Analysis of Third-Grade Fluid in Helical Screw Rheometer

Abstract

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 (𝛽), and with the increase in 𝛽, the velocity profiles increase progressively, which conclude that extrusion process increases with the increase in 𝛽. 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.

1. 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 [2–10]. 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. The Adomian decomposition method is comparatively easier to program in engineering problems than other series methods and provides immediate and visible solution terms without linearization, perturbation, or discretization of the problem, while the physical behavior of the solution remains unchanged. It provides analytical solution in the form of an infinite series in which each term can be easily determined [14–16]. If an exact solution exists for the problem, then the obtained series converges very rapidly to the solution. For concrete problems, where a closed-form solution is not obtainable, a truncated number of terms are usually used for numerical purposes [17].

The Helical Screw Rheometer (HSR) consists of a helical screw in a tight fitting cylinder, with the inlet and outlet parts closing the inner screw. Rotation of screw creates a pressure gradient along the axis of the screw. The HSR is being used for rheological measurements of fluid food suspensions. The geometry of an HSR is similar to a single-screw extruder [18]. Extrusion process is widely used in multigrade oils, liquid detergents, paints, polymer solutions and polymer melts [19], the injection molding process for polymeric materials, the production of pharmaceutical products, food extrusion, and processing of plastics [20]. Various food items in daily life, such as cookie dough, sevai, pastas, breakfast cereals, french fries, baby food, ready to eat snacks, and dry pet food, are most commonly manufactured using the extrusion process.

Knowledge of rheological properties is essential in the processing of fluid foods since these affect the flow behavior. During processing, physical and chemical changes can occur so it is desirable to monitor the process to achieve excellent output and quality control [21]. On-line rheological measurements in the food industry have been limited [22].

Bird et al. [23] presented an asymptotic solution and arbitrary values of the flow behavior index, for the power-law fluid in a very thin annulus. A brief discussion is given by Mohr and Mallouk [24] for the same problem considering Newtonian fluid in a screw extruder. Tamura et al. [18] also investigated the flow of Newtonian fluid in helical screw rheometer.

The objective of this paper is to study the flow of third-grade fluid in helical screw rheometer (HSR) where the effects of curvature and also of flights are neglected by assuming that the helical channel is “unwrapped.” The geometry is approximated as a shallow infinite channel, with ℎ/𝐵≪1, where 𝐵 denotes the channel width and ℎ is gap [18]. The formulation results in second-order nonlinear coupled differential equations which are reduced to first-order nonlinear differential equations by integrating and combined in single first-order differential equation using a transformation, the solution is obtained by using ADM. Analytical expressions are given for the velocity components in 𝑥-, 𝑧-directions and in direction of the screw axis. Volume flow rates are also obtained for all three types of velocities. The paper is organized as follows. Section 2 contains the governing equations of the fluid model. In Section 3, the problem under consideration is formulated. In Section 4, description of Adomian decomposition method is given. In Section 5, the governing equation of the problem is solved. In Section 6, results are discussed. Section 7 contains conclusion.

2. Basic Equations

The basic equations governing the motion of an incompressible fluid are 𝜌div𝐕=0,𝐷𝐕𝐷𝑡=𝜌𝐟+div𝐓, (1)(2) where 𝜌 is the constant fluid density, 𝐕 is the velocity vector, 𝐟 is the body force per unit mass, the 𝐷/𝐷𝑡 denotes the material time derivative defined as 𝐷(∗)=𝜕𝐷𝑡(∗)+(∗),𝜕𝑡𝐕⋅∇)( (3) and 𝐓 is the Cauchy stress tensor, given as 𝐓=−𝑃𝐈+𝐒, (4) where 𝑃 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 𝐒=𝜇𝐀₁+𝛼₁𝐀₂+𝛼₂𝐀²₁+𝛽₁𝐀₃+𝛽₂𝐀₁𝐀₂+𝐀₂𝐀₁+𝛽₃tr𝐀²₁𝐀₁, (5) where 𝜇 is the viscosity, 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂, and 𝛽₃ are the material constants, 𝐀₁, 𝐀₂, and 𝐀₃ are the first three Rivlin-Ericksen tensors defined as [19] 𝐀₁=grad𝐕+grad𝐕^𝑇,𝐀_𝑛+1=𝐷𝐀_𝑛+𝐀𝐷𝑡^𝑛grad𝐕+grad𝐕^𝑇𝐀_𝑛,(𝑛=1,2). (6)

