Study of Laminar Flow between Parallel Plates via Gupta Integral Transform

Abstract

Viscosity is the characteristic of a fluid by virtue of which viscous force becomes active when the fluid is in motion and opposes the relative motion of its different layers. This viscous force becomes active when the different layers of the fluid are moving with different velocities, and leads to shearing stress between the layers of the fluid in motion. This paper illustrates the application of Gupta integral transform for studying the unidirectional Laminar Flow between parallel plates directly without finding the general solution of a differential equation relating flow characteristics equation of the viscous liquid. In this paper, Gupta integral transform is applied for solving the differential equation relating flow characteristics of the viscous liquid to obtain the velocity and shear stress distributions of a unidirectional Laminar Flow between the stationary parallel plates as well as between the parallel plates having a relative motion.

Introduction

I. INTRODUCTION

The steady flow of a viscous fluid over a horizontal surface in the form of layers of different velocities in which the particles of the fluid move in a regular and well-defined paths is known as laminar flow. A velocity gradient exists between the two layers due to relative velocity and as a result, a shear stress acts on the layers. Seepage through soils, the flow of crude oil and highly viscous fluids through narrow passages are some of the examples of the laminar flow. In such a flow the fluid properties remain unchanged in the directions perpendicular to the direction of flow of the fluid [1-5].

Conclusion

In this paper, we have studied the unidirectional Laminar Flow between stationary parallel plates and have successfully obtained the velocity and shear stress distributions of a unidirectional Laminar Flow between stationary parallel plates as well as between parallel plates having a relative motion by solving the differential equation describing the flow characteristics of a viscous fluid via Gupta integral transform. Thus, Gupta integral transform has presented a powerful tool for obtaining the solution of the differential equation representing flow characteristics without finding the general solution. It is concluded that, in the case of unidirectional Laminar Flow with constant pressure gradient between stationary parallel plates, the velocity distribution is maximum at the midway between the parallel plates and decreases parabolically with maximum value at the midway between the parallel plates to a minimum value at the lower fixed plate as well as at the upper fixed plate but the shear stress varies linearly with a minimum value at the midway between the parallel plates to a maximum value at the lower fixed plate as well as at the upper fixed plate. In the case of laminar flow with constant pressure gradient between parallel plates having a relative motion, the velocity distribution is parabolic with a minimum at the lower fixed plate but the shear stress varies linearly and at the midway between the parallel plates having a relative motion it is equal to the mean of the values of the shear stresses at the lower fixed plate and at the uniformly moving upper plate, and having a constant value even if there is no pressure gradient between parallel plates having a relative motion.

References

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Copyright

Copyright © 2022 Rohit Gupta, Shivam Sharma, Rahul Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.