It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.
Citation: Fei Jiang. Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids[J]. Electronic Research Archive, 2021, 29(6): 4051-4074. doi: 10.3934/era.2021071
It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.
[1] | H. Bénard, Les tourbillons cellulaires dans une nappe liquide, Revue Générale des Sciences Pures et Appliquées, 45 (1900), 1261–71 and 1309–28. |
[2] | Convection in a viscoelastic fluid layer in hydromagnetics. Physics Letters A (1971) 37: 419-420. |
[3] | Thermal instability in a viscoelastic fluid layer in hydromagnetics. J. Math. Anal. Appl. (1973) 41: 271-283. |
[4] | Rayleigh–Taylor instability in a viscoelastic binary fluid. J. Fluid Mech. (2010) 643: 127-136. |
[5] | On the mathematical modelling of a compressible viscoelastic fluid. Arch. Rational Mech. Anal. (2012) 205: 1-26. |
[6] | Y. Cai, Uniform bound of the highest-order energy of the 2D incompressible elastodynamics, arXiv: 2010.08718, (2020). |
[7] | S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Oxford, Clarendon Press, 1961. |
[8] | Linear stability of compressible vortex sheets in two-dimensional elastodynamics. Adv. Math. (2017) 311: 18-60. |
[9] | The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm. Partial Differential Equations (2006) 31: 1793-1810. |
[10] | Remarks on Oldroyd-B and related complex fluid models. Commun. Math. Sci. (2012) 10: 33-73. |
[11] | Rayleigh–Benard convection of viscoelastic fluid. Appl. Math. Comput. (2003) 136: 251-267. |
[12] | P. G. Drazin and W. H. Reid, Hydrodynamic Stability, 2nd, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616938 |
[13] | I. A. Eltayeb, Convective instability in a rapidly rotating viscoelastic layer, Zeitschrift für Angewandte Mathematik und Mechanik, 55 (1975), 599–604. doi: 10.1002/zamm.19750551008 |
[14] | Nonlinear thermal convection in an elasticoviscous layer heated from below. Proc. Roy. Soc. London Ser. A (1977) 356: 161-176. |
[15] | Mathematical problems arising in differential models for viscoelastic fluids. In: Rodrigues, J. F., Sequeira, A. (eds.). Mathematical Topics in Fluid Mechanics, Pitman Res. Notes Math. Ser. (1992) 274: 64-92. |
[16] | Dynamics near unstable, interfacial fluids. Commun. Math. Phys. (2007) 270: 635-689. |
[17] | Compressible, inviscid Rayleigh–Taylor instability. Indiana Univ. Math. J. (2011) 60: 677-712. |
[18] | Linear Rayleigh–Taylor instability for viscous, compressible fluids. SIAM J. Math. Anal. (2010) 42: 1688-1720. |
[19] | Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Ration. Mech. Anal. (2013) 207: 459-531. |
[20] | Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE (2013) 6: 1429-1533. |
[21] | On the stability of visco-elastic liquids in heated plane Couette flow. J. Fluid Mech. (1963) 17: 353-359. |
[22] | Hydrodynamic and hydromagnetic stability. J. Fluid Mech. (1962) 13: 158-160. |
[23] | Global existence of weak solutions to two dimensional compressible viscoelastic flows. J. Differential Equations (2018) 265: 3130-3167. |
[24] | Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data. Comm. Pure Appl. Math. (2016) 69: 372-404. |
[25] | Global existence for the multi-dimensional compressible viscoelastic flows. J. Differential Equations (2011) 250: 1200-1231. |
[26] | The initial-boundary value problem for the compressible viscoelastic flows. Discrete Contin. Dyn. Syst. (2015) 35: 917-934. |
[27] | Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows. SIAM J. Math. Anal. (2013) 45: 2815-2833. |