3. 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 𝑉 at an angle 𝜙 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 𝐵 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 𝑥-axis is perpendicular to the wall and 𝑧-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: 𝑈 is along 𝑥-axis and 𝑊 is along 𝑧-axis [24]. Under these assumptions the velocity field and cauchy stress tensor can be written as 𝐕=𝑢𝑦,0,𝑤𝑦,𝐓=𝐓𝑦. (7) On substituting (7) in (4) and (5), we obtain nonzero components of Cauchy stress 𝐓, 𝜏_𝑥𝑥=−𝑃+𝛼₂𝑑𝑢𝑑𝑦²,𝜏_𝑥𝑦=𝜏_𝑦𝑥=𝜇𝑑𝑢𝛽𝑑𝑦+2₂+𝛽₃𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑢,𝜏𝑑𝑦_𝑥𝑧=𝜏_𝑧𝑥=𝛼₂𝑑𝑢𝑑𝑦𝑑𝑤,𝜏𝑑𝑦_𝑦𝑦=−𝑃+2𝛼₁+𝛼₂𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²,𝜏_𝑦𝑧=𝜏_𝑧𝑦=𝜇𝑑𝑤𝛽𝑑𝑦+2₂+𝛽₃𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑤,𝜏𝑑𝑦_𝑧𝑧=−𝑃+𝛼₂𝑑𝑤𝑑𝑦², (8) where 𝐓=[𝜏_𝑖𝑗].

Figure 1 

The geometry of the “unwrapped” screw channel and barrel surface.

Using (7), (1) is identically satisfied, and (2) in the absence of body forces results in 0=−𝜕𝑃+𝜕𝜕𝑥×𝜇𝜕𝑦𝑑𝑢𝛽𝑑𝑦+2₂+𝛽₃𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑢,𝑑𝑦0=−𝜕𝑃+1𝜕𝑦22𝛼₁+𝛼₂𝜕𝜕𝑦𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²,0=−𝜕𝑃+𝜕𝜕𝑧×𝜇𝜕𝑦𝑑𝑤𝛽𝑑𝑦+2₂+𝛽₃𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑤.𝑑𝑦 (9) Define the modified pressure 𝑃 as 𝑃=𝑃−𝛼₁+12𝛼₂𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦², (10) which implies that 𝑃=𝑃(𝑥,𝑧) only; thus, (9) reduce to 𝑑²𝑢𝑑𝑦²+2𝛽₂+𝛽₃𝜇𝑑𝑑𝑦𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑢=1𝑑𝑦𝜇𝜕𝑃,𝑑𝜕𝑥₂𝑤𝑑𝑦²+2𝛽₂+𝛽₃𝜇𝑑𝑑𝑦𝑑𝑢𝑑𝑦²+𝑑𝑤𝑑𝑦²𝑑𝑤=1𝑑𝑦𝜇𝜕𝑃.𝜕𝑧 (11) The associated boundary conditions can be taken as (see Figure 1) 𝑢=0,𝑤=0,at𝑦=0,𝑢=𝑈,𝑤=𝑊,at𝑦=ℎ, (12) where 𝑈=−𝑉sin𝜙,𝑊=𝑉cos𝜙. (13) Introducing nondimensionalized parameters 𝑥^∗=𝑥ℎ,𝑦^∗=𝑦ℎ,𝑧^∗=𝑧ℎ,𝑢^∗=𝑢𝑊,𝑤^∗=𝑤𝑊,𝑃^∗=𝑃𝜇(),𝛽𝑊/ℎ^∗=𝛽₂+𝛽₃𝑊²𝜇ℎ², (14) in (11)-(12), takes the form 𝑑²𝑢^∗𝑑𝑦^∗2+𝛽^∗𝑑𝑑𝑦^∗𝑑𝑢^∗𝑑𝑦^∗2+𝑑𝑤^∗𝑑𝑦^∗2𝑑𝑢^∗𝑑𝑦^∗=𝜕𝑃^∗𝜕𝑥^∗,𝑑²𝑤^∗𝑑𝑦^∗2+𝛽^∗𝑑^∗𝑑𝑦^∗𝑑𝑢^∗𝑑𝑦^∗2+𝑑𝑤^∗𝑑𝑦^∗2𝑑𝑤^∗𝑑𝑦^∗=𝜕𝑃^∗𝜕𝑧^∗,𝑢^∗=0,𝑤^∗=0,at𝑦^∗𝑢=0,_∗=𝑈𝑊,𝑤^∗=1,at𝑦^∗=1. (15) Dropping "∗" from (15) onward and defining 𝐹=𝑢+𝜄𝑤,𝑉₀=𝑈𝑊+𝜄1,𝐺=𝑃_,𝑥+𝜄𝑃_,𝑧, (16) where 𝜕𝑃/𝜕𝑥=𝑃_,𝑥, 𝜕𝑃/𝜕𝑧=𝑃_,𝑧 in (15) reduce to 𝑑²𝐹𝑑𝑦²=𝐺−𝛽𝑑𝐹𝑑𝑦²𝑑²𝐹𝑑𝑦²+2𝑑𝐹𝑑𝑑𝑦²𝐹𝑑𝑦²𝑑𝐹,𝑑𝑦 (17) where 𝐹 is the complex conjugate of 𝐹.

