameter increases the thermal boundary layer thickness and the temperature on the surface of the sheet increases. This is because the thermophoresis parameter is directly proportional to the heat transfer coefficient associated with the fluid.

The concentration profiles decrease with increasing values of thermophoresis parameters. Concentration is a decreasing function of thermophoresis parameter. For hot surfaces, thermophoresis tends to blow the nano-particle volume fraction boundary layer away from the surface since hot surface repels the submicron-sized particles from it, thereby forming a relatively particle-free layer near the surface.

Figures 19 and 20 represents the \(f{\prime }\left(\eta \right)\) and \(\theta \left(\eta \right)\), for various values of Prandtl numbers of the fluid. An increase in the Prandtl numbers leads to a decrease in velocity profiles. A higher Prandtl number implies that the fluid has a higher viscosity relative to its thermal conductivity which dampen the fluid velocity. This is because a higher viscosity can resist the flow of the fluid whereas a higher thermal conductivity enhances the transfer of heat through the fluid.

An increase in the Prandtl numbers corresponds to an increase in the ratio of thermal diffusivity to momentum diffusivity, indicating that the fluid is less efficient at transferring heat compared to momentum. A higher Prandtl number signifies that the fluid is less efficient at transferring heat, which result in a lower temperature gradient in the system.

Figure 21 manifest the Lewis number on the \(\varphi \left(\eta \right)\) of the fluid. A higher Lewis number indicates that thermal diffusivity is greater than mass diffusivity which implies that the rate of heat transfer in the fluid is higher compared to the rate of mass transfer. As a result, the temperature of the fluid increases faster than the concentration of the nano-particles leading to a decrease in the concentration profiles of the nano-particles. Thermal energy in the fluid is transferred more easily than the nano-particles due to the difference in their diffusivities.

Table 1

The numerical values of \({f}^{{\prime }{\prime }}\left(0\right)\), \(\theta {\prime }\left(0\right)\), \(\varphi {\prime }\left(0\right)\) for different values of temperature-dependent viscosity, E.

E | \(f{\prime }{\prime }\left(0\right)\) | \(\theta {\prime }\left(0\right)\) | \(\varphi {\prime }\left(0\right)\) |

1.0 | 1.51940 | -0.77990 | -2.35010 |

2.0 | 1.92866 | -0.80970 | -2.33274 |

3.0 | 2.18075 | -0.82658 | -2.30530 |

5.0 | 2.46402 | -0.84498 | -2.24707 |

7.0 | 2.58328 | -0.85274 | -2.21220 |

Table 1 represents values of skin-friction coefficient, Nusselt number and Sherwood number for various values of temperature-dependent viscosity. It shows that an increase in the values of temperature-dependent viscosity increases skin-friction coefficient and Nusselt number while the Sherwood number decreases.

Table 2

The numerical values of \({f}^{{\prime }{\prime }}\left(0\right)\), \(\theta {\prime }\left(0\right)\), \(\varphi {\prime }\left(0\right)\) for different values of temperature-dependent thermal conductivity, F.

F | \(f{\prime }{\prime }\left(0\right)\) | \(\theta {\prime }\left(0\right)\) | \(\varphi {\prime }\left(0\right)\) |

1.0 | 1.51940 | -0.77990 | -2.35010 |

2.0 | 1.64856 | -0.66447 | -2.36230 |

3.0 | 1.75190 | -0.58200 | -2.36990 |

5.0 | 1.90390 | -0.46961 | -2.38360 |

7.0 | 2.00550 | -0.39810 | -2.39440 |

Table 2 represents values of skin-friction coefficient, Nusselt number and Sherwood number for various values of temperature-dependent thermal conductivity. It shows that an increase in the values of temperature-dependent thermal conductivity increases skin-friction coefficient and Sherwood number while the Nusselt number decreases.