A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid - Ahmed, Mushtaq; Naeem, Rana Khalid; Ali, Syed Anwer (International Journal of Applied Mathematical Research, 2018)
This paper is to indicate a class of new exact solutions of the equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in the presence of body force. The class consists of the stream function $ψ$ characterized by equation $θ=f(r)+ a ψ + b $ in polar coordinates $r$, $θ$ , where a continuously differentiable function is $f(r)$ and $a eq 0 , b $ are constants. The exact solutions are determined for given one component of the body force, for both the cases when $f(r)$ is arbitrary and when it is not. When $f(r)$ is arbitrary, we find $a=1$ and we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for the cases when $R_{e}P_{r}=1$ and when $R_{e}P_{r} eq 1$ where $R_{e}$ represents Reynolds number and $P_{r}$Prandtl number. For the case when $f(r)$ is not arbitrary we can find solutions for the cases $R_{e}P_{r} eq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary.Â