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

Authors

• Mushtaq Ahmed University of Karachi
• Waseem Ahmed Khan University of Karachi

2018-04-13

Keywords:

Some Exact Solutions in the Presence of Body Force, Exact Solutions to the Flow Equations of Incompressible Fluids, Exact Solutions of Var-iable Viscosity Fluids, Navier-Stokes Equations with Body Force.

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  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.

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