臺灣博碩士論文加值系統

本文透過利用介面擴散法應用於Hele-Shaw流場中，配合高精準度之數值離散法並同時針對可互溶與不可互溶流場進行流場分析。透過將旋轉式Hele-Shaw流場的模擬結果發現，流場中因為黏性差異產生的介面不穩定現象(亦稱黏性指狀物)，其所產生的最大數量與無因次化參數Bond Number (Bo)相關(Bond Number為表面張力與離心力之比值)，其關係式為N=√(1⁄((3*〖Bo〗_e)))。另外在模擬的結果分析上，配合高精準度的數值離散方法我們成功地模擬出黏性指狀物在相關實驗室所呈現的各種特徵。同時透過比較不同的兩流體黏性比值(Atwood Number, A)，發現說當作用流體的黏性係數比環境流體小的時候(意即A<0)，其朝向圓心成長的黏性指狀物長度幾乎為定值，但朝外成長之黏性指狀物長度則較為混亂；反之，當作用流體黏性係數較環境流體高的時候(意即A>0)，則朝外成長之黏性指狀物長度較為統一。最後考量當柯式力存在於旋轉式Hele-Shaw流場所產生的影響。此時透過比較流場流線圖的結果發現，柯式力提供了在角度方向上的速度分佈，因此黏性指狀物在成長過程的產生了彎曲的現象，並且斷裂的情況也較多。另外在同時存在科式力情況之下，當Atwood Number越大時則科式力的影響越微弱。為了達到操控黏性指狀物成長的效應，在抽取式以及注入式Hele-Shaw流場中，在相同抽取/注入流量以及時間的前提下，我們考量了相關文獻中所提出的定量流率以及線性流率兩種不同的流率來進行分析。針對抽取式而言，不論是在可互溶流場或是不可互溶流場，兩種不同的流率所產生的黏性指狀物結果幾乎沒有差別。但是在透過分析後發現，不可互溶流場的抽取過程中所產生的黏性指狀物數量與文獻上所提到的結果(n_max≈√((1⁄3)*(1+((Ca*exp⁡(R)*r))⁄8)))非常吻合。而針對注入式的流場而言，在有表面張力的不可互溶流場以及考量到Korteweg Stress的可互溶流場的結果發現，線性注入流率的確提供了抑制指狀物成長的效果；但是對於完全可互溶流場而言卻無法達到抑制的效果。

In this study, we successfully use diffuse-interface method with high-order numerical scheme to simulate the Hele-Shaw flows. Diffuse-interface method helps us to simulate both miscible and immiscible flows by changing the interface energy profile. To verify our numerical scheme and diffuse-interface method, a rotating Hele-Shaw flow is simulated. In the results, we found that the number of viscous finger is a function of a dimensionless parameter, Bond number (Bo), which is a ratio of surface tension to centrifugal force, the function can be described as: N=√(1⁄((3*〖Boe〗))). Also, some features of viscous fingers what mentioned in references are captured in our simulations. We also compare the results of different fluid viscosity ratio, which is well-known dimensionless parameter Atwood number (A). When the viscosity of inner fluid is smaller than outer fluid (A<0), the length of each inward fingers keeps almost same; otherwise, when A>0, the outward fingers’ length almost keeps same. In the end, we try to find out the effect of Coriolis force. According to the comparison between the result with and without Coriolis force, the Coriolis force acts along the azimuthal direction. Because this azimuthal direction, the fingering patterns present a bending growing process, and increase the number of pinch-off tips. To control the growing of viscous fingers in suction and injection Hele-Shaw flows, a constant and linear flow rate, which have same flow volume and flow time, are considered in our simulations. For suction, the results show us that the linear flow rate cannot inhibit the viscous fingers’ growing, not only in immiscible condition, but also in fully miscible and partial miscible conditions. It is worth to noticed that the number of fingers of immiscible fluid flow follows the formula from the reference, which described as: n_max≈√((1⁄3)*(1+((Ca*exp(R)*r))⁄8)). For injection flow, the linear flow rate provides a great job of inhibit the fingers’ growing when flow exist the surface tension or Korteweg stress effect, which means the immiscible and partial miscible flow. Without the Korteweg stress, the linear flow rate does constrain the flow direction of viscous fingers, but it does not inhibit the growing process.