The boundary conditions become 𝐹=0at𝑦=0,𝐹=𝑉₀at𝑦=1. (18) 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 𝑥- and 𝑧-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.

4. Description of Adomian Decomposition Method

Consider equation 𝐽[𝐹(𝑦)]=𝑔(𝑦), where 𝐽 represents a general nonlinear ordinary or partial differential operator including both linear and nonlinear terms. The linear terms are decomposed into 𝐿+𝑅, where 𝐿 is invertible. 𝐿 is taken as the highest-order derivative to avoid difficult integrations, and 𝑅 is the remainder of the linear operator. Thus, the equation can be written as 𝐿(𝐹)+𝑅(𝐹)+𝑁(𝐹)=𝑔𝑦, (19) where 𝑁(𝐹) indicates the nonlinear term and 𝑔(𝑦) is forcing function. Since 𝐿 is invertible, so 𝐿⁻¹ exist. The above equation can be written as 𝐿⁻¹𝐿(𝐹)=𝐿⁻¹𝑔𝑦−𝐿⁻¹𝑅(𝐹)−𝐿⁻¹𝑁(𝐹). (20) If 𝐿(=𝑑²/𝑑𝑦²) is a second-order operator, 𝐿⁻¹(=∫∫(∗)𝑑𝑦𝑑𝑦) is a twofold indefinite integral. Equation (20) becomes 𝐹=𝐶₁+𝐶₂𝑦+𝐿⁻¹𝑔𝑦−𝐿⁻¹𝑅(𝐹)−𝐿⁻¹𝑁(𝐹), (21) where 𝐶₁ and 𝐶₂ are constants of integration and can be determined by using boundary or initial conditions. ADM assumes that the solution 𝐹 can be expanded into infinite series as 𝐹=∑^∞_𝑛=0𝐹_𝑛; also, the nonlinear term 𝑁(𝐹) will be written as 𝑁(𝐹)=∑^∞_𝑛=0𝐴_𝑛, where 𝐴_𝑛 are special Adomian polynomials which can be defined as 𝐴_𝑛=1𝑑𝑛!𝑑𝜆^𝑛𝑛𝑁^∞_𝑖=0𝜆^𝑖𝐹_𝑖𝜆=0𝑛=0,1,2,3,…, (22) finally, the solution can be written as ^∞_𝑛=0𝐹_𝑛=𝐹₀−𝐿⁻¹𝑅^∞_𝑛=0𝐹_𝑛−𝐿^−1∞_𝑛=0𝐴_𝑛, (23) where 𝐹₀=𝐶₁+𝐶₂𝑦+𝐿⁻¹𝑔𝑦 (24) is initial solution and 𝐹_𝑛=−𝐿⁻¹𝑅𝐹_𝑛−1−𝐿⁻¹𝐴_𝑛−1,𝑛≥1, (25) and (25) is 𝑛th-order solution. The practical solution will be the 𝑛-term approximation 𝜙_𝑛=^𝑛−1_𝑖=0𝐹_𝑖, (26) and by definition [25–27], lim_𝑛→∞𝜙_𝑛=^𝑛−1_𝑖=0𝐹_𝑖=𝐹. (27)

5. Solution of the Problem

Adomian decomposition method describes that in the operator form (17) can be written as 𝐿(𝐹)=𝐺−𝛽𝑑𝐹𝑑𝑦²𝑑²𝐹𝑑𝑦²+2𝑑𝐹𝑑𝑑𝑦²𝐹𝑑𝑦²𝑑𝐹,𝑑𝑦 (28) where 𝐿 is the differential operator taken as the highest-order derivative to avoid difficult integrations, assuming that 𝐿 is invertible, which implies that 𝐿⁻¹=∫∫(∗)𝑑𝑦𝑑𝑦 exist.

