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Edited by: Muhammad Mubashir Bhatti, Shanghai University, China

Reviewed by: R. Ellahi, University of California, Riverside, United States; Arash Asadollahi, Southern Illinois University Carbondale, United States

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The steady non-isothermal convective heat transfer in magnetohydrodynamic micropolar fluid flow over a non-linear extending wall is examined. The fluid flow is treated with strong magnetic field. The influence of magnetic field, Hall current, and couple stress are mainly focused in this work. The fluid flow problem is solved analytically. The impact of developing dimensionless parameters on primary, secondary, and angular velocity components and temperature profile are determined through graphs. The primary velocity component has reduced throughout the flow study. The greater magnetic parameter, Hall parameter and couple stress parameter have increased the secondary velocity component while the local Grashof number has reduced the secondary velocity component. The greater magnetic parameter and Hall parameter have reduced the angular velocity component. The greater magnetic parameter has increased the temperature profile while the Hall parameter and local Grashof number have decreased the temperature profile. The impact of developing dimensionless parameters on skin friction coefficient and local Nusselt number are determined through Tables.

The flow of non-Newtonian fluids has plentiful importance in industries and modern technology. Recently, the couple stress fluid among non-Newtonian fluid has acquired the exceptional position due to the spin field in the fluid. The elementary concept of couple stress was established by Stokes [

In recent times, the researchers have got interest in megnetohydrodynamic (MHD) owing to plentiful applications in industrial, engineering, and medical devices. Rudolf et al. [

In view of the above mentioned literature survey, the authors are in position to examine the three-dimensional MHD micropolar fluid flow over an extending wall with couple stress, Hall current and viscous dissipation influences. Section of Problem Formulation agrees with problem formulation. In the section of Solution by HAM, the recommended model is solved by HAM. Results section includes the results of the problem and the section of Discussion of the problem is presented independently. The final observations are obtainable in the section of Conclusion.

We assume the incompressible, steady, and electrically conducting couple stressed flow of micropolar fluid and heat transfer in the near wall zone of MHD Hall generator. The wall is considered as non-linearly stretching and concerned with _{0} is functional in

Geometrical illustration of the micropolar fluid flow.

The principal equations for the fluid flow can be written as [

with

Here, the positive _{e}, applied uniform magnetic field-_{0}, Hall parameter-_{2}, thermal conductivity-κ, Eringen spin gradient viscosity-_{1}, specific heat-_{p}, γ, and

To transform the coordinate system to a non-dimensional one and this is achieved readily via non-similar transformations, simultaneously eliminating one of the independent variables and reducing the PDEs into ODEs, the following transformation variables are defined.

The transformed equations are defined as:

with transformed boundary conditions:

Here,

For primary and secondary velocity components, the skin frication are defined as:

Using Equation (7), the skin fraction coefficients for primary and secondary velocities are reduced as:

The Nusselt number is specified by:

To solve the Equations (8)–(11) using boundary conditions (12), we proceed HAM with the following manners.

with the following properties:

where _{i}(

The consequential non-linear operators _{f}, _{g}, _{h}, and _{θ} are specified as:

The zeroth-order problems from Equations (8)–(11) are:

The equivalent boundary conditions are:

When Θ = 0 and Θ = 1 we have:

By Taylor's series expansion

where

The secondary constraints ħ_{f}, ħ_{g}, ħ_{h} and ħ_{θ} are nominated in such a way that the series (31) converges at Θ = 1, changing Θ = 1 in Equation (31), we get:

The ^{th}−order problem satisfies the following:

The equivalent boundary conditions are:

Here,

where,

Electrically conducting steady non-isothermal convective heat transfer in magnetohydrodynamic micropolar fluid flow over a non-linear extending wall is examined. Modeled equations are solved analytically through HAM. The impact of obtained important parameters

Impact of

In this section we have discussed the effects of obtained parameter which are shown graphically and numerically through tables. The greater Hartmann number strongly reduced the primary and angular velocity profile owing to the Lorentz drag force components as appear in Equations (8) and (9). The components are negative and positive and thus inhibit the fluid flow. According to the secondary Lorentz drag force is truthfully positive and is assistive to secondary momentum development when the magnetic field is positive. These impacts are depicted in

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

Impact of

_{fx} and _{fz} are shown in _{fx} where _{fx}. The higher value of _{fz} where, _{fz}. The influence of _{x} while, remaining parameter reduces the heat flux _{x}. It should be noted that _{x}.

Influence of _{fx}.

