Analysis of Third-grade Fluid in Helical Screw Rheometer

The steady flow of an incompressible, third-grade fluid in helical screw rheometer (HSR) is studied by " unwrapping or flattening " the channel, lands, and the outside rotating barrel. The geometry is approximated as a shallow infinite channel, by assuming that the width of the channel is large as compared to the depth. The developed second-order nonlinear coupled differential equations are reduced to single differential equation by using a transformation. Using Adomian decomposition method, analytical expressions are calculated for the the velocity profiles and volume flow rates. The results have been discussed with the help of graphs as well. We observed that the velocity profiles are strongly dependant on non-Newtonian parameter (̃ í µí»½), and with the increase in ̃ í µí»½, the velocity profiles increase progressively, which conclude that extrusion process increases with the increase in ̃ í µí»½. We also observed that the increase in pressure gradients in x-and z-direction increases the net flow inside the helical screw rheometer, which increases the extrusion process. We noticed that the flow increases as the flight angle increase.


Introduction
In real life, there are many materials that exhibit the mechanical characteristics of both elasticity and viscosity.These materials are known as non-Newtonian fluids.These fluids cannot be described satisfactorily by the theory of elasticity or viscosity but by a combination of both.Due to the rheological behavior of these fluids, many constitutive equations are proposed [1].In most fluid food products, the shear stress is dependent on the share rate; hence, nonlinear flow curve results and a unique viscosity are no longer adequate to characterize the fluid.Many fluids such as molten plastics, polymers, and slurries are non-Newtonian in their flow behavior.The basic governing equations for such fluids motion are highly nonlinear differential equations having no general solution, and only a limited number of exact solutions have been established for particular problems.To solve practical problems in engineering and mathematics, researchers and scientists have developed numerous numerical techniques, that is, finite difference method (FDM), finite volume approach, control-volume-based finite element method (CVFEM), lattice Boltzmann method (LBM), and analytical techniques, that is, variational iteration method (VIM), perturbation method (PM), homotopy perturbation method (HPM), HPM-Pade technique, homotopy analysis method (HAM), optimal homotopy analysis method (OHAM), optimal homotopy perturbation method (OHPM), and some other techniques, to overcome nonlinearity and get numerical and analytical solutions [2][3][4][5][6][7][8][9][10].A brief review on analytical techniques is presented by [11].In recent years in the area of series solutions, an iterative technique Adomian decomposition method [12,13] has received much attention.A considerable amount of research work has been invested in the application of this method to a wide class of linear, nonlinear, and partial differential equations and integral equations.Many interesting problems in applied science and engineering have been successfully solved by using ADM to their higher degree of accuracy.A useful quality of the ADM is that it has proved to be a competitive alternative to the Taylor series method and other series techniques.This method has been used in obtaining analytic and approximate solutions to a wide class of linear and nonlinear, differential and integral equations, homogeneous or inhomogeneous, with constant coefficients or with variable coefficients.

Basic Equations
The basic equations governing the motion of an incompressible fluid are div V = 0, where  is the constant fluid density, V is the velocity vector, f is the body force per unit mass, the / denotes the material time derivative defined as and T is the Cauchy stress tensor, given as where  denotes the dynamic pressure, I denotes unit tensor, and S denotes the extra stress tensor.The constitutive equation for third-grade fluid is defined as where  is the viscosity,  1 ,  2 ,  1 ,  2 , and  3 are the material constants, A 1 , A 2 , and A 3 are the first three Rivlin-Ericksen tensors defined as [19] A 1 = (grad V) + (grad V)  ,

Problem Formulation
Consider the steady flow of an isothermal, incompressible and homogeneous third-grade fluid in helical screw rheometer (HSR) in such a way that the curvature of the screw channel is ignored, unrolled and laid out on a flat surface.The barrel surface is also flattened.Assume that the screw surface, the lower plate, is stationary, and the barrel surface, the upper plate, is moving across the top of the channel with velocity  at an angle  to the direction of the channel Figure 1.The phenomena is the same as the barrel held stationary and the screw rotates.The geometry is approximated as a shallow infinite channel, by assuming that the width  of the channel is large compared with the depth ℎ; edge effects in the fluid at the land are ignored.The coordinate axes are positioned in such a way that the -axis is perpendicular to the wall and -axis is in down channel direction.The liquid wets all the surfaces and moves by the shear stresses produced by the relative movement of the barrel and channel.For simplicity, the velocity of the barrel relative to the channel is broken up into two components:  is along -axis and  is along -axis The geometry of the "unwrapped" screw channel and barrel surface.[24].Under these assumptions the velocity field and cauchy stress tensor can be written as On substituting (7) in ( 4) and (5), we obtain nonzero components of Cauchy stress T, where Using (7), ( 1) is identically satisfied, and (2) in the absence of body forces results in Define the modified pressure P as which implies that P = P(, ) only; thus, (9) reduce to The associated boundary conditions can be taken as (see Figure 1) where Introducing nondimensionalized parameters in ( 11)-( 12), takes the form Dropping " * " from ( 15) onward and defining where / =  , , / =  , in (15) reduce to where  is the complex conjugate of .
The boundary conditions become Equation ( 17) is second-order nonlinear ordinary differential equation, and the exact solution seems to be difficult.In the following section, we use Adomian decomposition method to obtain the approximate solution.To obtain the expressions for the velocity components in and -directions, (17) together with the boundary conditions ( 18) is solved up to the second component approximations by using the symbolic computation software Wolfram Mathematica 7.

