A class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force

Mushtaq Ahmed,Waseem Ahmed Khan

doi:10.14419/ijamr.v7i2.9998

Abstract

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation Â in polar coordinates Â and Â where Â and Â are continuously differentiable functions and the function Â is such that Â where a non-zero constant is Â and overhead prime represents derivative with respect to . When Â or Â we show exact solutions for given one component of the body force for both the cases when the function Â is arbitrary and when it is not. For the arbitrary function case, Â appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes Â and provides viscosity as derivative of temperature function for Â and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.

Highlights

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations
The basic system of partial differential equations (PDE) for the motion of a viscous fluid consists of the equation of continuity, Navier-Stokes equations (NSE) and energy equation
We can select many possible forms of F and F leading to the solution of equation (49) for R (r), we find that not all arbitrarily selected forms lead to the solution of the momentum equations

Summary

Introduction

The basic system of partial differential equations (PDE) for the motion of a viscous fluid consists of the equation of continuity, Navier-Stokes equations (NSE) and energy equation. As the Navier-Stokes equations have base on Newton’s law, it allows us to add body forces term to right-hand side of it in addition to surface force. The examples of the body forces are constant gravity force, coriolis force, centrifugal force etc. In the presence of body force the basic dimensionless form of system of PDE’s for the steady motion of incompressible fluid of variable viscosity in tensor notation are Continuity where. The non-dimensional parameters used in equations (2-3) with constant thermal conductivity k are mentioned in [1,2]. The solution of the equation (4) demands the existence of a stream function (x , y ) such that u

B 2 4A 2

Fundamental flow equations in Martin’s system

Exact solutions

(89) References