# Group theoretical analysis for MHD flow fields: a numerical result

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

## Abstract

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.

## Keywords

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

$$\mu_{0}$$

Zero shear rate viscosity

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

Extra stress tensor

$$\Gamma$$

Time-dependent material constant

$$n$$

Power law index

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

Velocity field

$$\rho$$

Fluid density

$$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\leftrightarrow}}}{\text{T}}$$

Stress tensor

$${\vec{\text{B}}}$$

Body force

$$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\leftrightarrow}}}{\text{I}}$$

Identity tensor

$$\nu_{1}$$

Kinematic viscosity

$$\sigma$$

Fluid electrical conductivity

$$B$$

Applied magnetic field strength

$$k$$

Thermal conductivity

$$c_{\text{p}}$$

Specific heat at constant pressure

$$Q_{1}$$

Heat generation/absorption coefficient

$$\hat{T}$$

Fluid temperature

$$\hat{T}_{\infty }$$

Ambient temperature

$$\hat{C}$$

Fluid concentration

$$\hat{C}_{\infty }$$

Ambient concentration

$$\hat{T}_{w}$$

Surface temperature

$$\hat{C}_{w}$$

Surface concentration

$$D_{c}$$

Mass diffusivity

$$b$$

Stretching rate

$$L_{1}$$

Velocity slip factor

$$D_{1}$$

Thermal slip factor

$$\theta$$

Dimensionless temperature

$$\phi$$

Dimensionless concentration

$$\psi$$

Stream function

$$\varepsilon$$

Small parameter

$$\lambda_{k = 1, \ldots ,6}$$

Real numbers

$$W_{\text{b}}$$

Weissenberg number

$$\gamma$$

Magnetic field parameter

Pr

Prandtl number

$$Q^{ + }$$

Heat generation parameter

$$Q^{ - }$$

Heat absorption parameter

$$Sc$$

Schmidt number

$$\alpha_{1}$$

Velocity slip parameter

$$\alpha_{2}$$

Thermal slip parameter

$$C_{\text{F}}$$

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

## Notes

### Acknowledgement

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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© 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