DURGUN POROZ ORTAMDAN LİNEER VE LİNEER OLMAYAN VİSKOELASTİK AKIŞKANLARIN AKIŞI VE ISI TRANSFERİ İÇİN MATEMATİKSEL MODEL DENKLEMLERİ

Year 2007, Issue: 014, 97 - 106, 17.12.2007

Bayram Alakuş Ramazan Köse

Abstract

Geçirgen heterojen ortamdaki akışkanın matematiksel modellemesi, Resin Transfer Molding, Injection Molding,

vs. gibi birçok imalat proseslerinde önemli bir yer tutar. Bu araştırmada, Local Volume Averaging Technique

metodu ile elde edilen 3 boyutlu, zamana bağlı izotermal olmayan genel bir matematiksel model elde edildi. Model,

ortalama hız, basınç, polimerik gerilmeleri ve sıcaklık değerlerini kütle, momentum ve enerji korunum kanunlar ile

polimerik gerilme modellerini kullanarak elde etmektedir. Polimerik akışkan gerilmeleri için lineer (UCM ve

Oldroyd-B modelleri) ve lineer olmayan (Giesekus ve PTT modelleri) modeller kullanılarak polimerik akışkanın

viskoelastik karakteristikleri ortaya konulabilir.

Keywords

Mathematical modeling, porous media, non-isothrmal viscoelastic fluid flows

GOVERNING EQUATIONS FOR QUASI-LINEAR AND NONLINEAR VISCOELASTIC FLUID FLOWS AND HEAT TRANSFER THROUGH STATIONARY POROUS MEDIA

Abstract

Mathematical modeling involving porous heterogeneous media is important in a number of composite

manufacturing processes, such as resin transfer molding (RTM), injection molding and the like. In this research, a

mathematical model by utilizing the local volume averaging technique to establish 3-D, time dependent and nonisothermal

governing equations is presented. The developments should be able to predict the averaged velocity,

pressure, polymeric stress and temperature fields by modeling the conservation laws (e.g. mass, momentum and

energy) of the flow field coupled with constitutive equations for polymeric stress field. The governing equations

of the flow are averaged for the fluid phase. Furthermore, the model target a variety of viscoelastic models (e.g.

Newtonian model, Upper-Convected-Maxwell Model, Oldroyd-B model, Giesekus model and PTT modl) to

provide a fundamental understanding of the elastic effects on the flow field. The present research is focused on

non-isothermal considerations and a variety of constitutive models accounting for the viscoelastic flow behaviors

