The recent development of theoretical and experimental rheology, coupled with the increasing performance of computers, now allows us to have a different approach and to envisage numerical predictions on complex geometries. Unfortunately, with current differential models, simulations of viscoelastic fluids in complex geometries still run up against the limits of memory resources and prohibitive computational times. In this study, the commercial software Fluent used in combination with a calculation code developed in C++, via sub-programs defined by User Defined Functions and User Defined Scalars. The purpose of this study is to compare the results with the analytical solution; which makes it possible to validate the numerical results by using the code developed in C++ and also, to give an assurance to use this code in the numerical simulation of several problems in UCM fluid, which does not exist on the data base of the Fluent software. The results obtained in this study, shows the effectiveness of the code developed in C++.
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
The behavior of fluids in confined devices such as the flow between two cylinders, two cores, or two spheres, is studied in the measurement of fluids. The flow between two coaxial cylinders has resulted in more than 2000 articles
[15]
A. Dowson, A generalized Reynolds equation for fluid film lubrication, Int. J. Mech. Sc., 4: 159–170, (1962).
K. P. Gertzos, P. G. Nikolakopoulos, C. A. Papadopoulos, CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant. J. Tribology (2008),
D. RH. Gwynllyw, T. N. Phillips, The effect of viscoelasticity on the performance of journal bearings. Springer. (2006) 176-186.
[15-18]
since Mr Couette’s. Because of its simplicity, this system has a rich variety of systems and records: laminar (basic flow), instabilities and chaos. The modes of transition these registers of rotation of the two cylinders, the characteristics of the system (ratio of rays, aspect ratio), and the properties of the fluid used (viscosity, elasticity, etc.).
The importance of thermal effects in hydrodynamic lubrication of Journal bearings (two eccentric cylinders) was highlighted in the first scientific studies, however, the consideration of these effects for the calculation of mechanisms is recent
[8]
M. Guemmadi, Convection Forcée de Fluides Viscoélastiques en Milieu Confiné – Cas de la Lubrification Hydrodynamique, Doctoral thesis, University of Boumerdes, (2019).
[18]
D. RH. Gwynllyw, T. N. Phillips, The effect of viscoelasticity on the performance of journal bearings. Springer. (2006) 176-186.
[21]
Shimiao Lian and all, Thermal and viscoelastic coupled influences analysis on mixed lubrication in dynamic journal bearings, Physics of Fluids 36, 047106 (2024),
The geometry of two coaxial cylinders is named after Maurice Couette, the first researcher to use this system in 1884, whose purpose was to measure the viscosity of fluids. It rotated the outer cylinder and measured the torque of the inner cylinder suspended by a twisting wire. He observed that the viscosity remains constant as long as the speed of rotation does not exceed a critical value; this is the basic circular flow: laminar speed. Four years later, A. Mallock (1888) observed similar characteristics by rotating the inner cylinder. In 1890, Couette
[9]
M. Couette. Etude sur le frottement des liquides. Ann. Chem. Phys. 21, p. 433, 1890.
[9]
remarked that in the situation where the inner cylinder rotates and the outer cylinder remains fixed, the speed conditions for which the movement of the fluid can be considered as purely circular are more restrictive than the mobile outer cylinder. These first authors did not then explain the phenomenon encountered in these experiments.
Viscoelastic fluids are generally obtained by adding a long molecular chain polymer to a Newtonian solvent. Subjected to shearing, the resulting solution exhibits non-Newtonian behaviors such as rheofluidification or the appearance of differences in normal stresses induced by the elongation of macromolecules in the flow. These viscoelastic properties modify the nature and characteristics of instabilities (flow regimes) observed for a Newtonian fluid.
Exact solutions of the equations of motion including the constituent equation of viscoelasticity are rare when the field of flow does not consist of homogeneous shear or when it presents elongation (i. e. normal constraints are not homogeneous). In this case, non-linearity in the constituent equation requires that the flow be determined by the numerical solution
[8]
M. Guemmadi, Convection Forcée de Fluides Viscoélastiques en Milieu Confiné – Cas de la Lubrification Hydrodynamique, Doctoral thesis, University of Boumerdes, (2019).
