Group theoretical analysis for MHD flow fields: a numerical result

  • Khalil Ur RehmanEmail author
  • M. Y. Malik
  • Iffat Zehra
  • M. S. Alqarni
Technical Paper


The physical phenomena having inhomogeneity subject to both the Newtonian and non-Newtonian fluid models yield the complex structured mathematical equations. It is well known that the exact solution in this direction is impossible. Therefore, the current pagination contains a systematic approach to present numerical solution of non-Newtonian fluid model. To be specific, the nonlinear mathematical problem is developed with the aid of fundamental laws involved in the field of fluid science. A group theoretic approach is implemented, and the obtained Lie point of transformation is used to step down the mathematical equations in terms of independent variables. The resultant system is solved by using shooting method conjectured with Runge–Kutta scheme. The impacts of involved parameters, namely power law index, magnetic field parameter, Weissenberg number, Prandtl number, Schmidt number, velocity slip parameter and thermal slip parameter are examined on dimensionless quantities in both the magnetized and non-magnetized flow fields. The obtained observations in this regard are provided by way of graphs. It is noticed that the fluid velocity is lesser in magnitude in a magnetized frame as compared to non-magnetized flow field.


MHD Tangent hyperbolic fluid Shooting method Group theoretic method 

List of symbols

\((\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} )\)

Space variables

\((\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{u} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{v} )\)

Velocity components

\(\mu_{\infty }\)

Infinite shear rate viscosity


Zero shear rate viscosity

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {s}\)

Extra stress tensor


Time-dependent material constant


Power law index

\({\vec{\text{V}}} = (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{u} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{v} )\)

Velocity field


Fluid density


Stress tensor


Body force


Identity tensor


Kinematic viscosity


Fluid electrical conductivity


Applied magnetic field strength


Thermal conductivity


Specific heat at constant pressure


Heat generation/absorption coefficient


Fluid temperature

\(\hat{T}_{\infty }\)

Ambient temperature


Fluid concentration

\(\hat{C}_{\infty }\)

Ambient concentration


Surface temperature


Surface concentration


Mass diffusivity


Stretching rate


Velocity slip factor


Thermal slip factor


Dimensionless temperature


Dimensionless concentration


Stream function


Small parameter

\(\lambda_{k = 1, \ldots ,6}\)

Real numbers


Weissenberg number


Magnetic field parameter


Prandtl number

\(Q^{ + }\)

Heat generation parameter

\(Q^{ - }\)

Heat absorption parameter


Schmidt number


Velocity slip parameter


Thermal slip parameter


Skin friction coefficient

\(Nu_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} }}\)

Local Nusselt number

\(Shu_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x} }}\)

Local Sherwood number

\(z_{i = 1, \ldots ,7}\)

Dummy variables

\(\frac{{{\text{d}}F(\xi )}}{{{\text{d}}\xi }}\)

Dimensionless fluid velocity



The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups programme under Grant Number R.G.P-1/77/40.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Khalil Ur Rehman
    • 2
    Email author
  • M. Y. Malik
    • 1
  • Iffat Zehra
    • 2
  • M. S. Alqarni
    • 1
  1. 1.Department of Mathematics, College of SciencesKing Khalid UniversityAbhaSaudi Arabia
  2. 2.Department of MathematicsAir UniversityIslamabadPakistan

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