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 - Ahmed, Mushtaq; Khan, Waseem Ahmed (International Journal of Applied Mathematical Research, 2018)
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.