On applying 𝐿⁻¹ to both sides of (28) results in 𝐹=𝐴+𝐵𝑦+𝐿⁻¹(𝐺)−𝛽𝐿⁻¹×𝑑𝐹𝑑𝑦²𝑑²𝐹𝑑𝑦²+2𝑑𝐹𝑑𝑑𝑦²𝐹𝑑𝑦²𝑑𝐹,𝑑𝑦 (29) where 𝐴 and 𝐵 are constants of integration and can be determined by using boundary conditions. According to procedure of Adomian decomposition method 𝐹 and 𝐹 can be written in component form as: 𝐹=^∞_𝑛=0𝐹_𝑛,𝐹=^∞_𝑛=0𝐹_𝑛. (30) Thus, (29) takes the form ^∞_𝑛=0𝐹_𝑛=𝐴+𝐵𝑦+𝐿⁻¹(𝐺)−𝛽𝐿⁻¹×𝑑𝑑𝑦^∞_𝑛=0𝐹_𝑛²𝑑²𝑑𝑦^2∞_𝑛=0𝐹_𝑛𝑑+2𝑑𝑦^∞_𝑛=0𝐹_𝑛𝑑²𝑑𝑦^2∞_𝑛=0𝐹_𝑛×𝑑𝑑𝑦^∞_𝑛=0𝐹_𝑛. (31) Adomian also suggested that the nonlinear terms can be explored in the form of Adomian polynomials, say, 𝐴_𝑛 and 𝐵_𝑛 as ^∞_𝑛=0𝐴_𝑛=𝑑𝑑𝑦^∞_𝑛=0𝐹_𝑛²𝑑²𝑑𝑦^2∞_𝑛=0𝐹_𝑛,_∞𝑛=0𝐵_𝑛𝑑=2𝑑𝑦^∞_𝑛=0𝐹_𝑛𝑑²𝑑𝑦^2∞_𝑛=0𝐹_𝑛𝑑𝑑𝑦^∞_𝑛=0𝐹_𝑛. (32) Equation (31) yields ^∞_𝑛=0𝐹_𝑛=𝐴+𝐵𝑦+𝐿⁻¹(𝐺)−𝛽𝐿^−1∞_𝑛=0𝐴_𝑛+^∞_𝑛=0𝐵_𝑛. (33) The associated boundary conditions (18) will be ^∞_𝑛=0𝐹_𝑛=0,𝑦=0,_∞𝑛=0𝐹_𝑛=𝑉₀,𝑦=1. (34) The recursive relation in (33) and (34) gives the component problems 𝐹₀=𝐴+𝐵𝑦+𝐿⁻¹(𝐺), (35) along with boundary conditions 𝐹₀𝐹=0,𝑦=0,₀=𝑉₀𝐹,𝑦=1,_𝑗+1=−𝛽𝐿⁻¹𝐴_𝑗+𝐵_𝑗,𝑗≥0 (36)(37) together with the boundary conditions ^∞_𝑛=1𝐹_𝑛=0,𝑦=0,_∞𝑛=1𝐹_𝑛=0,𝑦=1. (38) The ADM solution to (33) along with the boundary conditions (34) will be 𝐹=^∞_𝑛=0𝐹_𝑛. (39)

5.1. Zeroth Component Solution

The relations (35) and (36) give the zeroth component problem 𝐹₀=𝐴+𝐵𝑦+𝐿⁻¹(𝐺), (40) and the boundary conditions are 𝐹₀𝐹=0at𝑦=0,₀=𝑉₀at𝑦=1, (41) which gives the solution 𝑢₀=𝑈𝑊1𝑦+2𝑃_,𝑥𝑦²,𝑤−𝑦₀1=𝑦+2𝑃_,𝑧𝑦²−𝑦, (42)(43) which are the linearly viscous solutions to the problem.

5.2. First Component Solution

Equations (37) and (38) give 𝐹₁=−𝛽𝐿⁻¹𝐴₀+𝐵₀,𝐹₁𝐹=0at𝑦=0,₁=0at𝑦=1, (44)(45) where the remainder term 𝑅 of the linear part is zero and 𝐴₀=𝑑𝐹₀𝑑𝑦²𝑑²𝐹₀𝑑𝑦²,𝐵₀=2𝑑𝐹₀𝑑𝑑𝑦²𝐹₀𝑑𝑦²𝑑𝐹₀𝑑𝑦 (46) are Adomian polynomials. Using (45)-(46) in (44) results in 𝑢₁=−𝛽𝐿₁₁𝑦²−𝑦+𝐿₁₂𝑦³−𝑦+𝐿₁₃𝑦⁴𝑤−𝑦,₁=−𝛽𝑇₁₁𝑦²−𝑦+𝑇₁₂𝑦³−𝑦+𝑇₁₃𝑦⁴−𝑦, (47)(48) where 𝐿₁₁, 𝐿₁₂, 𝐿₁₃, 𝑇₁₁, 𝑇₁₂, and 𝑇₁₃ are constants given in appendix.