摘要......I
ABSTRACT......II
CONTENTS......III
LIST OF FIGURES......V
NOMENCLATURE......XI
CHAPTER 1. INTRODUCTION......1
1.1. HELE-SHAW FLOW......1
1.2. VISCOUS FINGERING STRUCTURES......1
CHAPTER 2. PHASE-FIELD APPROACH......5
2.1. DIFFUSE-INTERFACE METHOD......5
2.2. IMMISCIBLE HELE-SHAW FLOW......7
2.3. FULLY MISCIBLE HELE-SHAW FLOW......8
2.4. PARTIAL MISCIBLE HELE-SHAW FLOW......8
2.5. NUMERICAL SCHEME......9
CHAPTER 3. FINGERING DYNAMICS......13
3.1. ROTATING HELE-SHAW FLOW......13
3.1.1. Physical Model......13
3.1.2. Validation......16
3.1.3. Fingering Patterns......17
3.1.3.1. Fingering dynamics in the absence of the Coriolis force......18
3.1.3.2. Influence of the Coriolis force......19
3.2. RADIAL HELE-SHAW FLOW WITH SUCTION......20
3.2.1. Physical Model......20
3.2.2. Validation......24
3.2.3. Fingering Patterns......26
3.2.3.1. Constant suction scheme......26
3.2.3.2. Linear suction scheme......27
3.2.3.3. Qualitative measurements......28
CHAPTER 4. FINGERING CONTROL......60
4.1. Physical Model......60
4.2. Constant Injection Flow Rate......64
4.3. Linear Injection Flow Rate......66
4.4. Qualitative Measurement......68
CHAPTER 5. CONCLUSION......84
REFERENCE......85
APPENDIX......88
RESUME......90

[1]. A. Fick, “#westeur029#ber Diffusion”, Ann. der. Physik, 94, 1855.
[2]. Anatoliy Vorobev, “Boussinesq approximation of the Cahn - Hilliard - Navier - Stokes equations”, Phys. Rev. E, 82, 2010.
[3]. B. Jha, L. Cueto-Felgueroso, and R. Juanes, “Fluid mixing from viscous fingering”, Physical Review Letters, 106, 2011.
[4]. C. T. Tan and G. M. Homsy, “Simulation of nonlinear viscous fingering in miscible displacements.”, Phys. Fluids, 31(6), pp.1330-1338, 1988
[5]. C. T. Tan and G. M. Homsy, “Stability of miscible displacements in porous media: radial source flow.”, Phys. Fluids, 30(5), pp.1239-1987, 1987
[6]. C.-Y. Chen and E. Meiburg, “Miscible porous media displacements in the quarter five-spot configuration.”, J. Fluid Mech., 371(233), 1998
[7]. C.-Y. Chen and S.-W. Wang, “Miscible droplet in a porous medium and the effects of Korteweg stresses”, Fluid Dyn. Res., 30(315), 2002
[8]. C.-Y. Chen, C.-H. Chen, and J. A. Miranda, “Numerical study of pattern formation in miscible rotating Hele-Shaw flows.”, Phys. Rev. E, 73, 2006
[9]. C.-Y. Chen, C.-W. Huang, H. Gadelha, and Miranda, J., “Radial viscous fingering in miscible Hele-Shaw flows: A numerical study”, Phys. Rev. E, 78(1), 2008.
[10]. C.-Y. Chen, Li-Lin Wang, and Meiburg, Eckart, “Miscible Droplets in a Porous Medium and the Effect of Korteweg Stresses”, Phys. Fluids, 13(9) pp.2447~2456, 2001
[11]. C.-Y. Chen, C.-W. Huang and L.-C. Wang, “Controlling radial fingering patterns in miscible confined flows”, Phys. Rev. E, 82, 2010.
[12]. Chui J, De Anna P, HuanesR ,”The impact of miscible viscous fingering on mixing.”, Bull Am Phys Soc, 59(20), 2014
[13]. Coutinho ALGA, Alves J, “Finite element simulation of nonlinear viscous fingering in miscible displacements with anisotropic dispersion and nonmonotonic viscosity profiles.”, Comput Mech, 23,pp.108–116, 1999.
[14]. D. P. Jackson and J. A. Miranda, “Orientational Preference and Predictability in a Symmetric Arrangement of Magnetic Drops”, Phys. Rev. E, 67, 2003
[15]. Dzyaloshinskii, I E, Lifshitz, E M, Pitaevskii, "GENERAL THEORY OF VAN DER WAALS' FORCES". Soviet Physics Uspekhi, 4 (2), 1961
[16]. E. Alvarez-Lacalle, H. Gad#westeur043#lha and J. A. Miranda, “Coriolis effects on fingering patterns under rotation”, Phys. Rev. E, 78(2), 2008.
[17]. E. Alvarez-Lacalle, J. Ort#westeur046#n and J. Casademunt, “Low viscosity contrast fingering in a rotating Hele-Shaw cell”, Phys. Fluids., 16(4), pp.908-924, 2004.
[18]. E. Alvarez-Lacalle, J. Ort#westeur046#n and J. Casademunt, “Relevance of dynamic wetting in viscous fingering patterns”, Phys. Rev. E, 74(2), 2006.
[19]. E. Meiburg and C.-Y. Chen, “High-accuracy implicit finite-difference simulations of homogeneous and heterogeneous miscible-porous-medium flows”, SPE J., 5(2), pp.129-137, 2000.