[28] | The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit. Commun. Math. Phys. (2016) 343: 1039-1113. |
[29] | On linear instability and stability of the Rayleigh–Taylor problem in magnetohydrodynamics. J. Math. Fluid Mech. (2015) 17: 639-668. |
[30] | On the stabilizing effect of the magnetic fields in the magnetic Rayleigh-Taylor problem. SIAM J. Math. Anal. (2018) 50: 491-540. |
[31] | F. Jiang and S. Jiang, Nonlinear stability and instability in the Rayleigh–Taylor problem of stratified compressible MHD fluids, Calc. Var. Partial Differential Equations, 58 (2019), Art. 29, 61 pp. doi: 10.1007/s00526-018-1477-9 |
[32] | On magnetic inhibition theory in non-resistive magnetohydrodynamic fluids. Arch. Ration. Mech. Anal. (2019) 233: 749-798. |
[33] | Strong solutions of the equations for viscoelastic fluids in some classes of large data. J. Differential Equations (2021) 282: 148-183. |
[34] | Nonlinear Rayleigh–Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete Contin. Dyn. Syst. Ser. S (2016) 9: 1853-1898. |
[35] | On the Rayleigh–Taylor instability for the incompressible viscous magnetohydrodynamic equations. Comm. Partial Differential Equations (2014) 39: 399-438. |
[36] | On stabilizing effect of elasticity in the Rayleigh–Taylor problem of stratified viscoelastic fluids. J. Funct. Anal. (2017) 272: 3763-3824. |
[37] | Instability of the abstract Rayleigh–Taylor problem and applications. Math. Models Methods Appl. Sci. (2020) 30: 2299-2388. |
[38] | Nonlinear stability of the viscoelastic Bénard problem. Nonlinearity (2020) 33: 1677-1704. |
[39] | On exponential stability of gravity driven viscoelastic flows. J. Differential Equations (2016) 260: 7498-7534. |
[40] | Non-linear overstability in the thermal convection of viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics (1995) 58: 331-356. |
[41] | Small scale creation for solutions of the incompressible two-dimensional Euler equation. Ann. of Math. (2) (2014) 180: 1205-1220. |
[42] | P. Kumar, H. Mohan and R. Lal, Effect of magnetic field on thermal instability of a rotating Rivlin-Ericksen viscoelastic fluid, Int. J. Math. Math. Sci., 2006 (2006), Art. ID 28042, 10 pp. doi: 10.1155/IJMMS/2006/28042 |
[43] | Well-posedness of surface wave equations above a viscoelastic fluid. J. Math. Fluid Mech. (2011) 13: 481-514. |
[44] | Global well-posedness of incompressible elastodynamics in two dimensions. Comm. Pure Appl. Math. (2016) 69: 2072-2106. |
[45] | Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. (2018) 188: 371-398. |
[46] | Well-posedness of the free boundary problem in incompressible elastodynamics. J. Differential Equations (2019) 267: 6604-6643. |
[47] | Three-dimensional thermal convection of viscoelastic fluids. Physical Review E (2005) 71: 066305. |
[48] | Some analytical issues for elastic complex fluids. Comm. Pure Appl. Math. (2012) 65: 893-919. |
[49] | On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. (2005) 58: 1437-1471. |
[50] | On the initial-boundary value problem of the incompressible viscoelastic fluid system. Comm. Pure Appl. Math. (2008) 61: 539-558. |
[51] | On the formulation of rheological equations of state. Proc. Roy. Soc. London Ser. A (1950) 200: 523-541. |
[52] | Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. London Ser. A (1958) 245: 278-297. |
[53] | Alternative derivation of differential constitutive equations of the Oldroyd-B type. J. Non Newton. Fluid Mech. (2009) 160: 40-46. |
[54] | Thermal convection thresholds in a Oldroyd magnetic fluid. Journal of Magnetism and Magnetic Materials (2011) 323: 691-698. |