5.3. Second Component Solution

The relations (37) and (38) give 𝐹₂=−𝛽𝐿⁻¹𝐴₁+𝐵₁,𝐹₂𝐹=0at𝑦=0,₂𝐴=0at𝑦=ℎ,₁=𝑑𝐹₀𝑑𝑦²𝑑²𝐹₁𝑑𝑦²+2𝑑𝐹₀𝑑𝑦𝑑𝐹₁𝑑𝑑𝑦²𝐹₀𝑑𝑦²,𝐵₁=2𝑑𝐹₀𝑑𝑑𝑦²𝐹₀𝑑𝑦²𝑑𝐹₁+𝑑𝑦𝑑𝐹₀𝑑𝑑𝑦²𝐹₁𝑑𝑦²𝑑𝐹₀+𝑑𝑦𝑑𝐹₁𝑑𝑑𝑦²𝐹₀𝑑𝑦²𝑑𝐹₀,𝑑𝑦 (49)(50) where 𝐴₁ and 𝐵₁ are Adomian polynomials.

Using (50) in (49), we get 𝑢₂=𝛽²𝐿₁₄𝑦²−𝑦+𝐿₁₅𝑦³−𝑦+𝐿₁₆𝑦⁴−𝑦+𝐿₁₇𝑦⁵−𝑦+𝐿₁₈𝑦⁶,𝑤−𝑦₂=𝛽²𝑇₁₄𝑦²−𝑦+𝑇₁₅𝑦³−𝑦+𝑇₁₆𝑦⁴−𝑦+𝑇₁₇𝑦⁵−𝑦+𝑇₁₈𝑦⁶−𝑦, (51)(52) where 𝐿₁₄, 𝐿₁₅, 𝐿₁₆, 𝐿₁₇, 𝐿₁₈, 𝑇₁₄, 𝑇₁₅, 𝑇₁₆, 𝑇₁₇, and 𝑇₁₈ are constants mentioned in appendix.

5.4. Velocity Profiles

5.4.1. Velocity Profile in 𝑥-Direction

Equations (42), (47), and (51) give the ADM solution for the velocity profile in the transverse plane 𝑈𝑢=𝑊1𝑦+2𝑃_,𝑥+𝛽𝐿₁₁+𝛽²𝐿₁₄𝑦²+−𝑦𝛽𝐿₁₂+𝛽²𝐿₁₅𝑦³−𝑦+𝛽𝐿₁₃+𝛽²𝐿₁₆𝑦⁴+𝛽−𝑦₂𝐿₁₇𝑦⁵−𝑦+𝛽²𝐿₁₈𝑦⁶−𝑦. (53)

5.4.2. Velocity Profile in 𝑧-Direction

Equations (43), (48) and (52) give the ADM solution for the velocity profile in the down channel direction 1𝑤=𝑦+2𝑃_,𝑧+𝛽𝑇₁₁+𝛽²𝑇₁₄𝑦²+−𝑦𝛽𝑇₁₂+𝛽²𝑇₁₅𝑦³−𝑦+𝛽𝑇₁₃+𝛽²𝑇₁₆𝑦⁴+𝛽−𝑦₂𝑇₁₇𝑦⁵−𝑦+𝛽²𝑇₁₈𝑦⁶−𝑦. (54)

5.4.3. Velocity in the Direction of the Axis of Screw

The velocity in the direction of the axis of the screw at any depth in the channel can be computed from (53) and (54) as 1𝑠=𝑤sin𝜙+𝑢cos𝜙,𝑠=𝑦+2𝑃_,𝑧+𝛽𝑇₁₁+𝛽²𝑇₁₄𝑦²+−𝑦𝛽𝑇₁₂+𝛽²𝑇₁₅𝑦³+−𝑦𝛽𝑇₁₃+𝛽²𝑇₁₆𝑦⁴−𝑦+𝛽²𝑇₁₇𝑦⁵+𝛽−𝑦₂𝑇₁₈𝑦⁶+𝑈−𝑦sin𝜙𝑊1𝑦+2𝑃_,𝑥+𝛽𝐿₁₁+𝛽²𝐿₁₄𝑦²+−𝑦𝛽𝐿₁₂+𝛽²𝐿₁₅𝑦³+−𝑦𝛽𝐿₁₃+𝛽²𝐿₁₆𝑦⁴−𝑦+𝛽²𝐿₁₇𝑦⁵+𝛽−𝑦₂𝐿₁₈𝑦⁶−𝑦cos𝜙, (55)(56) which shows the resultant velocity of the flow.

