THE SUBORDINATION OF THE THREE- DIMENSIONAL FLOW INSTALLATION IN THE CONVERGING CHANNEL ON RHEOLOGICAL CHARACTERISTICS OF POLYMER STREAM

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International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 13, December 2018, pp. 949-956, Article ID: IJCIET_09_13_095 Available online at http://www.iaeme.com/ijciet/issues.

1 International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 13, December 2018, pp , Article ID: IJCIET_09_13_095 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed THE SUBORDINATION OF THE THREE- DIMENSIONAL FLOW INSTALLATION IN THE CONVERGING CHANNEL ON RHEOLOGICAL CHARACTERISTICS OF POLYMER STREAM Haider Nadhom Azziz Al Joda University of Kerbala, Iraq Grigory Pshnograi Altai State Technical University, Barnaul, Russia ABSTRACT Characteristic of hydro-dynamic for 3D stream of polymer flux in a flat-parallel channel are calculated. The channel is suddenly converged. A nonlinear viscoelastic fluid flow in the channel is considered in this paper. The problem in the original formulation is reduced for partial differential equations to the discrete analogs by the process of referee dimension with division of physical operation. Solution of numerical for the problem is obtained utilize the GPU-setup on technology of equivalent calculating. Pressure and velocity fields of polymer melt are predicted of two specimen of polyethylene flux. Case of revolving stream at an, inlet of cut out channel are cleared. Such it can be shown so as to the vortex zone space depends firstly on stream rheology. Keywords: flux, polymer stream, parallel computing, rheological, three-dimensional flows. Cite this Article: Haider Nadhom Azziz Al Joda and Grigory Pshnograi, the Subordination of the Three-Dimensional Flow Installation in the Converging Channel on Rheological Characteristics of Polymer Stream, International Journal of Civil Engineering and Technology, 9(13), 2018, pp INTRODUCTION In This paper may be reason characteristic of the hydro-dynamic in 3D stream for thermoplastics flux in a flat such as channel for all of a sudden convergence.experimental investigation of flows in the various converging channels is studied in a large number of papers (Hertel et al., 2008; Mitsoulis et al., 2003). These works describe results for different polymers: editor@iaeme.com

2 Haider Nadhom Azziz Al Joda and Grigory Pshnograi a low density polyethylene (Hertel et al., 2008; Mitsoulis et al., 2003), a linear low density polyethylene (Hertel et al., 2008) or a linear high density polyethylene (Bagley and Birks, 1960; Boger et al., 1986; Munstedt et al., 2005).Our methodology relies on two steps. Firstly, we shall select and justify the rheological model. Secondly, we shall take place algorithmic application of the acquired for equations of the numerical testing. However, an effort to utilize this numerical method for calculating flat channels come off. For calculating actual flux on the foundation, in this pattern at the following equations from mass and momentum conservation have to be added for (1) + = ik, =0. (1) Where - the polymer density; i-the velocity of vector component. In the system for one equation (1) is closes at consideration to parameters ik,,. For solve the case of initial and boundary value are wanted. Firstly compare the computational range, shown in Figure 1 to applicate these conditions. Figure 1. The computational domain and its dimensions (mm) 2. EXPERIMENTAL AND THEORETICAL STATIONARY SHEAR VISCOSITY SUBORDINATION. The contradiction between the experimental and theoretical curves may be explained by the one -method quality of the rheological sample (1). Its formulation takes into account only one slowest relaxation process. Consideration of multiple relaxation processes, as showed Merzlikina et al. (2013), can describe more accurately viscometric functions, but leads to a significant complication of the model. 3. THE NUMERICAL METHOD In the numerical method is structured for making finite-difference of the channel is applied to get the solution. In the discretization of the equations, an exploded difference scheme was used, shown in Fig editor@iaeme.com

