Study of Second Grade Fluid over a Rotating Disk with Coriolis and Centrifugal Forces

Rotating disk flow plays an important role in the field of engineering and industry. Centrifugal pumps are extensively used in petroleum industry to transport high viscosity fluids such as waxy crude oils. The first study of rotating disk was introduced by Von Karman in 1921 [1]. He was able to found that the rotating disk flow is a type of boundary layer flow and there is no depend of radial distance on the boundary thickness. In 1934, cochran [2], provided asymptotic solution to the ordinary differential equations derived by Von Karman. Although the analysis was simple but valuable in the field of rotating field. The work of cochran was extended by Benton [3], in 1966. He provided better solutions and solved the unsteady problem.


Introduction
Rotating disk flow plays an important role in the field of engineering and industry. Centrifugal pumps are extensively used in petroleum industry to transport high viscosity fluids such as waxy crude oils. The first study of rotating disk was introduced by Von Karman in 1921 [1]. He was able to found that the rotating disk flow is a type of boundary layer flow and there is no depend of radial distance on the boundary thickness. In 1934, cochran [2], provided asymptotic solution to the ordinary differential equations derived by Von Karman. Although the analysis was simple but valuable in the field of rotating field. The work of cochran was extended by Benton [3], in 1966. He provided better solutions and solved the unsteady problem.
In recent years, much attention has been given to the rotating flow of non-Newtonian fluids concerning to its applications in industries. The steady flow of non-Newtonian fluid over rotating disk with uniform suction was considered by Mithal [4], in 1961. His solutions were valid for small values of non-Newtonian problems.
Later, Attia [5] in 2003 extended the idea of Mithal and studied the same problem to the transient state with heat transfer. Their solutions were valid for the whole range of the parameters. In addition the reader may consult [6,7] for the studies of non-Newtonian fluids.
Boundary layer flow equations are developed by Reynolds number Re → ∞ in the boundary layer region combine with the use of the order of ε 2 Re≈1, where ε 2 → 0. A challenging mathematical model is developed with the nonlinearity in the term involving maximum order derivation. Most of analytic methods such as Adomian Decomposition Method, Differential Transform Method, Variation Iterative Method and Optimal Homotapy Asymptotic Methods fails to solve this problem. We handle this problem by HAM BVPh2.0 package [8] using 20thorder of approximations.
The following strategy is applied to the rest of the paper. In section 3 the basic governing equations for the motion are formulated in cylindrical coordinates. Section 4 is the solution by homotopy analysis method. Section 5 is the error analysis. Section 6 contains the numerical results and their discussion for different values of physical parameters. Finally, our conclusion follows in section 7.

Formulation of the Problem
Let us consider the steady incompressible flow of a Rivlin-Erickson type fluid produced by the rotation of an insulated disk of radius R with angular speed Ω and radial stretching. The disk is stretching in radial direction which has a velocity u ω (r). The co-ordinate system (r,θ,z) is adopted whose origin is taken at the center of the disk. In which r-axis is along the radius of the disk, z-axis is perpendicular to the disk and θ is oriented in the direction of rotation. Assuming flow is laminar, axisymetric and its density ρ, is constant.