5.5. Volume Flow Rates

Volume flow rate in 𝑥-direction per unit width is 𝑄^∗_𝑥=¹₀𝑢𝑑𝑦, (57) where 𝑄^∗_𝑥=𝑄_𝑥/𝑊ℎ𝐵, and (57) gives 𝑄^∗_𝑥=𝑈−12𝑊612𝑃_,𝑥+𝛽𝐿₁₁+𝛽²𝐿₁₄−14𝛽𝐿₁₂+𝛽²𝐿₁₅−310𝛽𝐿₁₃+𝛽²𝐿₁₆−13𝛽²𝐿₁₇−5𝛽14²𝐿₁₈. (58) Volume flow rate in 𝑧-direction per unit width is 𝑄^∗_𝑧=¹₀𝑤𝑑𝑦, (59) where 𝑄^∗_𝑧=𝑄_𝑧/𝑊ℎ𝐵, and (59) gives 𝑄^∗_𝑧=12−1612𝑃_,𝑧+𝛽𝑇₁₁+𝛽²𝑇₁₄−14𝛽𝑇₁₂+𝛽²𝑇₁₅−310𝛽𝑇₁₃+𝛽²𝑇₁₆−13𝛽²𝑇₁₇−5𝛽14²𝑇₁₈. (60) 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 𝑄^∗=𝑛sin𝜙¹₀𝑠𝑑𝑦, (61) where 𝑄^∗=𝑄/𝑊ℎ𝐵 and 𝑛 is the number of parallel flights in a multiflight screw.

Equation (61) gives 𝑄^∗=𝑛1sin𝜙2−1612𝑃_,𝑧+𝛽𝑇₁₁+𝛽²𝑇₁₄−14𝛽𝑇₁₂+𝛽²𝑇₁₅−310𝛽𝑇₁₃+𝛽²𝑇₁₆−13𝛽²𝑇₁₇−5𝛽14²𝑇₁₈+𝑈sin𝜙−12𝑊612𝑃_,𝑥+𝛽𝐿₁₁+𝛽²𝐿₁₄−14𝛽𝐿₁₂+𝛽²𝐿₁₅−310𝛽𝐿₁₃+𝛽²𝐿₁₆−13𝛽²𝐿₁₇−5𝛽14²𝐿₁₈cos𝜙, (62) which can be written as 𝑄^∗=𝑛𝑄sin𝜙^∗𝑧sin𝜙+𝑄^∗_𝑥cos𝜙. (63)

6. Results and Discussion

In the present work, we have considered the steady flow of an incompressible, isothermal, and homogeneous third-grade fluid in helical screw rheometer (HSR). Using Adomian decomposition method, solutions are obtained for velocity profiles in 𝑥-, 𝑧-directions and also in the direction of the axis of the screw 𝑠(𝑦). The volume flow rates are also calculated by using the velocities in 𝑥, 𝑧 and in the direction of the axis of the screw. Here we discussed the effect of dimensionless parameters 𝛽, 𝑈/𝑊=−tan𝜙, 𝑃_,𝑥, and 𝑃_,𝑧 where 𝜙=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 𝑥-direction 𝑢(𝑦), 𝑧-direction 𝑤(𝑦), and the resultant velocity 𝑠(𝑦) are plotted against 𝑦 for different values of non-Newtonian parameter 𝛽 and constant pressure gradients 𝑃_,𝑥=−2.0, 𝑃_,𝑧=−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 𝑥-, 𝑧-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.

Figure 2 

Profile of the nondimensional velocity 𝑢(𝑦) for different values of 𝛽, keeping 𝑃,𝑥=−2.0, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figure 3 

Profile of the nondimensional velocity 𝑤(𝑦) for different values of 𝛽, keeping 𝑃,𝑥=−2.0, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figure 4 

Profile of the nondimensional velocity 𝑠(𝑦) for different values of 𝛽, keeping 𝑃,𝑥=−2.0, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figures 5, 7, and 9 are sketched for the velocity profiles 𝑢(𝑦), 𝑤(𝑦), and 𝑠(𝑦) against 𝑦 for different values of 𝑃_,𝑥, keeping 𝛽=0.3 and 𝑃_,𝑧=−2.0 fixed; enlightened escalation is noted in the velocity profiles with increase in pressure gradient in 𝑥-direction.

Figure 5 

Profile of the nondimensional velocity 𝑢(𝑦) for different values of 𝑃,𝑥, keeping 𝛽=0.3, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figures 6, 8, and 10 are sketched for the velocity profiles 𝑢(𝑦), 𝑤(𝑦), and 𝑠(𝑦) against 𝑦 for different values of 𝑃_,𝑧, keeping 𝛽=0.3 and 𝑃_,𝑥=−2.0 fixed, it is observed that parabolicity of the velocity profiles increases with increase in pressure gradient in 𝑧-direction.

