Rayleigh-Taylor Instability of Viscoelastic Soft Solids in Hypergravity
-
摘要: 在超重力环境下，受限黏弹性软固体的自由表面可能发生Rayleigh-Taylor失稳(Rayleigh-Taylor instability, RTI)，其演化行为同时受到材料流变特性与几何约束的显著影响. 本文以受限圆柱形黏弹性软固体为研究对象，基于线性黏弹性本构关系建立了自由表面扰动的线性稳定性分析框架，通过在频域中求解控制方程，推导得到了扰动增长率与波数之间的色散关系，从而系统刻画了超重力、表面张力、材料压缩性及黏性耗散等因素对失稳特征的影响机制. 进一步考虑有限几何尺寸的作用，在柱坐标系下引入环向边界条件，将连续波数离散化，构建了适用于受限体系的失稳模态描述方法，明确了径厚比在失稳临界值与模态选择行为中的关键作用. 在此基础上，结合有限元特征值分析与非线性数值模拟，对理论预测进行了验证，并用于探讨失稳模态与后期形貌演化之间的关联. 本文的研究为分析超重力条件下受限黏弹性软固体的界面稳定性问题提供了一种系统的理论与数值研究思路，可为相关实验设计及软材料失稳形貌调控提供参考.
-
关键词:
- Rayleigh-Taylor失稳 /
- 超重力 /
- 黏弹性 /
- 模态选择 /
- 软材料
Abstract: Under hypergravity conditions, the free surface of confined viscoelastic soft solids can become unstable due to Rayleigh-Taylor instability, with the evolution behavior governed by both material rheology and geometric confinement. The confined cylindrical viscoelastic soft solids were studied, and a linear stability analysis for free-surface perturbations was developed based on linear viscoelastic constitutive relations. The governing equations were formulated and solved in the frequency domain, to deduce the dispersion relation between the perturbation growth rate and the wavenumber. Then, the roles of hypergravity, surface tension, material compressibility and viscous dissipation were systematically investigated in the instability process. Finite geometric effects were incorporated through introduction of circumferential boundary conditions in a cylindrical coordinate system, to discretize the admissible wavenumbers and reveal the effects of finite confinement on the instability critical values and mode selections. Furthermore, the finite element method was used to verify the theoretical predictions and to investigate the relationship between instability modes and the subsequent evolutions of surface patterns. This study provides a coherent theoretical and numerical approach for analyzing interfacial stability in confined viscoelastic soft solids under hypergravity, and offers a guidance for experiment design and pattern control of soft materials.-
Key words:
- Rayleigh-Taylor instability /
- hypergravity /
- viscoelasticity /
- mode selection /
- soft material
-
图 1 超重力环境下受约束的黏弹性软固体示意图
Figure 1. Schematic diagram of a constrained viscoelastic soft solid in a supergravity environment
图 2 纯弹性软固体的Rayleigh-Taylor失稳色散曲线(τ=0)
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. RTI dispersion curves for pure elastic soft solids (τ=0)
图 3 纯弹性软固体的Rayleigh-Taylor失稳临界荷载分析(τ=0)
Figure 3. Critical load analysis of RTI critical loads of pure elastic soft solids (τ=0)
图 4 黏弹性软固体的Rayleigh-Taylor失稳参数分析$(\Sigma=0, \tilde{\nu}=0)$
Figure 4. The parameter analysis of viscoelastic soft solids $(\Sigma=0, \tilde{\nu}=0)$
图 5 纯弹性软固体临界荷载αc与径厚比R/H的关系$(\Sigma=0, \tilde{\nu}=0, \tau=0)$
Figure 5. The relationship between αc and R/H of pure elastic soft solids $(\Sigma=0, \tilde{\nu}=0, \tau=0)$
图 6 纯弹性软固体的模态选择$(\alpha=8, \tilde{\nu}=0, \tau=0)$
Figure 6. Modal selections of pure elastic soft solids $(\alpha=8, \tilde{\nu}=0, \tau=0)$
图 7 黏弹性软固体的模态选择$(\alpha=8, \Sigma=0, \tilde{\nu}=0, \tau=0.1, n=0.5)$
Figure 7. Modal selections of viscoelastic soft solids $(\alpha=8, \Sigma=0, \tilde{\nu}=0, \tau=0.1, n=0.5)$
表 1 圆柱形黏弹性软固体的Rayleigh-Taylor失稳模态
Table 1. RTI modes of cylindrical viscoelastic soft solids
R/H 0.5 1.0 1.5 2.0 τ=0 
τ=0.1 
R/H 2.5 3.0 3.5 4.0 τ=0 
τ=0.1 
下载: 导出CSV
表 2 圆柱形黏弹性软固体的Rayleigh-Taylor失稳演化过程
Table 2. Evolutions of RTI of cylindrical viscoelastic soft solids
R/H 4(N=5) 7(N=10) t/s 50 75 100 50 75 100 τ=0.1 
τ=0.2 
τ=0.3 
R/H 12(N=15) 15(N=20） t/s 50 75 100 50 75 100 τ=0.1 
τ=0.2 
τ=0.3 
下载: 导出CSV
-