3 The Subordination of the Three-Dimensional Flow Installation in the Converging Channel on Rheological Characteristics of Polymer Stream Figure 2. Spaced difference grid. At the initial stage of the computations, the flow structure undergoes quite strong changes. To improve the stability of the difference analogs without the need to significantly reduce the time step, it is assumed that the polymer liquid is compressible. The melt density in this case is calculated by the formula: ρ ( p ) = ρ(0)(1 + kp). 3 Where ρ ( 0) = 918 kg / m - densities are for room temperature, p - hydrostatic pressure, k - value of compressibility factor. k - Small parameter of problem, in the calculations, it 6 ranged from to MATHEMATICAL MODELING FOR FLOWS IN THE SLIT CHANNEL. Let us consider the application of model (1) in order to determine a steady velocity profile of the polymer stream in the gap between the parallel plates under constant pressure gradient =. =! " #2+ % &! " ' + () * #+ 6# +12hy 8 +, 33 = 1 & #2 1 ) + * #2 * ) 24 (2) 22 = ) 12 #2, Where A=! " 4 " and # =1(mm) is channel width. Here in the second equation of y=β \8^#2y:0. It means non-zero differential pressure exists in the side orthogonal to the inflow velocity. Nevertheless, it does not cause minor stream. In this pressure gradient may cause the swelling effect of the jet at the channel outlet and can be explained by unacceptability of the boundary condition for calculation of steady velocity profile editor@iaeme.com

4 Haider Nadhom Azziz Al Joda and Grigory Pshnograi Integrating the first equation (2) with respect to y from the subordination of the flow on the pressure gradient we obtain the following: > ; =< dy? = 12! " h + + % 60! " ' + () * h@ (3) The first terms in (2, 3) are well known for Newtonian media, and the terms with A and B provide amendments to non-newtonian behavior. Comparison of the calculated subordination found on the basis of the analytical solution, and the values obtained during the experiment showed qualitative agreement. However, due to the nature of approximation an approximate analytical solution cannot be used at high pressure gradients, which is of great practical interest. Therefore, it is necessary to find more accurate expressions in the combination the stress tensor and velocity. Next we should define depending expressions for the components of the stress tensor and velocity will take into account thata = 1.2B (Oishi et al., 2011). The greatest interest in this case is the subordination of the velocity profile and the stress tensor components 22, 33 on pressure gradient. All calculations indicate that functions are symmetrically relative to the axis of the channel. The effect of pressure gradient on (Fig.3) and subordination 33 (Fig. 4) and (Fig. 5) on the components of tensor anisotropy has been investigated. It has shown that increasing pressure gradient leads to deviations of absolute values 33 h 2, 22 h 2 and h 2 which is also increased and the behavior of fluid deviates from the Newtonian law. Figure 3. The effect of pressure gradient Ē on the velocity profile at F=G.36 andh=g.i editor@iaeme.com

5 The Subordination of the Three-Dimensional Flow Installation in the Converging Channel on Rheological Characteristics of Polymer Stream Figure 4. The effect of pressure gradient E (between 1 and 2.5) on subordination zz y at F=G.36 and H= Figure 5. The effect of pressure gradient E (between 1 and 2.5) on subordination yy y at F=G.36 and H=G.I. Of particular interest is the investigation of dependencies for >> 10. For example, the behavior of anisotropy of tensor components changes abruptly already at >5 (Fig. 6), while at the boundaries of the channel the fixed level alternated by sharply zero value to the center channel extends editor@iaeme.com

6 Haider Nadhom Azziz Al Joda and Grigory Pshnograi Figure 6. The effect of pressure gradient E (between 1 and 10) on yy y at F=G.36 and H=G.I. Figure. 7. Subordination of the volume rate on the pressure gradient E at different values of parameters F andh. Compare the values of the velocity profile with the experimental data (Munstedt et al., 2000) presented in Figure 8. In this case, calculations show that for the shape of the velocity profile changes slightly. Note that similar results are obtained Munstedt et al. (2000) on the basis of another rheological model. Also, the calculations carried out for show that the growth of value increases the maximum speed and its deviation from the parabolic profile corresponding to Poiseuille profile editor@iaeme.com