Figure 6 

Profile of the nondimensional velocity 𝑢(𝑦) for different values of 𝑃,𝑧, keeping 𝛽=0.3, 𝑃,𝑥=−2.0, and 𝜙=45∘.

Figure 7 

Profile of the nondimensional velocity 𝑤(𝑦) for different values of 𝑃,𝑥, keeping 𝛽=0.3, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figure 8 

Profile of the nondimensional velocity 𝑤(𝑦) for different values of 𝑃,𝑧, keeping 𝛽=0.3, 𝑃,𝑥=−2.0, and 𝜙=45∘.

Figure 9 

Profile of the nondimensional velocity 𝑠(𝑦) for different values of 𝑃,𝑥, keeping 𝛽=0.3, 𝑃,𝑧=−2.0, and 𝜙=45∘.

Figure 10 

Profile of the nondimensional velocity 𝑠(𝑦) for different values of 𝑃,𝑧, keeping 𝛽=0.3, 𝑃,𝑥=−2.0, and 𝜙=45∘.

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

Figure 11 

Profile of the nondimensional velocity 𝑠(𝑦) for different values of 𝑃,𝑧, keeping 𝛽=0.3, 𝑃,𝑥=−2.0, and 𝜙=45∘.

7. 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 𝑥- and 𝑧-direction. Thus, the extrusion process strongly depends on the involved parameters.

