Flow of Third Grade Fluid and Vogel’s Model Viscosity in Cylindrical Pipe

Abstract

The flow of non-Newtonian fluid and Vogel’s model viscosity in cylindrical pipe is treated. The coupled nonlinear equations of motion solved and the effects of sundry parameters of the non-Newtonian fluid and the Vogel model viscosity examined. Results show that the gravitational parameter has a great influence on the flow field. It is observed that as the viscosity index increases, the velocity of the fluid flow reduces. This indicates that the shear strain which reduces the flow velocity, increases the viscosity. Results further show that increase in increases the temperature of the cylinder indicates a low viscosity. This is because high temperature means that particles have more thermal energy and are easily able to overcome the attractive forces holding them together.

Introduction

I. INTRODUCTION

In the past years, generalization of the Navier-Stoke’s model which is highly non-linear constitutive laws have been proposed and examined by many researchers. Because of the deficiency of the classical Navier-Stokes theory in describing rheological complex fluids such as paints, blood, oils and greases and the applications in engineering and technology as well as the pulp industries, has led to the development of many theories of non-Newtonian fluids.

Many researchers have done some works to explain some of the complex nature of the non-Newtonian fluids of the differential types, amongst whom are: Rivlin and Erickson [12], on stress deformation relations for isotropic materials. Fosdic and Rajagopal [4], on thermodynamics, stability of fluids of third grade. Okedayo et al [7] carried out  computational study of reactive flow of an electrically conducting fluid with temperature dependent viscosity and axial magnetic field using the semi-implicit finite difference scheme.

Aksoy and Pakdemirli [2]. They dealt with the flow of a non-Newtonian fluid through a porous medium in between two parallel plates at different temperatures. They considered different cases: constant viscosity, Reynold’s model viscosity and Vogel’s model viscosity and derived the criteria for validity, for approximate solutions. Obi et al [8], analyzed the flow of  incompressible MHD third grade fluid in an inclined   rotating cylindrical pipe with isothermal wall and joule heating. Obi [9], applied perturbation technique to analyze magnetohydrodynamic flow of third grade fluid in an inclined cylindrical pipe.

Hayat et al [5] applied homotopy perturbation and numerically obtained the solution of the third grade  fluid past a porous channel with suction and injection at the walls. Massoudi and Christie [6] examined the effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe. Shirkhani et al [13], examined the unsteady time-dependent incompressible Newtonian fluid flow between two parallel plates by homotopy analysis method (HAM), homotopy perturbation method (HPM) and collocation method (CM). They transformed the Navier-Stokes equation into ordinary differential equation using similarity transformation and investigated the effects of Reynolds number and suction or injection characteristic parameter on the velocity field.

Pakdemirli and Yilbas [10] examined entropy generation in a pipe due to non-Newtonian fluid flow, a case of constant viscosity. They formulated the entropy generation number due to heat transfer and fluid friction. The influences of non-Newtonian parameters and Brinkman number on entropy generation number were examined and results revealed that increase in the non-Newtonian parameter reduces the fluid friction in the region close to the wall of the pipe, given rise to low entropy generation. They further discovered that increase in the Brinkman number enhances the fluid friction and heat transfer rate thereby increases the entropy number.

Aiyesimi  et al  [1] on the analysis of unsteady  MHD thin film flow of a third grade fluid with heat transfer down an inclined plane. They discovered that the variation of the velocity and temperature profiles with the magnetic fields and gravitational field parameters depended on time. Ayub  et al [3] examined the exact flow of third grade fluid past  a porous plat e. the applied homotopy perturbation method for their analysis.

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Conclusion

The non-Newtonian fluid flow and Vogel’s model viscosity in cylindrical pipe is treated. The coupled nonlinear equations of motion solved and the effects of sundry parameters of the non-Newtonian fluid and the Vogel model viscosity examined. Results show that the gravitational parameter has a great influence on the flow field. It is observed that as the viscosity index increases, the velocity of the fluid flow reduces. This indicates that the rate of shear strain which reduces the flow velocity, increases the viscosity. Results further show that increase in increases the temperature of the cylinder indicates a low viscosity. This is because high temperature means that particles have higher thermal energy and are more easily able to pull through the attractive forces holding them together. A. Declarations 1) Funding: Not applicable 2) Informed Consent Statement: Not applicable 3) Data Availability: Not applicable 4) Conflict of Interest Statement: No conflict of interest

References

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Copyright

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