Finite difference code for velocity and surface traction of a Fluid between Two Eccentric Spheres

The Numerical study of the flow of a fluid in the annular region between two eccentric sphere susing PHP Code is investigated. This flow is created by considering the inner sphere to rotate with angular velocity 1  and the outer sphere rotate with angular velocity 2  about the axis passing through their centers, the z-axis, using the three dimensional Bispherical coordinates ) , , (    .The velocity field of fluid is determined by solving equation of motion using PHP Code at different cases of angular velocities of inner and outer sphere. Also Finite difference code is used to calculate surface tractions at outer sphere.


Introduction
The determination of the rheological properties of fluids depends, in general, on solution of suitable boundary value problems based on a specific rheological model that represents the fluid under consideration. The theoretical and experimental studies concerning the flow of viscous or viscoelastic fluids in the annular narrow gaps between two rotating bodies are very interesting boundary value problems in rheology. These problems represent the keystone in the high developing today industries and technology such as the flow in rotation turbo machinery, in journal-bearing lubrication, socket joints, petroleum and so on.
One of these problems, for two concentric spheres the numerical and experimental studies are carried out, say by Wimmer [1] and Yamaguchi et. al. [2,3,4] . A large number of theoretical and experimental works are done on the viscous flow between two eccentric spheres; Jeffery [5], Stimson and Jeffery [6], Majumdar [7], Munson [8], Menguturk and Munson [9]. The analytical study of the flow of viscoelastic fluid between two eccentric spheres is investigated by Abu-El Hassan et .al. [10,11] . Force and torque at outer stationary sphere are studied using neural network system and genetic programming by Mostafa Y.Elbakry et .al. [12,13].
Finite difference method one of the important numerical methods in solving many problems in newtonian and nonnewtonian flow in fluid mechanics [14][15][16][17][18].
The present work is concerned with the numerical solution of this boundary value problem using PHP programming. The velocity field of a fluid between two eccentric spheres is investigated. Also surface tractions at outer stationary sphere is calculated using Finite difference code.

Formulation of the problem
A viscous fluid is assumed to perform steady and isochoric motion in the annular region between two eccentric spheres . This flow is created by considering the inner sphere to rotate with angular velocity  about the axis connecting their centers, the z-axis, while the outer sphere is kept at rest.
where the components The equation of continuity , is satisfied identically if u and v are derivable from a stream function  by the expression Or in the compact form we can write The Cauchy dynamical equation of motion for a stationary flow is being where T is the stress tensor, E T is the extra stress tensor,  is the hydrostatic pressure function and  is the density of the fluid .
The constitutive equation for a viscous fluid is stated as follow where  is the coefficient of viscosity.

Substitution from equations (1)and (3b)into equation (5)the extra stress tensor becomes
Substitution from equation (6) into equation (4) we get two equations of velocity components, The axial velocity satisfies the harmonic equation, with the boundary conditions , This boundary value problem has the solution which determined by finite difference iteration method.
On the other hand, the stream function  satisfies the boundary value problem, with the boundary conditions 1 0 2 , It can be easily shown that the only solution for the boundary value problem defined by equations (9) and (10) is the trivial The velocity field reduces to, 3Surface Tractions at outer stationary sphere The surface traction at the boundary 2    .when the inner sphere rotate with angular velocity  and the outer sphere at rest, is defined by This expression describes the stress vector per unit area on the surface of a spherical shell 2    The constitutive equation for a fluid of grade two is defined by the relation, [21], Where, The velocity field, as expressed by equation (11) where Substituting from equations (16), (17) and (18) where The surface traction at the stationary outer sphere, 2    in equations (22) and (23) can be normalized as follows

4.1Numerical calculation:
In our numerical calculation we used the finite differences schema to handle this boundary value problem, this schema is Where h and k are the increments in α and β directions and i,j =0,1,2…….n We used the PHP code to carry the numerical calculation, we found this code is useful and very fast to perform the calculations.
The following algorithm can be written in a different programing languages

5Results and discussion
The present work represents a numerical investigation of the isochoric and isothermal flow of a viscous fluidin the annular region between two eccentric spheres using finite difference method. The results show that the axial component,W(  , )is determined for different cases of angular velocities of two spheres, while the planar secondary velocity field U vanishes.
The surface tractions at stationary sphere is obtained.

We have different cases of flow as follows:
Case (1)      and distance between two poles c=4cm.

Surface tractions
We have two components of normalized surface tractions 