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Nanofluids are used as coolants in heat transport devices like heat exchangers, radiators, and electronic cooling systems (like a flat plate) because of their improved thermal properties. The preeminent perspective of this study is to highlight the influence of combined convection on heat transfer and pseudoplastic non-Newtonian nanofluid flow towards an extendable Riga surface. Buongiorno model is incorporated in the present study to tackle a diverse range of Reynolds numbers and to analyze the behavior of the pseudoplastic nanofluid flow. Nanofluid features are scrutinized through Brownian motion and thermophoresis diffusion. By the use of the boundary layer principle, the compact form of flow equations is transformed into component forms. The modeled system is numerically simulated. The effects of various physical parameters on skin friction, mass transfer, and thermal energy are numerically computed. Fluctuations of velocity increased when modified Hartmann number and mixed convection parameter are boosted, where it collapses for Weissenberg number and width parameter. It can be revealed that the temperature curve gets down if modified Hartmann number, mixed convection, and buoyancy ratio parameters upgrade. Concentration patterns diminish when there is an incline in width parameter and Lewis number; on the other hand, it went upward for Brownian motion parameter, modified Hartmann, and Prandtl number.

Mixed convection flow or a combination of free and forced convections exists in numerous electrify operations; both certainly occur in many engineering applications. Such applications mostly appear in heat exchangers, nuclear reactors, nanotechnology, atmospheric boundary layer flow, electronic equipment, and so on. These operations arise at some stage in the outcomes of buoyancy forces in combined convections or the effects of forced flow in free convection become significant [

The research of heat transmission triggered employing boundary layer flow of an incompressible fluid towards a stretched surface had received extended interest from the scientists and researchers due to its advantages in industry and engineering. Presently, huge exertion has been made to concentrate on this subject to the frame of reference of its different industrial and engineering applications. The effects of combined convection MHD on the boundary layer flow in the aspect of heat transport of Hematite-water nanofluid on a stretchable surface are demonstrated in [

Nowadays, to improve the abilities of heat transport of habitual fluids like water, glycerin, and engine oil because of the decay of the thermal effects, nanofluid is the classic bestowal to enhance the thermal conductivity. Nano liquid is a liquid that was invented by nanometer-sized particles and microfibers having a diameter less than 100 nm. Nanofluids are used in many operations to fulfill industrial requirements, like propellant combustion, drug delivery, cooling of automotive engines, extraction of geothermal forces, and heat transfer. Al-Hossainy and Eid [

Lorentz force theoretically [

The current examination adds a novel era for scientists to find the characteristics of pseudoplastic nanofluid. Here, we inspect heat transfer analysis and mixed convection stagnation-point flow of a pseudoplastic non-Newtonian nanofluid over a convectively heated flexible Riga plate. Obtained equations are solved by bvp4c numerically. The consequences of the given problem are demonstrated by parameters such as modified Hartmann number, mixed convection parameter, Brownian motion parameter, buoyancy ratio parameter, thermophoresis parameter, width parameter, stretching parameter, Weissenberg, Biot, Prandtl, and Lewis numbers to the velocity, thermal energy, and mass transmission towards the vertical elastic Riga plate.

We inspected the incompressible, two-dimensional, steady pseudoplastic nanofluid flow and heat transfer improvement examination over the vertical extendable Riga surface as shown in Figure

Sketch of the flow problem. (a) Physical geometry of the problem. (b) Riga plate.

The elemental equation of pseudoplastic fluid is [

In the above equations,

The subjected boundary conditions are

Invoking similarity variables are defined as,

By using the upper relationships, the continuity equation (

The skin friction coefficient

The obtained flow of nonlinear differential equation (

Consequences of

Result of

Influence of

Impact of

Variation of

Upshot of

Effect of

Result of

Influence of

Consequences of

Impact of

Upshot of

Effect of

Impact of

Deviation of

Influence of

Impact of

Consequences of

Deviation of concentration field can be revealed in Figures

Impact of

Effect of

Influence of

Consequence of

Influence of

Result of

Upshot of

Consequences of

Streamlines for

Streamlines for

Streamlines for

Streamlines for

Streamlines for

Streamlines for

Deviation of

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 7 | 0.09524 | 0.09550 | 0.09598 |

0.5 | — | — | — | — | — | — | — | — | — | — | — | 0.09524 | 0.09550 | 0.09598 |

1 | — | — | — | — | — | — | — | — | — | — | — | 0.09524 | 0.09550 | 0.09598 |

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 7 | 0.09524 | 0.09550 | 0.09598 |

— | — | 1 | — | — | — | — | — | — | — | — | — | 0.09526 | 0.09551 | 0.09600 |

— | — | 2 | — | — | — | — | — | — | — | — | — | 0.09528 | 0.09553 | 0.09601 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 2 | 0.5 | 1 | 7 | 1.02513 | 1.04844 | 1.10026 |