[1] FOYART G, RAMOS L, MORA S, et al. The fingering to fracturing transition in a transient gel[J]. Soft Matter, 2013, 9(32): 7775-7779. doi: 10.1039/c3sm51320c [2] LIN S, COHEN T, ZHANG T, et al. Fringe instability in constrained soft elastic layers[J]. Soft Matter, 2016, 12(43): 8899-8906. doi: 10.1039/C6SM01672C [3] DU Y K, LV C F, LIU C S, et al. Prescribing patterns in growing tubular soft matter by initial residual stress[J]. Soft Matter, 2019, 15(42): 8468-8474. doi: 10.1039/C9SM01563A [4] GRZELKA M, BOSTWICK J B, DANIELS K E. Capillary fracture of ultrasoft gels: variability and delayed nucleation[J]. Soft Matter, 2017, 13(16): 2962-2966. doi: 10.1039/C7SM00257B [5] STYLE R W, BOLTYANSKIY R, CHE Y, et al. Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses[J]. Physical Review Letters, 2013, 110(6): 066103. doi: 10.1103/PhysRevLett.110.066103 [6] DERVAUX J, BEN AMAR M. Mechanical instabilities of gels[J]. Annual Review of Condensed Matter Physics, 2012, 3: 311-332. doi: 10.1146/annurev-conmatphys-062910-140436 [7] ANDREOTTI B, SNOEIJER J H. Statics and dynamics of soft wetting[J]. Annual Review of Fluid Mechanics, 2020, 52: 285-308. doi: 10.1146/annurev-fluid-010719-060147 [8] RAYLEIGH L. On the instability of a cylinder of viscous liquid under capillary force[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1892, 34(207): 145-154. doi: 10.1080/14786449208620301 [9] MORA S, PHOU T, FROMENTAL J M, et al. Gravity driven instability in elastic solid layers[J]. Physical Review Letters, 2014, 113(17): 178301. doi: 10.1103/PhysRevLett.113.178301 [10] CHOU Y J, SHAO Y C. Numerical study of particle-induced Rayleigh-Taylor instability: effects of particle settling and entrainment[J]. Physics of Fluids, 2016, 28(4): 043302. doi: 10.1063/1.4945652 [11] LORENZ K T, EDWARDS M J, GLENDINNING S G, et al. Accessing ultrahigh-pressure, quasi-isentropic states of matter[J]. Physics of Plasmas, 2005, 12(5): 056309. doi: 10.1063/1.1873812 [12] GORCZYK W, VOGT K. Tectonics and melting in intra-continental settings[J]. Gondwana Research, 2015, 27(1): 196-208. doi: 10.1016/j.gr.2013.09.021 [13] BURROWS A. Supernova explosions in the universe[J]. Nature, 2000, 403(6771): 727-733. doi: 10.1038/35001501 [14] BEN AMAR M, JIA F. Anisotropic growth shapes intestinal tissues during embryogenesis[J]. Proceedings of the National Academy of Sciences of the United States of America, 2013, 110(26): 10525-10530. [15] LEGOFF L, LECUIT T. Mechanical forces and growth in animal tissues[J]. Cold Spring Harbor Perspectives in Biology, 2016, 8(3): a019232. doi: 10.1101/cshperspect.a019232 [16] CHAKRABARTI A, MORA S, RICHARD F, et al. Selection of hexagonal buckling patterns by the elastic Rayleigh-Taylor instability[J]. Journal of the Mechanics and Physics of Solids, 2018, 121: 234-257. doi: 10.1016/j.jmps.2018.07.024 [17] ZHENG Y, LAI Y, HU Y, et al. Rayleigh-Taylor instability in a confined elastic soft cylinder[J]. Journal of the Mechanics and Physics of Solids, 2019, 131: 221-229. doi: 10.1016/j.jmps.2019.07.006 [18] RICCOBELLI D, CIARLETTA P. Rayleigh-Taylor instability in soft elastic layers[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017, 375(2093): 20160421. doi: 10.1098/rsta.2016.0421 [19] TAMIM S I, BOSTWICK J B. A dynamic analysis of the Rayleigh-Taylor instability in soft solids[J]. Extreme Mechanics Letters, 2020, 40: 100940. doi: 10.1016/j.eml.2020.100940 [20] PIRIZ S A, PIRIZ A R, TAHIR N A, et al. Magneto-Rayleigh-Taylor instability in an elastic-medium slab[J]. Physics of Fluids, 2018, 30(11): 111703. doi: 10.1063/1.5050800 [21] BRUN P T. Shape formation in interfacial flows[J]. Physical Review Fluids, 2024, 9(11): 110501. doi: 10.1103/PhysRevFluids.9.110501 [22] CHRISTENSEN. Theory of Viscoelasticity[M]. New York: Academic Press, 1982. [23] KARPITSCHKA S, DAS S, VAN GORCUM M, et al. Droplets move over viscoelastic substrates by surfing a ridge[J]. Nature Communications, 2015, 6: 7891. doi: 10.1038/ncomms8891 [24] SWEENEY H, KERSWELL R R, MULLIN T. Rayleigh-Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions[J]. Journal of Fluid Mechanics, 2013, 734: 338-362. doi: 10.1017/jfm.2013.492 -
计量
- 文章访问数: 119
- HTML全文浏览量: 44
- PDF下载量: 32
- 被引次数: 0
渝公网安备50010802005915号