A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid

Abstract

<jats:p>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 $\psi$ characterized by equation $\theta=f(r)+ a \psi + b $ in polar coordinates $r$, $\theta$ , where a continuously differentiable function is $f(r)$ and $a\neq 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 $R_{e}P_{r}\neq 1$ 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 $R_{e}P_{r}\neq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary. </jats:p>

A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

“…For the exact solutions of fundamental equations, we follow [4][5][6][7]. First, we write the basic equations in some manageable form by introducing the vorticity function w and the total energy function L defined by…”

“…In [5][6][7], we used a relation between the two functions A and B for the solution of compatible equation; however we find that such a momentous relation from equations (19) and (20) for this communication is not possible. Therefore, we consider the following two cases Case I: 0 = A Case II: 0 = B…”

“…We are using coordinate transformations technique on non-dimensional equations for determining the exact solutions in this communication. The fundamental non-dimensional equations for plane steady motion of incompressible variable viscosity fluid in the presence of external force in Cartesian space ) , ( y x , given in [5][6][7] are following…”

On Two-Dimensional Variable Viscosity Fluid Motion with Body Forcefor Intermediate Peclet Number Via von-Mises Coordinates

“…The solutions of momentum and energy equations are there through dimension analysis methods and coordinates transformation techniques [1][2][3][4][5][6]. For solution of these equations when NSE includes body force some transformations technique are applied [7][8][9][10]. Further, solutions are there for very small and very large e P ′ where as the solution with intermediate e P ′ is challenging [11][12][13][14].…”

A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates

“…The body force term appears in the study of magneto-hydrodynamic and in geophysical fluid dynamics [6][7][8] for example. The exact solution of fundamental equations with body force by setting the arbitrary coordinates of the Martin's system in radial directionare there in [9][10][11][12][13]. Further, the solution of the basic system of equations is found for very large and very small e P ′ where as the solution for moderate e P ′ is challenging.…”