On Plane Motion of Incompressible Variable Viscosity Fluids with Moderate Peclet Number in Presence of Body Force Via Von-Mises Coordinates

The aim of this article is to use von-Mises coordinates to find a class of new exact solutionsof the equations governing the plane steady motion with moderate Peclet number of incompressible fluid of variable viscosity in presence of body force. An equation relating a differentiable function and a stream function characterizes the class under consideration. When the differentiable function is parabolic and when it is not, in both the cases, it finds exact solutions for given one component of the body force. This discourse shows an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for moderate Peclet number in presence of body force. Moreover, for parabolic case, it obtains viscosity as a function of temperature distribution for moderate Peclet number.


Introduction
In general, a moving fluid element experiences both the surface and body forces. The momentum of moving fluid element is given by the Navier-Stokes equations (NSE). The non-linear terms in NSE offers a great difficulty for its exact solution show ever, some transformation techniques and dimension analysis methods are workable. A variety of techniques/methods and references given there are practical for some exact solutions of NSE without body force [1][2][3][4][5][6]. Moreover recently Mushtaq A. et. al., applied a new technique for exact solution of variable viscosity fluids without body force term [7,10]. Body force term like coriolis force is considered by Giga, Y. et. al. in [8] and Gerbeau, J. et. al. gives a fundamental remark on NSE with body force in [9] where as Mushtaq A. et. al. has applied successive transformation technique for exact solution for flow of incompressible variable viscosity fluids in presence of body force in [11][12][13][14].
To achieve the aim of this letter successive transformation technique is applied. According to this method the basic non-dimensional flows equations with body force in Cartesian space ( , ) x y are transformed into Martin's coordinates ( ) , φ ψ then to von-Mises coordinates ( , ) x ψ . In Martin's coordinates, the curvilinear coordinates ( ) , φ ψ are such that the coordinate lines . const ψ = are streamlines and the coordinate lines constant φ = are arbitrary [15]. Whereas in the von-Mises coordinates, the arbitrary coordinate lines of Martin's system is taken along the x axis − . Thus, the function x φ = and stream function ψ of Martin's coordinates as independent variables instead of y and x [16]. Further, the characteristic equation for streamlines of the class of flows under consideration is: Where 0 m ≠ , n are constants and a differentiable function is ( ) g x . Without loss of generality the equation (1) implies ( ) ( ) y g x ν ψ = + (2) where ( ) m n ν ψ ψ = + .
Paper's organization is follow: Section (2) givescentral flow equations in non-dimensional form and transforms them into Martin system ( , ) φ ψ . Section (3) retransforms the basic equations to von-Mises coordinates. The exact solutions to the problem in presence of body force are given in section 4. Conclusions are given at the end.

Basic Non-dimensional Equations in Martin's Coordinates
The equation of continuity, NSE and energy equation, for the steady plane motion of incompressible fluid of variable viscosity with constant thermal conductivity in the presence of unknown external force, in non-dimensional form are respectively following The solution of the remaining system of equations (4-6), as experience teaches, offers a great difficulty because of the presence of the non-linear term. These equations are managed by introducing the total energy function x T and the vorticity function Ω defined by: Utilizing equation (8)(9) in equations (4-6), we have Consider the allowable change of coordinates: ( , ) y y φ ψ = (14) where the system ( , ) φ ψ are curvilinear coordinates in the ( , ) x y plane − such that the Jacobian ( , ) 0 ( , ) Let curvilinear coordinate ψ is stream function as defined in Martin [15]. Let λ be the angle between the tangent to the streamlines . const ψ = and the curves . const φ = as arbitrary at a point ( , ) P x y , then tan( ) The first fundamental form is (16) wherein: Differentiating equation (14) with respect to x and y , and solving the resulting equations, one finds: Application of trigonometric identities on equation (15) and equation (18) provides The integrability conditions: where K is called the Gaussian curvature. Now equations (10-11), on substituting equation (15), equation (18), equation (20) andequations (22)(23) simplifies to following According to differential geometry [23], the expression x y uT vT + in equation (12) Therefore, the energy equation (12)becomes The magnitude of velocity vector ( , ) u v = q is 2 2 q u v = + and it simplifies to: The equation (13) on substitute values from equations (18-23), provides ( , ) ( , ) The vorticity function Ω in Martin's system is Equation (32) on substituting equation (15), equation (18)and equation (20) provides The fundamental system of equations transformed to Martin's system as momentum equations (25-26), the energy equation (28) for moderate Peclet number together with equations (30-31) and equation (33).

Basic Equations in Von-Mises Coordinates
Since the purpose of this communication is to determine a class of exact solutions to flow equations in von-Mises coordinates therefore the definition of von-Mises coordinates in [16] demands to set The equation (15), equation (17), equations (19), equations (30-31) and equation (33) reduces to The equations (25-26) and equation (28) on utilizing equations (34-40), give and 2 1 N q m x Applying the integrability condition x Through the solution of equation (45), the viscosity µ from either equation (38) or equation (39), the generalized energy function L from equations (41-42)for pressure p from Equation (9), the temperature distribution T for moderate Peclet number from equation (43), the velocity components from equation (7)streamlines from equation (2) are found.

Exact Solutions in Presence of Body Force
The compatibility equation (45) Use of equation (46) in equation (45), gives The equation (50)  The search for the appropriate form of 1 Insertion of equation (52) in equation (50) keeps ( ) R x and ( ) S ψ arbitrary and provides where 1 ( ) P x is function of integration. Solution of equations (41-42) for L , on substituting equations (52-53), is following The energy equation (43), on utilizing equation (46), equation (49) and equations (55-56) becomes The right-hand side of equation (57) suggests seeking solution of the form Let us differentiate equation (59) with respect to ψ .
Since x and ψ are independent variables therefore the right-hand side of equation (60) where 2 c , 3 c , 4 c and 5 c are constant of integration and ( ) Utilization of equations (65-68) in equation (58) provides the temperature T formoderate e P ′ and the back substitution gives the viscosity µ from equation (56), the velocity components from equation (7), the pressure p from equation (9) using equation (55), and streamlines from equation (2) and 1 ( ) where the function of integration is 2 ( ) P x .
In view of equation (72) for moderate e P ′ . It is now easy to find the velocity components from equation (7), the pressure p from equation (9) using equation (77), and streamlines from equation (2) for ( ) g x given by equation (72).

Conclusion
The following dimensionless parameters are used to obtain the non-dimensional form of the basic equations for the twodimensional steady motion of incompressible fluid of variable viscosity in the presence of body force constants. In both the cases, an infinite set of velocity components, viscosity function, generalized energy function, temperature distribution for moderate Peclet number in presence of body force can be constructed and graph of streamlines can easily be drawn through computer algebra system software to observe the streamline patterns. For parabolic case, viscosity is obtained as a function of temperature distribution for moderate Peclet number.