[8]
.
The recent development of theoretical and experimental rheology, coupled with the ever-increasing performance of computers, now allows us to have a different approach and to consider numerical predictions on complex geometries. Unfortunately, with current differential-type models, simulations of viscoelastic fluids in complex geometries
[5]
H. K. Versteeg, W. Malalasekera, An introduction to computational fluid dynamics. Longman Scientific & Technical (1995). ISBN: 0-470-23515-2.
[5]
and
[18]
D. RH. Gwynllyw, T. N. Phillips, The effect of viscoelasticity on the performance of journal bearings. Springer. (2006) 176-186.
[19]
Amit Chauhan, Circular bearing performance parameters with isothermal and thermo-hydodynamic approach using CFD, International Journal of Research in Advent Technology, vol 2, N07, E-ISSN: 2321-9637, (2014).
[20]
Juliana Javorova et Jordanka Angelova, On the modified Reynolds equation for journal bearings in case of non-Newtonian rabinowitsch fluid model, volume 145-(2018),
still come up against memory resource limits and prohibitive calculation times.
This work is devoted firstly to analytical and numerical studies of the Couette flow between two concentric cylinders. Secondly, comparisons between the results obtained numerically and those obtained analytically. The fluid used in these studies is a Maxwell viscoelastic fluid. The Fluent software is used to which a C++ calculation code is integrated.
The main aim of these studies is to validate the calculation code in order to use it in the field of hydrodynamic lubrication of journal bearings by viscoelastic fluids obeying the Maxwell model
[8]
M. Guemmadi, Convection Forcée de Fluides Viscoélastiques en Milieu Confiné – Cas de la Lubrification Hydrodynamique, Doctoral thesis, University of Boumerdes, (2019).
[8]
.
2. Analytical Study of Couette Flow
The fluid is between the inner cylinder of radius R1, rotating at speed , and the outer cylinder of radius R2 is fixed, the velocity field is of the form .
In cylindrical coordinate system (), the equations of dynamic equilibrium, strain tensor, velocity gradient tensor and stress tensor are given as follows:
Or: ij: the stress tensor. : the relaxation time. is the convective derivative of the stress tensor
[1]
J. F. Agassant, P. Avenas, J. Ph. Sergent, B. Vergnes, M. Vincent, La mise en forme des matières plastiques. 3e édition, Technique et Documentation (1996).
[1]
.
2.2. Dynamic Equilibrium
Let the dynamic equilibrium system in the general case as follows
[3]
N. Midoux, Mécanique et Rhéologie des fluides en génie chimique. Technique et Documentation (1999).
[3];
(Flow is permanent).
(2)
2.3. Energy Equation
In the general case, the energy equation in cylindrical coordinates is written as follows
[4]
Loan C. Popa, Modélisation numérique du transfert thermique – Méthode des volumes finis. Maison d’édition university of Craiova, (2002).
[4]
:
(3)
2.4. Simplifying Assumptions
Flow is permanent.
By reason of symmetry relative to we can write:
No flow in the z direction, so:
Taking into account the simplifying assumptions and the boundary conditions. Finally, we arrive to the following expressions:
(4)
.(5)
(6)
(7)
(8)
3. Numerical Procedure
3D flow between two coaxial cylinders is modeled by Fluent. The distance C (R2-R1) between the cylinders is equal to 0.05 m and the range considered is of length L such as L=5C. The inner cylinder rotates with a rotation speed .
The simulation concerns a viscoelastic fluid flow (Maxwell model) of density and dynamic viscosity µ. The precision on the convergence is taken 10-7 and the coefficient of under relaxation of momentum equation is 0.5,
[10]
M. Guemmadi, A. Ouibrahim, Thin Film Analysis of Viscoelastic Journal Bearing Lubrication Using a CFD as a Tool, (ICMSA2023), Khenchela- Algérie.