Keywords

Mathematical modeling, porous media, non-isothrmal viscoelastic fluid flows

References

• [1] T. W. Lambe, R. V. Whitman, “Soil Mechanics”, J. Wiley & Sons, NY,1969.
• [2] E. Scheidegger, “Physics of Flow Through Porous Media”, U Toronto Press, 3rd ed., Toronto, 1974
• [3] H. Darcy, “Les Fontaines Publiques de la ville de dijon”, Dalmont, Paris, 1856.
• [4] R. A. Greenkorn, “Flow Phenomena in porous Media”, Marcel Dekker, NY, 1983.
• [5] R. J. Dave, J. L. Kardos, M. P. Dudukovic, “Model for resin flow during composite processing: Part 1-General mathematical development” Polym. Compos, 8(1): 29-38, 1987.
• [6] T. G. Gutowski, M. Tadahiko, C. Zhong. “Consolidation of laminate composites”, J. Compos. Mater.,21(2): 172-188, 1987.
• [7] Z. Tadmor, C. G. Gogos. “Principles of Polymer Processing”, J. Wiley & Sons, NY, 1979.
• [8] K. L. Adams, B. Miller, L. Rebenfeld. “Forced in-plane flow of an epoxy resin in fibrous networks”,Polym. Eng. Sc., 26(20): 1434-1441,1986
• [9] J. P. Coulter and S. I. Guceri. “Resin impregnation during the manufacturing of composite materials subject to prescribed injection rate”, J. Reinf Plast & Comp. 7(3): 200-219, 1988.
• [10] G. Q. Martin and J. S. Son. “Fluid mechanics of mold filling for fiber reinforced plastics”,
• [11] Publ by ASM Int, Metals Park, OH, USA. p 149-157, 1986
• [12] J. Kozeny. Sitzungsberichte Wiener Akademie der Wissenschaft, 136, 1927.
• [13] P. C. Carman. “Fluid flow through granular beds”, Trans. Inst. Chem. Eng. 15, 1937
• [14] E. Scheidegger, “Physics of Flow through Porous Media”, 3rd ed., University of Toronto Press, Toronto, 1974.
• [15] R. C. Lam, J. L. Kardos. “Flow of resin through aligned and cross-plied fiber beds during
• [16] processing of composites”, ANTEC, 1989.
• [17] R. Dave, J. L. Kardos, M. P. Dudokovic. ”Model for resin flow during composite processing part 2:Numerical analysis for unidirectional graphite/epoxy laminates”, Polym. Composites, 8(2): 123-132, 1987.
• [18] L. Skartsis, J. L. Kardos. Proocedings of the American Society for Composites 5th Technical Conference, 1990.
• [19] Chiemlewski, C. A. Petty, K. Jayaraman. Proocedings of the American Society for Composites 5th Technical Conference, 1990.
• [20] T. G. Gutowski, Z. Cai S. Bauer, D. Boucher, J. Kingery, S. Wineman. “Consolidation experiments for laminate composites”, J. Compos. Mater., 23(1): 650-669, 1987.
• 21] M. V. Bruschke. “A predictive model for permeability and non-isothermal flow of viscous and shearthinning fluids in anisotropic fibrious media”, PhD thesis, University of Delaware, 1992.
• [22] J. Happel, H. Brenner. “Low Reynolds Number Hydrodynamics”, Boston, 1986.
• [23] S. Sangani, A. Acrivos. ”Slow flow through a periodic array of spheres”, Int. J. Multiphase Flow,8(4):343-360, 1982. [24] S. Sangani, A. Acrivos. “Slow flow past periodic array of cylinders with application to heat transfer”, Int. J. Multiphase Flow, 8(3): 193-206, 1982.
• [25] J. E. Drummond, M. I. Tahir. “Laminar viscous flow through regular arrays of parallel solid cylinders”,Int. J. Multiphase Flow, 10(5): 515-540, 1984.
• 26] P. Yu, T. T. Soong. “A random cell model for pressure drop prediction in fibrous filters”, J. Appl. Mech.,42(2): 301-304, 1975.
• [27] S. Sangani, C. Yao. “Transport processes in random arrays of cylinders. II. Viscous flow”, Phys. Fluids,31(9): 2435-2444, 1988.
• [28] G. Neale, J. H. Masliyah. “Flow perpendicular to mats of randomly arranged cylindrical fibers (importance of cell models)”, AIChE J., 21(4): 805-807, 1975.
• [29] H.C. Brinkman. “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles”, Applied Science Research, A1:27:34, 1947
• [30] T.S. Lungdren. “Slow flow through stationary random beds and suspensions of spheres”, Journal of Fluid Mechanics, 51 (part 2):273-299, 1972
• [31] G.S. Beavers and D.D. Joseph. “Boundary conditions at a naturally permeable wall”, Journal of Fluid D.P.Ü. Fen Bilimleri Enstitüsü Governing Equations for Quasi-Linear and Non-Linear Viscoelastic Fluid Flows and 14. Sayı Aralık 2007 Heat Transfer Through Stationary Porous MediaB.ALAKUŞ & R.KÖSE 1 0 6 Mechanics, 30: 197-207, 1967