— | — | — | 1 | — | — | — | — | — | — | — | — | 1.04508 | 1.06400 | 1.10922 |

— | — | — | 3 | — | — | — | — | — | — | — | — | 1.07624 | 1.09004 | 1.12598 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.5 | 1 | 7 | 0.09530 | 0.09554 | 0.09601 |

— | — | — | — | 0.3 | — | — | — | — | — | — | — | 0.09533 | 0.09556 | 0.09602 |

— | — | — | — | 0.5 | — | — | — | — | — | — | — | 0.09536 | 0.09558 | 0.09603 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.5 | 1 | 7 | 0.09530 | 0.09554 | 0.09601 |

— | — | — | — | — | — | — | — | 0.5 | — | — | — | 0.40213 | 0.40619 | 0.41437 |

— | — | — | — | — | — | — | — | 1 | — | — | — | 0.67496 | 0.68576 | 0.70853 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.5 | 1 | 7 | 0.09530 | 0.09554 | 0.09601 |

— | — | — | — | — | — | — | — | — | 1 | — | — | 0.09537 | 0.09559 | 0.09604 |

— | — | — | — | — | — | — | — | — | 1.5 | — | — | 0.09544 | 0.09564 | 0.09606 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.5 | 1 | 7 | 0.09530 | 0.09554 | 0.09601 |

— | — | — | — | — | — | — | — | — | — | 1.5 | — | 0.09528 | 0.09553 | 0.09601 |

— | — | — | — | — | — | — | — | — | — | 2 | — | 0.09527 | 0.09552 | 0.09600 |

0.1 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.5 | 1 | 6.5 | 0.09513 | 0.09539 | 0.09588 |

— | — | — | — | — | — | — | — | — | — | — | 8.5 | 0.09573 | 0.09594 | 0.09634 |

— | — | — | — | — | — | — | — | — | — | — | 10.5 | 0.09616 | 0.09633 | 0.09667 |

Variation of

0.1 | 0.1 | 0.01 | 0.1 | 0 | 1 | 1 | 1 | 0.1 | 0.1 | 1.5 | 7 | −0.61859 | 0.04466 | 2.05609 |

0.5 | — | — | — | — | — | — | — | — | — | — | — | −0.61406 | 0.04859 | 2.05910 |

1 | — | — | — | — | — | — | — | — | — | — | — | −0.60839 | 0.05350 | 2.06286 |

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 1 | 1 | 1 | 0.1 | 0.1 | 1.5 | 7 | −0.62883 | 0.03580 | 2.04931 |

— | — | 0.5 | — | — | — | — | — | — | — | — | — | −0.62427 | 0.03975 | 2.05233 |

— | — | 1 | — | — | — | — | — | — | — | — | — | −0.61859 | 0.04466 | 2.05609 |

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.1 | 1.5 | 7 | −0.62546 | 0.03869 | 2.05167 |

— | — | — | 1 | — | — | — | — | — | 1 | — | — | −0.30441 | 0.34063 | 2.32326 |

— | — | — | 3 | — | — | — | — | — | 1.5 | — | — | −0.12896 | 0.50656 | 2.47328 |

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 7 | −0.57767 | 0.07476 | 2.07641 |

— | — | — | — | — | — | — | — | — | — | 1 | — | −0.61658 | 0.04606 | 2.05726 |

— | — | — | — | — | — | — | — | — | — | 2 | — | −0.63130 | 0.03363 | 2.04763 |

The following main resulting ideas are taken into account:

Fluctuations of velocity were examined for some physical parameters; profile increases when

It can be revealed that thermal boundary layer width gets down when the values of

It can be easily observed that concentration patterns increase when the values of

When the values of the parameters

The numerical values of the skin friction coefficient increase by increasing the values of

Skin friction coefficient

Body forces

Ambient fluid concentration

Dimensionless concentration function

Biot number

Thermophoresis diffusion coefficient

Nanoparticles concentration

Brownian diffusion coefficient

Dimensionless velocity function

Gravity acceleration

Convection coefficient

Local Sherwood number

Local Nusselt number

Local Reynolds number

Thermophoresis parameter

Ambient temperature

Brownian motion parameter

Similarity variable

Thermal conductivity

Stretching sheet velocity

Hot fluid temperature

External flow velocity

Velocity components

Cartesian coordinate components

Weissenberg number

Time fluid parameter

Extra stress tensor

Power index

Fluid characteristics

Buoyancy ratio parameter

Width of the magnet between electrodes

Zero shear rate viscosity

Mixed convection parameter

Surface shear stress

The density of the nanoparticles

Surface mass flux

The density of the base fluid

Applied current density

Density

Prandtl number

Surface heat flux

Width parameter

Nanoparticle

Stretching parameter

Infinite shear rate viscosity

Magnetization of permanent magnets

Rivlin-Ericksen tensor

Temperature

The dimensionless heat temperature function

Thermal diffusivity

Modified Hartmann number

Lewis number

The data used to support the findings of the study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

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