Appendix

𝐿₁₁=12𝑃₁+3𝑈²𝑃₁𝑊²−3𝑈𝑃²₁𝑊+3𝑃³₁4+2𝑈𝑃₂𝑊−2𝑃₁𝑃₂−𝑈𝑃²₂𝑊+34𝑃₁𝑃²₂,𝐿₁₂=166𝑈𝑃²₁𝑊−3𝑃³₁+4𝑃₁𝑃₂+2𝑈𝑃²₂𝑊−3𝑃₁𝑃²₂,𝐿₁₃=1123𝑃³₁+3𝑃₁𝑃²₂,𝐿₁₄=12𝑃₁+9𝑈⁴𝑃₁𝑊⁴+10𝑈²𝑃₁𝑊²−27𝑈³𝑃²₁𝑊³−15𝑈𝑃²₁𝑊+13𝑃³₁3+21𝑈²𝑃³₁𝑊²−27𝑈𝑃⁴₁+4𝑊15𝑃⁵₁+168𝑈³𝑃₂𝑊³+8𝑈𝑃₂𝑊−10𝑃₁𝑃₂−34𝑈²𝑃₁𝑃₂𝑊²+25𝑈𝑃²₁𝑃₂𝑊−163𝑃³₁𝑃₂−5𝑈³𝑃²₂𝑊³−17𝑈𝑃²₂𝑊+383𝑃₁𝑃²₂+38𝑈²𝑃₁𝑃²₂3𝑊²−49𝑈𝑃²₁𝑃²₂+6𝑊158𝑃³₁𝑃²₂+25𝑈𝑃³₂−3𝑊163𝑃₁𝑃³₂−17𝑈𝑃⁴₂+12𝑊15𝑃16¹𝑃⁴₂,𝐿₁₅=1654𝑈³𝑃²₁𝑊³+30𝑈𝑃²₁𝑊−18𝑃³₁−90𝑈²𝑃³₁𝑊²+87𝑈𝑃⁴₁−2𝑊15𝑃⁵₁2+68𝑈²𝑃₁𝑃₂𝑊²−108𝑈𝑃²₁𝑃₂𝑊+1043𝑃³₁𝑃₂+10𝑈³𝑃²₂𝑊³+34𝑈𝑃²₂𝑊−54𝑃₁𝑃²₂−54𝑈²𝑃₁𝑃²₂𝑊²+157𝑈𝑃²₁𝑃²₂3𝑊−15𝑃³₁𝑃²₂−36𝑈𝑃³₂𝑊+1043𝑃₁𝑃³₂+53𝑈𝑃⁴₂−6𝑊152𝑃₁𝑃⁴₂+20𝑃₁𝑃₂,𝐿₁₆=11218𝑃³₁+90𝑈²𝑃³₁𝑊²−90𝑈𝑃⁴₁𝑊+45𝑃⁵₁2+108𝑈𝑃²₁𝑃₂𝑊−72𝑃³₁𝑃₂+54𝑃₁𝑃²₂+54𝑈²𝑃₁𝑃²₂𝑊²−108𝑈𝑃²₁𝑃²₂𝑊+45𝑃³₁𝑃²₂+36𝑈𝑃³₂𝑊−72𝑃₁𝑃³₂−18𝑈𝑃⁴₂𝑊+452𝑃₁𝑃⁴₂,𝐿₁₇=12060𝑈𝑃⁴₁𝑊−30𝑃⁵₁+48𝑃³₁𝑃₂+72𝑈𝑃²₁𝑃²₂𝑊−60𝑃³₁𝑃²₂+48𝑃₁𝑃³₂+12𝑈𝑃⁴₂𝑊−30𝑃₁𝑃⁴₂,𝐿₁₈=13015𝑃⁵₁+30𝑃³₁𝑃²₂+15𝑃₁𝑃⁴₂,𝑇₁₁=122𝑈𝑃₁𝑊−𝑃²₁+3𝑃₂+𝑈²𝑃₂𝑊²−2𝑈𝑃₁𝑃₂𝑊+34𝑃²₁𝑃₂−3𝑃²₂+34𝑃³₂,𝑇₁₂=162𝑃²₁+4𝑈𝑃₁𝑃₂𝑊−3𝑃²₁𝑃₂+6𝑃²₂−3𝑃³₂,𝑇₁₃=1123𝑃²₁𝑃₂+3𝑃³₂,𝑇₁₄=128𝑈³𝑃₁𝑊³+8𝑈𝑃₁𝑊−5𝑃²₁−17𝑈²𝑃²₁𝑊²+25𝑈𝑃³₁−3𝑊17𝑃⁴₁12+9𝑃₂+𝑈⁴𝑃₂𝑊⁴+10𝑈²𝑃₂𝑊²−10𝑈³𝑃₁𝑃₂𝑊³−34𝑈𝑃₁𝑃₂𝑊+383𝑃²₁𝑃₂+38𝑈²𝑃²₁𝑃₂3𝑊²−16𝑈𝑃³₁𝑃₂+3𝑊15𝑃16⁴¹𝑃₂−27𝑃²₂−15𝑈²𝑃²₂𝑊²+25𝑈𝑃₁𝑃²₂𝑊−496𝑃²₁𝑃²₂+21𝑃³₂+13𝑈²𝑃³₂3𝑊²−16𝑈𝑃₁𝑃³₂+3𝑊158𝑃²₁𝑃³₂−27𝑃⁴₂4+15𝑃⁵₂,𝑇16₁₅=1610𝑃²₁+34𝑈²𝑃²₁𝑊²−36𝑈𝑃³₁𝑊+53𝑃⁴₁6+20𝑈³𝑃₁𝑃₂𝑊³+68𝑈𝑃₁𝑃₂𝑊−54𝑃²₁𝑃₂−54𝑈²𝑃²₁𝑃₂𝑊²+104𝑈𝑃³₁𝑃₂−3𝑊152𝑃⁴₁𝑃₂+54𝑃²₂+30𝑈²𝑃²₂𝑊²−108𝑈𝑃₁𝑃²₂𝑊+1573𝑃²₁𝑃²₂−90𝑃³₂−18𝑈²𝑃³₂𝑊²+104𝑈𝑃₁𝑃³₂3𝑊−15𝑃²₁𝑃³₂+87𝑃⁴₂2−15𝑃⁵₂2,𝑇₁₆=11236𝑈𝑃³₁𝑊−18𝑃⁴₁+54𝑃²₁𝑃₂+54𝑈²𝑃²₁𝑃₂𝑊²−72𝑈𝑃³₁𝑃₂𝑊+452𝑃⁴₁𝑃₂+108𝑈𝑃₁𝑃²₂𝑊−108𝑃²₁𝑃²₂+90𝑃³₂+18𝑈²𝑃³₂𝑊²−72𝑈𝑃₁𝑃³₂𝑊+45𝑃²₁𝑃³₂−90𝑃⁴₂+45𝑃⁵₂2,𝑇₁₇=12012𝑃⁴₁+48𝑈𝑃³₁𝑃₂𝑊−30𝑃⁴₁𝑃₂+72𝑃²₁𝑃²₂+48𝑈𝑃₁𝑃³₂𝑊−60𝑃²₁𝑃³₂+60𝑃⁴₂−30𝑃⁵₂,𝑇₁₈=13015𝑃⁴₁𝑃₂+30𝑃²₁𝑃³₂+15𝑃⁵₂. (A.1)(A.2)

Acknowledgment

The first author is very thankful to Higher Education Commission (HEC) of Pakistan for funding his higher studies under the 5000 indigenous scholarship scheme Batch-IV.