On Two-Dimensional Variable Viscosity Fluid Motion with Body Forcefor Intermediate Peclet Number Via von-Mises Coordinates

Mushtaq Ahmed

doi:10.11648/j.ijfmts.20190503.13

Abstract

This article uses von-Mises coordinates to present a class of new exact solutions of the system of partial differential equations for the plane steady motion of incompressible fluid of variable viscosity in presence of body forcefor moderate Peclet number. This communication applies successive transformation technique and characterizes streamlines through an equation relating a differentiable function f(x) and a function of stream function. Considering the function of stream function satisfies a specific relation, the exact solutions for moderate Peclet number with body force are determined for given one component of the body force when f(x) takes a specific value and when it is not. In both the cases, it shows an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution for intermediate Peclet number in presence of body force. When f(x) takes a specific value, a relation between viscosity and temperature function is observed.

Highlights

For the motion of a variable viscosity fluidthe equation of conservation of mass, momentum and energy are known as a system of partial differential equations (PDE)
For other f (x), the solution of variable coefficient differential equation [60] is easy to find from computer algebra system (CAS) software
This communication finds a class of new exact solutions of the equations governing the two-dimensional steady motion with moderate Peclet number of incompressible fluid of variable viscosity in presence of body force in von-Mises coordinates

Summary

Introduction

For the motion of a variable viscosity fluidthe equation of conservation of mass, momentum and energy are known as a system of partial differential equations (PDE). Number Via von-Mises Coordinates where Fα (xα ) is the body force per unit mass, vα (xα ) the fluid velocity, p = p(xα ) is pressure, the coefficients of viscosity μ > 0 , the space coordinates xα and α, β ∈{1, 2, 3}. The solutions of momentum and energy equations are there through dimension analysis methods and coordinates transformation techniques [1-6]. For solution of these equations when NSE includes body force some transformations technique are applied [7-10]. This communicationapplies successive transformation scheme to meet the challenge of moderate Pe′ According to this scheme the basic non-dimensional flows equations with body force in Cartesian space (x, y) are first transformed into Martin’s coordinates (φ,ψ ) to von-Mises coordinates (x,ψ ).

Basic Equations to Martin’s System

Exact Solutions in von-Mises Coordinates

Conclusion