Open AccessA class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force
Author(s) -
Mushtaq Ahmed,
Waseem S. Khan
Publication year - 2018
Publication title -
international journal of applied mathematical research
Language(s) - English
Resource type - Journals
ISSN - 2227-4324
DOI - 10.14419/ijamr.v7i2.9998
Subject(s) - mathematics , mathematical analysis , ordinary differential equation , function (biology) , exact differential equation , exact solutions in general relativity , streamlines, streaklines, and pathlines , plane (geometry) , viscosity , partial differential equation , constant (computer programming) , stream function , differential equation , physics , geometry , vorticity , mechanics , computer science , quantum mechanics , evolutionary biology , vortex , biology , programming language
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 in polar coordinates and where and are continuously differentiable functions and the function is such that where a non-zero constant is and overhead prime represents derivative with respect to . When or we show exact solutions for given one component of the body force for both the cases when the function is arbitrary and when it is not. For the arbitrary function case, 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 and provides viscosity as derivative of temperature function for and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.