[11]
M. Guemmadi, Mathematical Modeling and Numerical Simulation of Viscoelastic Flow between two Eccentric Cylinders - Case of Journal Bearing, 3rd International Congress on Energetic and Environmental Systems (IEES-2023), Sousse- Tunisie.
[12]
M. Guemmadi, A. Ouibrahim, Etude Numérique de l’écoulement du fluide Viscoélastique de Maxwell entre deux cylindres excentriques- cas des paliers lisses. Congrès national en mécanique, matériaux et métallurgie, (NCMMM2023), Oran- Algérie.
[13]
M. Guemmad, F. Brahimi and A. Ouibrahim, Comparative Study between Newtonian and Viscoelastic lubricants using a CFD as a tool – Case of Journal Bearing, (GECS’2023), Djerba- Tunisie.
[14]
M. Guemmadi, F. Brahimi and A. Ouibrahim, Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects, (CNMM_2023), Boumerdes- Algeria.
[10-14]
.
The following table represents the different terms of the continuity equation, momentum equation, constitutive law and energy equation
[8]
M. Guemmadi, Convection Forcée de Fluides Viscoélastiques en Milieu Confiné – Cas de la Lubrification Hydrodynamique, Doctoral thesis, University of Boumerdes, (2019).
[8]
.
Table 1. The different source terms.
Continuity Equation
divu=SM
SM=0
Momentum Equation elong x
divρuu=-∂p∂x+divμgradu+SMx μ=0
SMx=∂τxx∂x+∂τxy∂y
Momentum Equation elong y
divρvu=-∂p∂y+divμgradv+SMy μ=0
SMy=∂τxy∂x+∂τyy∂y
Stress τxx
divρτxxu=divΓgradτxx+Sτxx Γ=0
Sτxx=ρ-1λ+2∂u∂xτxx+ρ2∂u∂yτxy+ρμλ∂u∂x μ=μp
Stress τyy
divρτyyu=divΓgradτyy+Sτyy Γ=0
Sτyy=ρ-1λ+2∂v∂yτyy+ρ2∂v∂xτxy+ρμλ∂v∂y μ=μp
Stressτxy
divρτxyu=divΓgradτxx+Sτxy Γ=0
Sτxy=ρ∂v∂xτxx+∂u∂yτyy+ρμλ∂u∂y+∂v∂x-ρλτxy μ=μp
Energy Equation
divρCpTu=divΓgradT+ST Γ=k
ST=τxx∂u∂x+τyy∂v∂y+τxy∂u∂y+∂v∂x
4. Results and Discussion
The figure 2 shows the velocity vectors in the annular space. The figure shows that the speed is maximized on the inner cylinder and zero on the outer cylinder.
The temperature distribution in the annular space is shown in Figure 3 and Figure 7. Note that the temperature decreases from Ta=127°C (the temperature of the inner cylinder) to the temperature of the outer cylinder (Tc).
Figure 4 shows the evolution of fluid velocity in the radial direction, obtained by analytical and numerical calculations under Fluent. Note that the speed is maximum at the inner cylinder and decreases to zero at the fixed outer cylinder.
Figure 5 shows the evolution of the shear stress as a function of r. It is noted that the variation is non-linear which is confirmed by the analytical solution obtained from equation (eq. 6).
Figure 6 shows the evolution of the analytically and numerically calculated normal stress as a function of r. It is noted that the normal stress varies not linearly according to r and decreases as a function of the radius of the inner cylinder towards the outer cylinder.
The results shown in figures 4, 5, 6 and 7 display a conformity between the analytical solution and the numerical solution obtained by the Fluent software.
Figure 7. Evolution of the temperature in annular space as a function of the radius r.
5. Effect of Relaxation Time on Temperature and Normal Stress
The analytical calculation shows through the expression (eq. 8) that the temperature does not depend on the relaxation time or that the normal stress varies according to λ. The numerical results shown in figures 8 and 9 are confirmed by the analytical results. The analytical and numerical results show that the viscoelastic fluid studied does not play a major role; as regards the dissipation of the heat of this fluid that flows between two concentric cylinders, whose temperature varies according to the viscosity, the speed of rotation of the inner cylinder as well as the radius r. The conformity between the results obtained numerically and analytically shows well the efficiency of the code developed in C++.