Boundary conditions
Due to no penetration the value of w  vanishes near the surface of the insulated disk. The tangential velocity v  have a value Ωr at the disk surface. The position vector is given by The velocity vector is The basic equations governing the flow of second grade fluid is the continuity equation and Navier-Stokes equations (NSE)  is shear stress, r  is the position vector V  is velocity vector and w  is the rotational vector given as After making use of assumptions and shear stress eqns. (10-12) takes the form Radial co-ordinate is scaled on a characteristic length along the radius of disk R, and the surface normal co-ordinate is further scaled on boundary layer thickness The radial and azimuthal velocity components are nondimensionalized using the local surface velocity ΩR, and the axial velocity component is non-dimensionalized with the local boundary layer angular velocity ( Figure 1).
where r is the radius, θ is the angle and z is a wall normal co-ordinate, , , u v w    are radial, azimuthal and axial velocity components respectively. The continuity equation for incompressible fluid is Where 1 is the material differential term. The Coriolis force terms 2 given as The pressure term 4 in component form is and 5 is stress tensor in a second grade fluid given as Where α 1 α 1 are material constants and 1 A   and 2 A  are Rivlin-Ericksen tensors, given as 1 ; .
The continuity equation in cylindrical coordinates has the form ( ) Momentum equations in cylindrical co-ordinates are r-component of momentum equation: θ-component of momentum equation: We scale the pressure as Using eqns. (16)(17)(18)(19) and making use of ( ) , the continuity equation (9) and momentum equations (13)(14)(15), after dropping asterisks, we obtain ( ) The terms 2 v ρ δ ν Ω are the projections of the Coriolis onto the axis r and θ, respectively, while the term 2 r ρ δ ν Ω is the projection of centrifugal force. where f(z) and g(z) can be considered as dimensionless velocities onto the r-axis depends only on z.
Boundary conditions in non-dimensional form are where A γ = Ω is the stretching parameter. To convert the partial differential equation into ordinary differential equation we make the following transformations.
The continuity equation satisfied identically and momentum equations take the following form after algebraic manipulation [ ] Boundary conditions becomes is Ekman number and S is the height.
The model applies strictly to an infinite disk, but can be applied to a finite disk of radius R, provided that R>δ is satisfied.

Solution by Homotopy Analysis Method
By the HAM method, the functions f(z)and g(z)as: where c i (i=1-5) are arbitrary constants. The zeroth order deformation problems can be obtain as: ; ; 2  3  2  2  3  2  3   3  3  3  2  2  3  2  3  3  3   3  3  3  2  2  3   ; ; where q is an embedding parameter, h f and h g are the non-zero auxiliary parameter and N f and â€¢ N are nonlinear operators. For q=0 and q=1 we have: Therefore, as the embedding parameter q increases from 0 to 1,

( )
; f z q and ( ) ; g z q varies from their initial guesses f 0 and g 0 to the exact solutions f(z) and g(z)respectively. Taylor's series expansion of these functions yields:  3  3  1  2  3  ,  3  3  0   2  3  3  1  2  3  2  3  0   3  3  3  2  3  1  3  3  0   3  3  3  2  1  2  3  3  3 ε can be minimized by increasing the order of approximations. Here, it can be seen that increasing the order of approximations the total squared residual errors are reduced. Table  2 illustrate the individual average squared residual error at different orders of approximations. Besides this Figure 2 also shows the maximum average squared residual error at different orders of approximation. It   can also be observed that the total averaged squared errors and average squared residual errors are decreasing as the order of approximation is increasing for different values of physical parameters α and β.

Results and Discussion
In this section, we present graphical results of the system of coupled nonlinear ODE's given in eqn. (26) and eqn. (27) corresponding the boundary conditions (28). Numerical Solution is obtained by means of the BVPh2.0, a HAM Mathematica package [9,10]. For better analysis,   To investigate the fluid velocity along azimuthal direction with and without Coriolis and centrifugal force Figure 9 is made. Near the disk there is no effect on the velocity component f and far away these forces effect f (z), the combine effect of these forces clearly effect the velocity field f. As expected, near the disk the Coriolis and centrifugal forces balance the effect of each other [16][17][18][19][20][21][22][23][24][25].  It is interesting to note that the radial velocity increases by taking the effect of these forces. The effect of centrifugal force near the surface of the disk is seems to be negligible and far away from the surface of the rotating disk its effect can be seen clearly. The influence of different values of viscoelastic parameter α=0.1 0.5, 0.8 for fixed values of β=0. 5 and γ=0.01 on radial velocity component are plotted in Figure 14. It is also interesting to note that increasing the parameter α the azimuthal velocity component also increases but for small values of β and γ this increase is small in magnitude [26][27][28][29][30][31][32]. Table 3     2. It is also concluded that increasing the rotation number β, the radial and axial components of velocity profile also increases.
3. Further more, increasing slip effect causes the radial velocity component to increase.
4. Moreover, the radial and axial velocity component increase by taking the effect of centrifugal and Coriolis forces in the momentum equations. These finding abstruse and enrich our understanding about the boundary layer flow of second grade fluid.