# Effect of the aspect ratio on the flow characteristics of magnetohydrodynamic (MHD) third grade fluid flow through a rectangular channel

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## Abstract

The magnetohydrodynamic (MHD) flow of a third grade fluid through a rectangular channel, considering the effect of aspect ratio, has been investigated. The flow considered is steady, laminar, incompressible and hydro-dynamically fully developed. The equation, describing the flow, is a highly non-linear partial differential equation (PDE) with remote possibility of having an exact solution and even numerical solution also is very difficult to obtain. A combination of the homotopy perturbation method (HPM) and integral method (IM) has been employed to solve the non-linear PDE which is scarce in open literature. The results of the present study are compared with the results obtained by the least square method (LSM) of the MHD third grade fluid flow through a rectangular channel, without the effect of aspect ratio and are found to be in close agreement. The results indicate that the flow field is significantly affected by the aspect ratio which should be considered for practical applications. In all the available literatures of the third grade fluid flow, the aspect ratio effect is neglected and this simplifying assumption reduces the highly complicated non-linear PDE to a non-linear ordinary differential equation (ODE). The novelty of the subject work lies in the inclusion of the effects of aspect ratio in the governing equation describing the flow of a third grade fluid through a channel and solving this by a combined analytical method (HPM and IM). Further, the effects of the Hartmann number and non-Newtonian third grade fluid parameter on the flow filed are discussed.

## Keywords

Third grade fluid aspect ratio homotopy perturbation method (HPM) integral method (IM) Hartmann number MHD flow## Notations

*A*third grade fluid parameter,

*A*_{1},*A*_{2},*A*_{3}kinematic tensors

*a*_{0,}*a*_{1},*b*_{0,}*b*_{1}Constants

*As*aspect ratio of the channel,

*B*applied magnetic field

*C*constant

*F*,*G*functions

*Ha*Hartmann number,

*J*current density respectively

*L*_{1},*L*_{2}half depth and half width of the channel respectively

*N*non-dimensional pressure gradient,

*R*Residual

*U*average velocity through the channel

*V*^{*}velocity vector

*c*_{1},*c*_{2}constants

*f*body force per unit volume

*p*dimensional static pressure,

*u*dimensional axial velocity

*v*dimensionless coordinate in the axial direction

*y*dimensionless coordinate in vertical direction

*z*dimensionless coordinate in the lateral direction

*q*an embedding parameter

*p*^{*}dimensional pressure

*u*^{*}dimensional axial velocity

*y*^{*}dimensional coordinate in the vertical direction

*z*^{*}dimensional coordinate in the lateral direction

*q*^{0}*, q*^{1}embedding parameter

*u*_{0}0

^{th}order solution for the velocity*u*_{1}1

^{st}order solution for the velocity

## Greek Symbols

- ρ
density of the fluid,

- τ
stress tensor

- σ
electrical conductivity

- μ
dynamic viscosity the fluid,

- α
_{1}, α_{2}, β_{1}, β_{2,}β_{3} material constants of the third grade fluid

## Notes

### Acknowledgements

The authors are thankful to the reviewers for their valuable comments and suggestions which have improved the paper.

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