Figure 9. Evolution of Normal stress as a function of r to <i></i>=0 for different values of <i></i>.
6. Conclusion
The study presented in this article aims to adapt Fluent with viscoelastic fluid presented by an Upper Convected Maxwell model that is not integrated into the database of this software.
To check the validity of the code developed in C++ and the integrated under Fluent, we calculated the velocity field, the pressure field, the normal stress and the shear stress in the case of viscoelastic fluid flow between two concentric cylinders (Couette flow). The results of the numerical simulation are compared by analytic computation. Then, we studied the thermal effect by integrating the energy equation during the numerical simulation, and we found that the results obtained show the efficiency of the calculation code developed in C++. It was also found that in the concentric case, the relaxation time has no effect on the fluid temperature.
J. F. Agassant, P. Avenas, J. Ph. Sergent, B. Vergnes, M. Vincent, La mise en forme des matières plastiques. 3e édition, Technique et Documentation (1996).
[2]
Guy Couarraz, Jean- Louis Grossiord, Initiation à la rhéologie. 3e edition, Technique et Documentation (2000).
[3]
N. Midoux, Mécanique et Rhéologie des fluides en génie chimique. Technique et Documentation (1999).
[4]
Loan C. Popa, Modélisation numérique du transfert thermique – Méthode des volumes finis. Maison d’édition university of Craiova, (2002).
[5]
H. K. Versteeg, W. Malalasekera, An introduction to computational fluid dynamics. Longman Scientific & Technical (1995). ISBN: 0-470-23515-2.
[6]
Ghania Benbelkacem, Viscoélasticité et écoulements de fluides structurés, Doctoral thesis, Nancy University, 2009.
[7]
M. Guemmadi, A. Ouibrahim, Generalized Maxwell Model as Viscoelastic Lubricant in Journal Bearing, Key Engineering Materials Vol. 478 (2011) pp 64-69.
M. Guemmadi, Convection Forcée de Fluides Viscoélastiques en Milieu Confiné – Cas de la Lubrification Hydrodynamique, Doctoral thesis, University of Boumerdes, (2019).
[9]
M. Couette. Etude sur le frottement des liquides. Ann. Chem. Phys. 21, p. 433, 1890.
[10]
M. Guemmadi, A. Ouibrahim, Thin Film Analysis of Viscoelastic Journal Bearing Lubrication Using a CFD as a Tool, (ICMSA2023), Khenchela- Algérie.
[11]
M. Guemmadi, Mathematical Modeling and Numerical Simulation of Viscoelastic Flow between two Eccentric Cylinders - Case of Journal Bearing, 3rd International Congress on Energetic and Environmental Systems (IEES-2023), Sousse- Tunisie.
[12]
M. Guemmadi, A. Ouibrahim, Etude Numérique de l’écoulement du fluide Viscoélastique de Maxwell entre deux cylindres excentriques- cas des paliers lisses. Congrès national en mécanique, matériaux et métallurgie, (NCMMM2023), Oran- Algérie.
[13]
M. Guemmad, F. Brahimi and A. Ouibrahim, Comparative Study between Newtonian and Viscoelastic lubricants using a CFD as a tool – Case of Journal Bearing, (GECS’2023), Djerba- Tunisie.
[14]
M. Guemmadi, F. Brahimi and A. Ouibrahim, Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects, (CNMM_2023), Boumerdes- Algeria.
[15]
A. Dowson, A generalized Reynolds equation for fluid film lubrication, Int. J. Mech. Sc., 4: 159–170, (1962).
K. P. Gertzos, P. G. Nikolakopoulos, C. A. Papadopoulos, CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant. J. Tribology (2008),
D. RH. Gwynllyw, T. N. Phillips, The effect of viscoelasticity on the performance of journal bearings. Springer. (2006) 176-186.