• [32] G.S. Beavers, E.M. Sparrow, and B.A. Masha. “Boundary condition at a porous surface which bounds a fluid flow”, A.I. Ch.E. Journal, 20(3): 596-597, 1974
• [33] E.M. Sparrow and A.L. Loeffler, Jr. “Longitudinal laminar flow between cylinders arranged in regular array”, A.I. Ch.E. Journal, 5(3): 325-330, 1959
• [34] A.S. Sangani and A. Acrivos. “Slow flow past periodic arrays of cylinders with application to heat transfer”, International Journal of Multiphase Flow, 8(3): 193-206, 1982
• [35] B.R. Gebart. “Permeability of unidirectional reinforcements for RTM”, Journal of Composite Materials, 26(8): 1100-1133, 1992
• [36] A.L. Berdichevsky, and Z. Cai. “Prediction of permeability of fibrous media using self-consistent method”, First Intl. Conf. Transport Phenom. Process, pages 1203-1212,1992
• [37] Z. Cai and A.L. Berdichevsky. “Improved self-consistent method for estimating the
• [38] permeability of a fiber assembly”, Polymer Composites, 14(4): 314-323, 1993
• [39] A.W. Chan, D.E. Larive and R.J. Morgan. “Anistropic permeability of fiber preforms: Constant flow rate measurement”, Journal of Composite Materials, 27(10):996-1008, 1993
• [40] R.S. Parnas and A.J. Salem. “A comparison of the unidirectional and radial in-plane flow of fluids through woven composite reinforcements”, Polymer Composites, 14(5):383-394, 1993
• [41] F.R. Phelan, Jr., Y. Leung, and R.S. Parnas. “Modeling of microscale flow in unidirectional fibrous porous media”, Journal of Thermoplastic Composite Materials, 7:208-218, 1994
• [42] S. Ranganathan, G.M. Wise, F.R. Phelan, Jr., R.S. Parnas, and S.G. Advani, Proceedings of the 10th Annual ASM/ESD Advanced Composites Conference and Exposition, 1994
• [43] W. Chang and N. Kikuchi. “Analysis of non-isothermal mold filling process in resin transfer molding and structural reaction injection molding”, Computer Methods in Applied Mechanics and Engineering, 112:41-68, 1994
• [44] J.B. Keller, “Nonlinear Partial Differential Equations in Engineering and Applied Science”, pages 429- 443, M. Dekker, 1980, New York
• [45] H.I. Ene, “Dynamics of Fluids in Hierarchical Porous Media”, pages 223-241 Academic Press Ltd., San Diego, 1990
• [46] W. Chang and N. Kikuchi. “Analysis of non-isothermal mold filling process in resin transfer molding (RTM) and structural reaction injection molding (SRIM)”, Computational Mechanics, 16:22-35, 1995
• [47] K. Vafai. “Convective flow and heat transfer in variable-porosity media” Journal of Fluid Mechanics, 147:233-259, 1984
• [48] K. Vafai and C.L. Tien. “Boundary and inertial effects on flow and heat transfer
• [49] in porous media”, Int. J. Heat Mass Transfer, 24:195-203, 1981
• [50] K. Vafai and C.L. Tien, “Boundary and inertial effects on convective mass transfer in porous media”, Int. J. Heat Mass Transfer, 25:1183-1190, 1982
• [51] S.J. Kim. “Interfacial interactions in heat transfer and fluid flow through porous media”, PhD Thesis,Ohio State University, 1989.
• [52] K. Vafai and M. Sozen. “ Investigation of a latent heat storage porous bed and condensing flow through it”, Transaction of the ASME, 112:1014-1022, 1990
• [53] Amiri and K. Vafai. “Analysis of dispersion effects and non-thermal equilibrium,
• [54] non-Darcian, variable porosity incompressible flow through porous media”, Int. J. Heat Mass Transfer, 37:939-954, 1994
• [55] C.L. Tucker and R.B. Dessenberger. “Governing equations for flow and heat transfer in stationary”, Flow and Rheology in Polymer Composites Manufacturing, by S.G. Advani, Chapter 8, pp 257-323, 1994
• [56] Alakus . “Finite element viscoelastic fluid flow computatıons through porous media employing quasilinear and nonlinear models”, PhD Thesis, The University of Minnesota, 2001
• [57] R. B. Bird, R. C. Armstrong, O. Hassager. “Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics “, 2nd ed., Wiley Interscience, 1987