7 The Subordination of the Three-Dimensional Flow Installation in the Converging Channel on Rheological Characteristics of Polymer Stream Figure. 8. Comparison of the velocity profile of with experimental data for different volume rates V. 6. CONCLUSION. 1- It has matched of the polymer stream flux with various rheological properties in the convergent channel for cross section a rectangular. 2- It has shown such a raise of the repose with the time for the polymer pattern allow lifting to vortex flow that is adjusted for the experiments. 3- It proves the applicability for appearing in dynamics for polymer stream in the zone at complex geometry. And we obtained in this state steady of the effective use in equivalent computing CUDA for technology in 3-dimentional flow of polymer stream. REFERENCES [1] Bagley, E.B. and Birks, A.M., 1960, Flow of polyethylene into a capillary, J. Appl. Phys., 31, [2] Bambaeva N.V., Blokhin A.M., 2012, The t-hyporboliscity of a nonstationary system governing flows of polymeric media, Journal of Mathematical Science, 188, [3] Bambayeva N.V., Blokhin A.N., 2014, Stationary solutions of equations of incompressible viscoelastic polymer liquid, Computational Mathematics and Mathematical Physics, 54, [4] Blokhin A.M., Tkachev D.L., 2014, Linear asymptotic instability of a stationary flow of a polymeric medium in a plane channel in the case of periodic perturbations, Journal of Applied and Industrial Mathematics, 8, [5] Blokhin A.M., Yegitov A.V., Tkachev D.L., 2015, Linear Instability of Solutions in a Mathematical Model Describing Polymer Flows in an Infinite Channel, Computational Mathematics and Mathematical Physics, 55, [6] Boger, D.V., Hur, D.U., and Binnington, R.J., 1986, Further observations of elastic effects in tubular entry flows, J. Non-Newtonian Fluid Mech., 20, [7] Hertel D., Münstedt H., 2008, Dependence of the secondary flow of a low-density polyethylene on processing parameters as investigated by laser-doppler velocimetry, J. Non-Newtonian Fluid Mech. 153, editor@iaeme.com

8 Haider Nadhom Azziz Al Joda and Grigory Pshnograi [8] Hertel D., Valette R., Münstedt H., 2008, Three-dimensional entrance flow of a low-density polyethylene (LDPE) and a linear low-density polyethylene (LLDPE) into a slit die, J. Non-Newtonian Fluid Mech. 153, [9] Merzlikina D.A., Philip P., Pivokonsky R., Pyshnogray G.V., 2013, Multimode rheological model and findings for simple shear and elongation, Mekhanika kompozitsionnykh materialov i konstruktsii, 19, [10] Merzlikina D.A., Pyshnograi G.V., Pivokonskii R., Filip P., 2016, Rheological model for describing viscometric flows melts of branched polymers, Journal of Engineering Physics and Thermophysics, 89, [11] Munstedt H., Schmidt M., Wassner E., 2000, Stick and slip phenomena during extrusion of polyethylene melts as investigated by laser-doppler velocimetry, J. Rheol., 44, [12] Mitsoulis E., Schwetz M, Münstedt H., 2003, Entry flow of LDPE melts in a planar contraction, J. Non-Newtonian Fluid Mech. 111, [13] Oishi C.M., Martins F.P., Tome M.F., Cuminato J.A., McKee S., 2011, Numerical Solution of the Extended Pom-Pom model for viscoelastic free surface flows, J. Non-Newtonian Fluid Mech. 166, editor@iaeme.com

TWO-DIMENSIONAL SIMULATIONS OF THE EFFECT OF THE RESERVOIR REGION ON THE PRESSURE OSCILLATIONS OBSERVED IN THE STICK-SLIP INSTABILITY REGIME

1 TWO-DIMENSIONAL SIMULATIONS OF THE EFFECT OF THE RESERVOIR REGION ON THE PRESSURE OSCILLATIONS OBSERVED IN THE STICK-SLIP INSTABILITY REGIME Eleni Taliadorou and Georgios Georgiou * Department of Mathematics