[19]
Amit Chauhan, Circular bearing performance parameters with isothermal and thermo-hydodynamic approach using CFD, International Journal of Research in Advent Technology, vol 2, N07, E-ISSN: 2321-9637, (2014).
[20]
Juliana Javorova et Jordanka Angelova, On the modified Reynolds equation for journal bearings in case of non-Newtonian rabinowitsch fluid model, volume 145-(2018),
Shimiao Lian and all, Thermal and viscoelastic coupled influences analysis on mixed lubrication in dynamic journal bearings, Physics of Fluids 36, 047106 (2024),
Guemmadi, M., Brahimi, F., Ouibrahim, A. (2024). Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. International Journal of Fluid Mechanics & Thermal Sciences, 10(2), 25-30. https://doi.org/10.11648/j.ijfmts.20241002.11
Guemmadi, M.; Brahimi, F.; Ouibrahim, A. Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. Int. J. Fluid Mech. Therm. Sci.2024, 10(2), 25-30. doi: 10.11648/j.ijfmts.20241002.11
Guemmadi M, Brahimi F, Ouibrahim A. Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. Int J Fluid Mech Therm Sci. 2024;10(2):25-30. doi: 10.11648/j.ijfmts.20241002.11
@article{10.11648/j.ijfmts.20241002.11,
author = {Messaouda Guemmadi and Faiza Brahimi and Ahmed Ouibrahim},
title = {Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects
},
journal = {International Journal of Fluid Mechanics & Thermal Sciences},
volume = {10},
number = {2},
pages = {25-30},
doi = {10.11648/j.ijfmts.20241002.11},
url = {https://doi.org/10.11648/j.ijfmts.20241002.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijfmts.20241002.11},
abstract = {The recent development of theoretical and experimental rheology, coupled with the increasing performance of computers, now allows us to have a different approach and to envisage numerical predictions on complex geometries. Unfortunately, with current differential models, simulations of viscoelastic fluids in complex geometries still run up against the limits of memory resources and prohibitive computational times. In this study, the commercial software Fluent used in combination with a calculation code developed in C++, via sub-programs defined by User Defined Functions and User Defined Scalars. The purpose of this study is to compare the results with the analytical solution; which makes it possible to validate the numerical results by using the code developed in C++ and also, to give an assurance to use this code in the numerical simulation of several problems in UCM fluid, which does not exist on the data base of the Fluent software. The results obtained in this study, shows the effectiveness of the code developed in C++.
},
year = {2024}
}
TY - JOUR
T1 - Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects
AU - Messaouda Guemmadi
AU - Faiza Brahimi
AU - Ahmed Ouibrahim
Y1 - 2024/09/20
PY - 2024
N1 - https://doi.org/10.11648/j.ijfmts.20241002.11
DO - 10.11648/j.ijfmts.20241002.11
T2 - International Journal of Fluid Mechanics & Thermal Sciences
JF - International Journal of Fluid Mechanics & Thermal Sciences
JO - International Journal of Fluid Mechanics & Thermal Sciences
SP - 25
EP - 30
PB - Science Publishing Group
SN - 2469-8113
UR - https://doi.org/10.11648/j.ijfmts.20241002.11
AB - The recent development of theoretical and experimental rheology, coupled with the increasing performance of computers, now allows us to have a different approach and to envisage numerical predictions on complex geometries. Unfortunately, with current differential models, simulations of viscoelastic fluids in complex geometries still run up against the limits of memory resources and prohibitive computational times. In this study, the commercial software Fluent used in combination with a calculation code developed in C++, via sub-programs defined by User Defined Functions and User Defined Scalars. The purpose of this study is to compare the results with the analytical solution; which makes it possible to validate the numerical results by using the code developed in C++ and also, to give an assurance to use this code in the numerical simulation of several problems in UCM fluid, which does not exist on the data base of the Fluent software. The results obtained in this study, shows the effectiveness of the code developed in C++.
