On the Exact Solutions of Couple Stress Fluids

Exact solutions of the momentum equations of couple stress fluid are investigated. Making use of stream function, the two-dimensional flow equations are transformed into non-linear compatibility equation, and then it is linearized by vorticity function. Stream functions and velocity distributions are discussed for various flow situations.


Introduction
The basic governing equations for the flow of a fluid of Newtonian type are the Navier-Stokes equations. This set of non-linear partial differential equations has no general solution, and only a limited number of exact solutions can be found in literature. On the other hand, exact solutions are very important not only they represent the basic flow phenomena but also solutions obtained by various techniques can be attested by these basic solutions. For situations where flow of fluid is non-Newtonian in nature, it becomes rather difficult to obtain exact solutions. This difficulty occurs because of the highly non-linear terms in the viscous part of the flow equations.
Most of the fluids in nature do not obey the linear relationship between the stress and the rate of strain. Such fluids represent the non-Newtonian class of fluids. Several models have been presented to explain the behavior of these fluids. In the theory of non-Newtonian fluids, the couple stress fluid introduced by Stokes [1] has gained considerable attention and has been widely studied by researchers [2][3][4][5][6][7]. Couple stress fluid has been the subject of interest, due to its numerous industrial and scientific applications such as extrusion of polymer fluids, solidification of liquid crystals and animal bloods.
In this paper, we investigate exact solutions of the couple stress fluid motion for different flow situations. Solutions are found by using appropriate conditions for the different cases of flow. Expressions for the stream functions and velocity distributions are derived for two-dimensional flow behind a grid; flow above a porous plate; flow due to stretching plate, and the corner flow.

Governing Equations
The equations of motion governing the flow of a couple stress fluid are given by [1] (1) (2) where V is the velocity vector,  the constant density,  the dynamic viscosity, p the pressure,  the couple stress parameter, f the force vector, denotes the Laplacian operator. Equation (9) is a non-linear partial differential equation of an incompressible couple stress fluid. In order to linearize (9) vorticity is assumed of the form (10) where A 1 , A 2 are constants. Substitution from (10) into (9) yields (11) (12) which is a linear partial differential equation in H, and results in a number of known exact solutions for different values of A 1 , A 2. Using the method of separation of variables and by setting  (16) in which C is the separation constant, f 1 (x) and f 2 (x) are unknown functions to be determined. We now investigate the significance of these equations in some special cases.

Exact Solutions for different Flow Geometries
In this part of the paper, we investigate some exact solutions of the couple stress fluid for various flow situations. For different values of the constants A 1 , A 2 and C equations (15) and (16) will turn out exact solutions for some physical flow situations.

Laminar Flow behind Two Dimensional Grid
We consider the flow of an incompressible couple stress fluid behind a two-dimensional grid with grid spacing l. To obtain a desired solution for this geometry, we assume , and where V is the uniform velocity and  is a constant. Introducing the above values of A 1 and A 2 and in equations (15)

Reverse Flow above a Plate
Consider steady flow of an incompressible couple stress fluid above a plate with suction at y = 0. In order to study the behavior of the fluid above the plate it is assumed that A 1 =0, A 2 =V and C=-σ 2 <0, where V is the uniform velocity and σ is a constant. Employing these values in equations (15)

Flow due to Stretching Plate
We consider the two-dimensional boundary layer flow of a couple stress fluid above a stretching surface placed at y = 0. The application of an external force along the xdirection results in stretching of the plate and hence the flow. The boundary conditions for the flow situation are:

Flow into a Corner
The flow of an incompressible couple stress fluid along a corner is considered with suction at both the walls which are placed perpendicular to each other. The relevant boundary conditions are (47) The constants A 1 , A 2 and C in (15)

Conclusions
Some exact solutions have been investigated for two dimensional equations of motion of the incompressible couple stress fluid by assuming vorticity as a function of stream. This study shows that each solution is strongly dependent on parameter β. Solutions for the stream functions and velocity distributions studied in this work demonstrate a good agreement with the already existing solutions for viscous fluid found in literature [10].