[55] | On the Rayleigh–Taylor instability for the two-phase Navier–Stokes equations. Indiana Univ. Math. J. (2010) 59: 1853-1871. |
[56] | A thermodynamic frame work for rate type fluid models. J. Non Newton. Fluid Mech. (2000) 88: 207-227. |
[57] | L. Rayleigh, Investigation of the character of the equilibrium of an in compressible heavy fluid of variable density, Scientific Paper, II, (1990), 200–207. |
[58] | Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. (1985) 65: 449-451. |
[59] | Thermal convection in a viscoelastic liquid. Journal of Non-Newtonian Fluid Mechanics (1986) 21: 201-223. |
[60] | C. Rumford, Of the Propagation of Heat in Fluids, Complete Works, 1,239, American Academy of Arts and Sciences, Boston, 1870. |
[61] | Viscoelastic convection: Few-modes model and numerical simulations of field equations for Maxwellian fluids. Physical Review E (2012) 86: 046312. |
[62] | Thermal instability of an Oldroydian visco-elastic fluid in porous medium. Engrg. Trans. (1996) 44: 99-111. |
[63] | Rayleigh–Taylor instability of two viscoelastic superposed fluids. Acta Physica Academiae Scientiarum Hungaricae, Tomus (1978) 45: 213-220. |
[64] | Global existence for three-dimensional incompressible isotropic elastodynamics. Comm. Pure Appl. Math. (2007) 60: 1707-1730. |
[65] | Convective stability of a general viscoelastic fluid heated from below. The Physics of Fluids (1972) 15: 534-539. |
[66] | Solvability of a nonstationary thermal convection problem for a viscoelastic incompressible fluid. Differ. Equ. (2000) 36: 1225-1232. |
[67] | M. S. Swamy and W. Sidram, Effect of rotation on the onset of thermal convection in a viscoelastic fluid layer, Fluid Dyn. Res., 45 (2013), 015504, 21 pp. doi: 10.1088/0169-5983/45/1/015504 |
[68] | The stability of liquid surface when accelerated in a direction perpendicular to their planes. Proc. Roy Soc. A (1950) 201: 192-196. |
[69] | Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids (2007) 19: 103. |
[70] | On a changing tesselated. Structure in certain liquids. Pro. Phil. Soc. Glasgow (1882) 13: 464-468. |
[71] | J. H. Wang, Two-Dimensional Nonsteady Flows and Shock Waves (in Chinese), Science Press, Beijing, China, 1994. |
[72] | Global existence for the 2D incompressible isotropic elastodynamics for small initial data. Ann. Henri Poincaré (2017) 18: 1213-1267. |
[73] | Y. Wang, Critical magnetic number in the MHD Rayleigh–Taylor instability, J. Math. Phys., 53 (2012), 073701, 22 pp. doi: 10.1063/1.4731479 |
[74] | Sharp nonlinear stability criterion of viscous non-resistive MHD internal waves in 3D. Arch. Ration. Mech. Anal. (2019) 231: 1675-1743. |
[75] | The viscous surface-internal wave problem: global well-posedness and decay. Arch. Rational Mech. Anal. (2014) 212: 1-92. |
[76] | Global solvability of a free boundary three-dimensional incompressible viscoelastic fluid system with surface tension. Arch. Ration. Mech. Anal. (2013) 208: 753-803. |
[77] | Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below. Physics of Fluids (2008) 20: 084103. |
[78] | Y. Zhao, W. Wang and J. Cao, Stability of the viscoelastic Rayleigh–Taylor problem with internal surface tension, Nonlinear Anal. Real World Appl., 56 (2020), 103170, 28 pp. doi: 10.1016/j.nonrwa.2020.103170 |
[79] | Global solution to the incompressible Oldroyd-B model in the critical $L^p$ framework: The case of the non-small coupling parameter. Arch. Ration. Mech. Anal. (2014) 213: 651-687. |
[80] | Exponential growth of the vorticity gradient for the Euler equation on the torus. Adv. Math. (2015) 268: 396-403. |