VL - 10
IS - 2
ER -
Department of Mechanical Engineering, University of M'hamed Bougara, Boumerdes, Algeria; Laboratory of Mechanical Energetics and Engineering, University of M'hamed Bougara, Boumerdes, Algeria
Department of Mechanical Engineering, University of M'hamed Bougara, Boumerdes, Algeria; Laboratory of Dynamic Motors and Vibro-Acoustic, University of M'hamed Bougara, Boumerdes, Algeria
Guemmadi, M., Brahimi, F., Ouibrahim, A. (2024). Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. International Journal of Fluid Mechanics & Thermal Sciences, 10(2), 25-30. https://doi.org/10.11648/j.ijfmts.20241002.11
Guemmadi, M.; Brahimi, F.; Ouibrahim, A. Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. Int. J. Fluid Mech. Therm. Sci.2024, 10(2), 25-30. doi: 10.11648/j.ijfmts.20241002.11
Guemmadi M, Brahimi F, Ouibrahim A. Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects. Int J Fluid Mech Therm Sci. 2024;10(2):25-30. doi: 10.11648/j.ijfmts.20241002.11
@article{10.11648/j.ijfmts.20241002.11,
author = {Messaouda Guemmadi and Faiza Brahimi and Ahmed Ouibrahim},
title = {Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects
},
journal = {International Journal of Fluid Mechanics & Thermal Sciences},
volume = {10},
number = {2},
pages = {25-30},
doi = {10.11648/j.ijfmts.20241002.11},
url = {https://doi.org/10.11648/j.ijfmts.20241002.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijfmts.20241002.11},
abstract = {The recent development of theoretical and experimental rheology, coupled with the increasing performance of computers, now allows us to have a different approach and to envisage numerical predictions on complex geometries. Unfortunately, with current differential models, simulations of viscoelastic fluids in complex geometries still run up against the limits of memory resources and prohibitive computational times. In this study, the commercial software Fluent used in combination with a calculation code developed in C++, via sub-programs defined by User Defined Functions and User Defined Scalars. The purpose of this study is to compare the results with the analytical solution; which makes it possible to validate the numerical results by using the code developed in C++ and also, to give an assurance to use this code in the numerical simulation of several problems in UCM fluid, which does not exist on the data base of the Fluent software. The results obtained in this study, shows the effectiveness of the code developed in C++.
},
year = {2024}
}
TY - JOUR
T1 - Analytical and Numerical Resolution of Viscoelastic Upper-Convected Maxwell Fluid in Couette Flow with Thermal Effects
AU - Messaouda Guemmadi
AU - Faiza Brahimi
AU - Ahmed Ouibrahim
Y1 - 2024/09/20
PY - 2024
N1 - https://doi.org/10.11648/j.ijfmts.20241002.11
DO - 10.11648/j.ijfmts.20241002.11
T2 - International Journal of Fluid Mechanics & Thermal Sciences
JF - International Journal of Fluid Mechanics & Thermal Sciences
JO - International Journal of Fluid Mechanics & Thermal Sciences
SP - 25
EP - 30
PB - Science Publishing Group
SN - 2469-8113
UR - https://doi.org/10.11648/j.ijfmts.20241002.11
AB - The recent development of theoretical and experimental rheology, coupled with the increasing performance of computers, now allows us to have a different approach and to envisage numerical predictions on complex geometries. Unfortunately, with current differential models, simulations of viscoelastic fluids in complex geometries still run up against the limits of memory resources and prohibitive computational times. In this study, the commercial software Fluent used in combination with a calculation code developed in C++, via sub-programs defined by User Defined Functions and User Defined Scalars. The purpose of this study is to compare the results with the analytical solution; which makes it possible to validate the numerical results by using the code developed in C++ and also, to give an assurance to use this code in the numerical simulation of several problems in UCM fluid, which does not exist on the data base of the Fluent software. The results obtained in this study, shows the effectiveness of the code developed in C++.
VL - 10
IS - 2
ER -
.
), the equations of dynamic equilibrium, strain tensor, velocity gradient tensor and stress tensor are given as follows:
(1)
is the convective derivative of the stress tensor 
(2)
(3)
we can write: 
(4)
.(5)
(6)
(7)
(8)
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