[20]. Eduardo O. Dias, Enrique Alvarez-Lacalle, M#westeur034#rcio S. Carvalho and Jos#westeur042# A. Miranda, “Minimization of Viscous Fluid Fingering: A Variational Scheme for Optimal Flow Rates”, Phys. Rev. L., 109, 2012
[21]. H. Thom#westeur042#, M. Rabaud, V. Hakim, and Y. Couder, Phys. Fluids A1, 224, 1989
[22]. Hyeong-Gi Lee and J. S. Lowengrub, “Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration”, Phys. Fluids, 14 (2), Feb 2002
[23]. J. A. Miranda and E. Alvarez-Lacalle, “Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows”, Phys. Rev. E 72, 2005.
[24]. J. A. Miranda and M. Widom, “Radial fingering in a Hele–Shaw cell: a weakly nonlinear analysis”, Physica D, 120, pp.315-328, 1998.
[25]. J. Kuang, T. Maxworthy and P. Petitjeans, “Miscible displacements between silicone oils in capillary tubes”, Eur. J. Mech. B: Fluids 22, pp.271-277, 2003.
[26]. J. Mathiesen, I. Procaccia, H. L. Swinney and M. Thrasher, “The Universality Class of Diffusion Limited Aggregation and Viscous Fingering.”, EPL 76(2), pp.257-263, 2006
[27]. J. V. Maher, “Development of Viscous Fingering Patterns”, Phys. Rev. Lett. 54(14), pp.1498-1501, 1985.
[28]. J.-D. Chen, “Growth of radial viscous fingers in a Hele–Shaw cell”, J. Fluid Mech. 201, pp.223-242, 1989.
[29]. Lincoln Paterson, “Radial fingering in a Hele Shaw cell”, J. Fluid Mech. 113, pp.513-529, 1981.
[30]. Leonard William Schwartz, “Instability and Fingering in a Rotating Hele-Shaw Cell or Porous Medium”, Phys. Fluids A, 1, pp.167-170, 1989.
[31]. Ll. Carrillo, F. X.Magdaleno, J. Casademunt and J. Ort#westeur046#n, Phys. Rev. E 54(6), pp.6260-6267 ,1996.
[32]. M. Ruith and E. Meiburg, “Miscible Rectilinear Displacement with Gravity Override.”, J. Fluid Mech., 420, pp.225-257, 2000.
[33]. O. Manikcam and G. M. Homsy, “Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles”, Physics of Fluids, 6(1), pp.95-107, 1994.
[34]. O. Praud and H. L. Swinney, “Fractal dimension and unscreened angles measured for radial viscous fingering”, Phys. Rev. E 72, 2005.
[35]. P. Fast and M. J. Shelley, “Moores law and the Saffman–Taylor instability”, J. Comput. Phys. 212(1), 2005.
[36]. P. G. Saffman , Geoffrey Taylor, “The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid”, Proc. R. Soc. London A 245, 1958.
[37]. P. Petitjeans and T. Maxworthy, “Miscible Displacement in Capillary Tubes. Part 1. Experiments.”, J. Fluid Mech. 326, pp.37-56, 1996.
[38]. Philippe Petitjeans, Ching-Yao Chen, Eckart Meiburg, Tony Maxworthy, “Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion”, Phys. Fluids. 11, pp.1705-1716 ,1999.
[39]. Pengtao Yue, Chunfeng Zhou and James J. Feng, “A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids.”, Phys. Fluids. 18, 2006.
[40]. Pengtao Yue, Chunfeng Zhou and James J. Feng, “Sharp interface limit of the Cahn-Hilliard model for moving contact lines”, J. Fluid Mech. 645, pp.279-294 , 2010.
[41]. Pengtao Yue, Zhou, C, Feng, J. J., Ollivier-Gooch, C. F. &; Hu, H. H., “Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing.”, J. Comput. Phys. 219, 47-67, 2006.
[42]. R. Folch, E. Alvarez-Lacalle, J. Ort#westeur046#n and J. Casademunt, “Pattern formation and interface pinch-off in rotating Hele-Shaw flows: A phase-field approach”, Phys. Rev. E 80, 2009.
[43]. S. W. Li, J. S. Lowengrub, and P. H. Leo, “A rescaling scheme with application to the long time simulation of viscous fingering in a Hele-Shaw cell”, J. Comput. Phys. 225, pp.554-567, 2007.
[44]. T. Maxworthy, “The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell”, J. Fluid Mech., 177, pp.207-232, 1987.
[45]. Yue PT, Feng J, Liu C, Shen J ,“A diffuse-interface method for simulating two-phase flows of complex fluids.”, J Fluid Mech 515:293–317,2004
[46]. Ed. A. Fasano and M. Primicerio, Free Boundary Problems: Theory and Applications, Pitman, Boston, 1983.
[47]. Langer, J. S. "Models of Pattern Formation in First-Order Phase Transitions". Directions In Condensed Matter Physics. ,(1), pp.165-186, 1986.
[48]. Gr#westeur042#tar Tryggvason and Hassan Aref, “Finger-interaction mechanisms in stratified Hele-Shaw flow”, J. Fluid Mech. 154, pp.287-301, 1985.