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A convergence criterion for tangent hyperbolic fluid along a stretching wall subjected to inclined electromagnetic field

  • Emran Khoshrouye GhiasiEmail author
  • Reza SalehEmail author
Article
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Abstract

The homotopy-based approach is a useful tool for solving nonlinear partial differential equations (PDEs) in physics and engineering. Our aim here is to optimize this approach by generating a convergence criterion for tangent hyperbolic fluid along a stretching wall with magnetic force. To this end, the governing partial differential equations (PDEs) get transformed to the dimensionless form via similarity variables. A comparison of the homotopy-based approach for the skin friction coefficient with different solution methodologies shows that the 9th-order approximate solution together with \( \hbar = - \,0. 5 2 3 \) will certainly achieve a very minor error for the present system.

Keywords

Homotopy-based approach Convergence CPU time Tangent hyperbolic fluid Magnetic strength 

Abbreviations

\( {\text{T}} \)

Cauchy stress tensor [Pa]

\( p \)

Hydrostatic pressure [Pa]

\( {\text{I}} \)

Identity tensor

\( n \)

Power-law index

\( u \), \( v \)

Velocity components along \( x \)- and \( y \)-directions, respectively [m s−1]

\( B_{0} \)

Magnetic field strength [kg s−2 A−1]

\( U_{w} \)

Velocity at the wall [m s−1]

\( b \)

Stretching rate [s−1]

\( f \)

Similarity function

\( We \)

Weissenberg number

\( M \)

Magnetic field parameter

\( C_{f} \)

Skin friction coefficient

\( Re_{x} \)

Reynolds number

Greek symbols

\( \varvec{\tau} \)

Viscous stress tensor [Pa]

\( \mu_{0} \)

Initial shear rate viscosity [kg m−1 s−1]

\( \mu_{\infty } \)

Infinite shear rate viscosity [kg m−1 s−1]

\( \varGamma \)

Time constant [s]

\( \dot{\gamma } \)

Shear rate [s−1]

\( \varPi \)

Second invariant of the viscous stress tensor

\( \upsilon \)

Kinematic viscosity [m2 s−1]

\( \sigma \)

Electrical conductivity [S m−1]

\( \rho \)

Fluid density [kg m−3]

\( \alpha \)

Inclination angle of the magnetic field

\( \eta \)

Similarity variable

\( \tau_{w} \)

Wall shear stress [Pa]

Subscripts

\( \infty \)

Condition at the infinite medium

\( i \), \( j \)

Tensor index

Mathematics Subject Classification

76W05 76D05 76M99 

Notes

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Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringCollege of Engineering, Mashhad Branch, Islamic Azad UniversityMashhadIran

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