A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

Mushtaq Ahmed,S M Shad Ahsen,Waseem Ahmed Khan

doi:10.14419/ijamr.v7i3.12326

Mushtaq Ahmed, S M Shad Ahsen + Show 1 more

Open Access

https://doi.org/10.14419/ijamr.v7i3.12326

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Abstract

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function Â characterized by equation , in polar coordinates , Â where , Â and Â are continuously differentiable functions, derivative of Â is non-zero but double derivative of Â is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions , Â and provides viscosity as function of temperature. Where as the second case fixes the function , leaves Â arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force.Â Â

Highlights

For the study of motion of a fluid element, we consider its momentum, the energy and conservation of mass equations
The complexity for exact solutions increases when we add more terms to the right-hand side of Navier-Stokes equations like body force, as it appears in geophysical fluid dynamics (GFD) due to the earth’s rotation and in magnetohydrodynamic (MHD) where the magnetic field induces currents in a moving fluid for creating force on the fluid [1,2,3]
For the exact solutions of these basic equations with and without body force, we find some appropriate dimension analysis method and coordinates transformation techniques in the available literature

Summary

Introduction

For the study of motion of a fluid element, we consider its momentum, the energy and conservation of mass equations. We are using coordinate transformations technique on non-dimensional equations for determining the exact solutions in this communication. In order to determine the solution of the flow equations (15-16), we follow [4,5,6,7] and form a compatibility equation applying the natural integrability condition Lr = L r on the function L. We have given the resulting fundamental equations in Martin’s coordinates system ( , ) in [5,6,7] considering a common point P(x, y) on the curves = const. Once a solution of this equation (24) is determined, the energy function L and temperature distribution T are obtained from equations (15-16) and (17) respectively, the viscosity is obtained either from equation (19) or (20), the pressure p from equation (10) and velocity components u , v from (5).

Exact solutions

Conclusion