On Plane Motion of Incompressible Variable Viscosity Fluids with Moderate Peclet Number in Presence of Body Force Via Von-Mises Coordinates

Mushtaq Ahmed

doi:10.11648/j.ijfmts.20190503.12

Abstract

The aim of this article is to use von-Mises coordinates to find a class of new exact solutionsof the equations governing the plane steady motion with moderate Peclet number of incompressible fluid of variable viscosity in presence of body force. An equation relating a differentiable function and a stream function characterizes the class under consideration. When the differentiable function is parabolic and when it is not, in both the cases, it finds exact solutions for given one component of the body force. This discourse shows an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for moderate Peclet number in presence of body force. Moreover, for parabolic case, it obtains viscosity as a function of temperature distribution for moderate Peclet number.

Highlights

A moving fluid element experiences both the surface and body forces
The momentum of moving fluid element is given by the Navier-Stokes equations (NSE)
A variety of techniques/methods and references given there are practical for some exact solutions of NSE without body force [1,2,3,4,5,6]

Summary

Introduction

A moving fluid element experiences both the surface and body forces. The momentum of moving fluid element is given by the Navier-Stokes equations (NSE). Body force term like coriolis force is considered by Giga, Y. et al in [8] and Gerbeau, J. et al gives a fundamental remark on NSE with body force in [9] where as Mushtaq A. et al has applied successive transformation technique for exact solution for flow of incompressible variable viscosity fluids in presence of body force in [11,12,13,14]. To achieve the aim of this letter successive transformation technique is applied According to this method the basic nondimensional flows equations with body force in Cartesian space (x, y) are transformed into Martin’s coordinates (φ,ψ ). Mushtaq Ahmed: On Plane Motion of Incompressible Variable Viscosity Fluids with Moderate Peclet Number in Presence of Body Force Via Von-Mises Coordinates y = g(x) +ν (ψ ).

Basic Non-dimensional Equations in Martin’s Coordinates

Basic Equations in Von-Mises Coordinates

Exact Solutions in Presence of Body Force

Conclusion