_{fx} |
|||||||||
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.2 | 0.3 | 0.2 | 1.1 | 0.1 | 0.72 | 1.1 | 0.1 | −1.233236 |

0.3 | −1.339327 | ||||||||

0.4 | −1.443959 | ||||||||

0.4 | −1.391409 | ||||||||

0.6 | −1.338869 | ||||||||

0.8 | −1.286370 | ||||||||

0.4 | −1.286278 | ||||||||

0.6 | −1.286189 | ||||||||

0.8 | −1.286101 | ||||||||

0.3 | −1.288813 | ||||||||

0.4 | −1.291526 | ||||||||

0.5 | −1.294238 | ||||||||

1.3 | −1.173870 | ||||||||

1.5 | −1.097241 | ||||||||

1.7 | −1.042032 | ||||||||

0.3 | −1.045267 | ||||||||

0.6 | −1.044867 | ||||||||

0.9 | −1.044467 | ||||||||

1.0 | −1.046198 | ||||||||

5.0 | −1.050270 | ||||||||

10.0 | −1.050834 | ||||||||

1.2 | −1.081639 | ||||||||

1.3 | -.1117715 | ||||||||

1.4 | −1.152781 | ||||||||

0.3 | −1.561382 | ||||||||

0.5 | −2.936200 | ||||||||

0.7 | −8.747216 |

Influence of _{fz}.

_{fz} |
|||||||||
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.2 | 0.3 | 0.2 | 1.1 | 0.1 | 0.72 | 1.1 | 0.1 | 0.210288 |

0.3 | 0.312368 | ||||||||

0.4 | 0.410969 | ||||||||

0.4 | 0.412875 | ||||||||

0.6 | 0.414780 | ||||||||

0.8 | 0.416684 | ||||||||

0.4 | 0.416776 | ||||||||

0.6 | 0.416866 | ||||||||

0.8 | 0.416953 | ||||||||

0.3 | 0.414227 | ||||||||

0.4 | 0.411502 | ||||||||

0.5 | 0.408776 | ||||||||

1.3 | 0.295179 | ||||||||

1.5 | 0.221561 | ||||||||

1.7 | 0.171285 | ||||||||

0.3 | 0.171296 | ||||||||

0.6 | 0.171313 | ||||||||

0.9 | 0.171330 | ||||||||

1.0 | 0.171273 | ||||||||

5.0 | 0.171144 | ||||||||

10.0 | 0.171127 | ||||||||

1.2 | 0.166810 | ||||||||

1.3 | 0.162582 | ||||||||

1.4 | 0.158615 | ||||||||

0.3 | 0.191738 | ||||||||

0.5 | 0.184492 | ||||||||

0.7 | 0.184488 |

Influence of _{x}.

_{x} |
|||||||||
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.2 | 0.3 | 0.2 | 1.1 | 0.1 | 0.72 | 1.1 | 0.1 | 1.567232 |

0.3 | 1.553232 | ||||||||

0.4 | 1.539852 | ||||||||

0.4 | 1.540408 | ||||||||

0.6 | 1.540960 | ||||||||

0.8 | 1.541507 | ||||||||

0.4 | 1.541508 | ||||||||

0.6 | 1.541508 | ||||||||

0.8 | 1.541508 | ||||||||

0.3 | 1.541500 | ||||||||

0.4 | 1.541493 | ||||||||

0.5 | 1.541486 | ||||||||

1.3 | 1.557281 | ||||||||

1.5 | 1.567616 | ||||||||

1.7 | 1.574715 | ||||||||

0.3 | 1.530638 | ||||||||

0.6 | 1.464505 | ||||||||

0.9 | 1.398349 | ||||||||

1.0 | 1.524106 | ||||||||

5.0 | 1.893057 | ||||||||

10.0 | 1.952805 | ||||||||

1.2 | 1.437652 | ||||||||

1.3 | 1.475901 | ||||||||

1.4 | 1.513178 | ||||||||

0.3 | 1.515541 | ||||||||

0.5 | 1.518194 | ||||||||

0.7 | 1.523959 |

In the current paper, the MHD micropolar boundary layer flow and heat transfer over a non-linear extending sheet infused by a strong magnetic field with couple stress, viscous dissipation and Hall impact have been determined.

The final observations are:

The primary velocity reduces with greater magnetic parameter, local Grashof number, Hall parameter and couples stress parameter.

The secondary velocity increases with greater magnetic parameter, Hall parameter and couple stress parameter.

The secondary velocity decreases with greater local Grashof number.

The angular velocity reduces with greater magnetic parameter and Hall parameter.

The angular velocity increases with greater local Grashof number.

The temperature profile increases with greater magnetic parameter.

The temperature profile increases with greater Hall parameter and local Grashof number.

The datasets generated for this study are available on request to the corresponding author.

ZS and PK developed the numerical method and led the manuscript preparation. AD contributed to the code development and to the article preparation. EA and PT contributed to the analysis and discussion of the results.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the research program International Research Common Laboratory in cooperation under Renewable Energy Research Centre (RERC)—King Mongkut's University of Technology North Bangkok (KMUTNB), Center of Excellence in Theoretical and Computational Science (TaCS-CoE)—King Mongkut's University of Technology Thonburi (KMUTT), and Groupe de Recherche en Energie Electrique de Nancy (GREEN)—Université de Lorraine (UL) under Grant KMUTNB-61-GOV-A-01.