Description of Adomian Decomposition Method
Consider equation [()] = (), where  represents a general nonlinear ordinary or partial differential operator including both linear and nonlinear terms.The linear terms are decomposed into  + , where  is invertible. is taken as the highest-order derivative to avoid difficult integrations, and  is the remainder of the linear operator.Thus, the equation can be written as where () indicates the nonlinear term and () is forcing function.Since  is invertible, so  −1 exist.The above equation can be written as If (=  2 / 2 ) is a second-order operator,  −1 (= ∫ ∫( * ) ) is a twofold indefinite integral.Equation ( 20) becomes (21) where  1 and  2 are constants of integration and can be determined by using boundary or initial conditions.ADM assumes that the solution  can be expanded into infinite series as  = ∑ ∞ =0   ; also, the nonlinear term () will be written as () = ∑ ∞ =0   , where   are special Adomian polynomials which can be defined as finally, the solution can be written as where is initial solution and and ( 25) is th-order solution.The practical solution will be the -term approximation and by definition [25][26][27],

Solution of the Problem
Adomian decomposition method describes that in the operator form (17) can be written as where  is the differential operator taken as the highest-order derivative to avoid difficult integrations, assuming that  is invertible, which implies that  −1 = ∫ ∫( * )  exist.
On applying  −1 to both sides of (28) results in where  and  are constants of integration and can be determined by using boundary conditions.According to procedure of Adomian decomposition method  and  can be written in component form as: Thus, (29) takes the form Adomian also suggested that the nonlinear terms can be explored in the form of Adomian polynomials, say,   and   as Equation (31) yields The associated boundary conditions (18) will be The recursive relation in (33) and (34) gives the component problems along with boundary conditions +1 = − β −1 (  +   ) ,  ≥ 0 (37) together with the boundary conditions The ADM solution to (33) along with the boundary conditions (34) will be and the boundary conditions are which gives the solution which are the linearly viscous solutions to the problem.

Second Component Solution.
The relations (37) and (38) give where (58) Volume flow rate in -direction per unit width is where  *  =   /ℎ, and (59) gives Equation ( 56) gives the resultant volume flow rate forward in the screw channel, which is the product of the velocity and cross-sectional area integrated from the root of the screw to the barrel surface where  * = /ℎ and  is the number of parallel flights in a multiflight screw.Equation (61) gives which can be written as

Results and Discussion
In the present work, we have considered the steady flow of an incompressible, isothermal, and homogeneous thirdgrade fluid in helical screw rheometer (HSR).Using Adomian decomposition method, solutions are obtained for velocity profiles in -, -directions and also in the direction of the axis of the screw ().The volume flow rates are also calculated by using the velocities in ,  and in the direction of the axis of the screw.Here we discussed the effect of dimensionless parameters β, / = − tan ,  , , and  , where  = 45 ∘ , on the velocity profiles given in ( 53), (54), and (56) with the help of graphical representation.Figures 2, 3, and 4 for the velocities in -direction (), -direction (), and the resultant velocity () are plotted against  for different values of non-Newtonian parameter β and constant pressure gradients  , = −2.0, , = −2.0,respectively.From these figures, it is seen that the velocity profiles are strongly dependant on the non-Newtonian parameter β, as we increase the value of β in the interval 0 to 0.9, the progressive increase in velocities in -, -direction and in the direction of the axis of screw found.
It is worthwhile to note that the extrusion process increases with the increase of the non-Newtonian parameter β. is noted in the velocity profiles with increase in pressure gradient in -direction.Figures 6, 8, and 10 are sketched for the velocity profiles (), (), and () against  for different values of  , , keeping β = 0.3 and  , = −2.0fixed, it is observed that parabolicity of the velocity profiles increases with increase in pressure gradient in -direction.

Conclusion
The steady flow of an isothermal, homogeneous and incompressible third-grade fluid is investigated in helical screw rheometer (HSR).The geometry of the problem under consideration gives second-order nonlinear coupled differential equations which are reduced to single differential equation by using a transformation.Adomian decomposition method is used to obtain analytical expressions for the flow profiles, volume flow rate.It is noticed that the zeroth component solution matches with solution of the linearly viscous fluid in HSR, and it is also found that the net velocity of the fluid is due to the pressure gradient as the expression for the net velocity is free from the drag term.Graphical representation shows that the velocity profiles are strongly dependant on non-Newtonian parameter ( β) and pressure gradients in and -direction.Thus, the extrusion process strongly depends on the involved parameters.

Appendix
Figure 1: The geometry of the "unwrapped" screw channel and barrel surface.

Figure 11 is
plotted for the velocity () against  for different values of , keeping β = 0.3,  , = −2.0,and  , = −2.0.It is observed that flow increases as the flight angle increases up to  = 45 ∘ .