doi: 10.3934/era.2021071
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Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids

College of Mathematics and Computer Science, Fuzhou University, Key Laboratory of Operations Research and Control of Universities in Fujian, Fuzhou 350108, China

* Corresponding author: Fei Jiang

Received  June 2021 Revised  July 2021 Early access September 2021

Fund Project: The research of Fei Jiang was supported by NSFC (Grant No. 12022102) and the Natural Science Foundation of Fujian Province of China (2020J02013)

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. Electronic Research Archive, doi: 10.3934/era.2021071
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show all references

References:
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[12]

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[14]

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[15]

C. Guillopé and J.-C. Saut, 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., 274 (1992), 64-92.   Google Scholar

[16]

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[17]

Y. Guo and I. Tice, Compressible, inviscid Rayleigh–Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-712.  doi: 10.1512/iumj.2011.60.4193.  Google Scholar

[18]

Y. Guo and I. Tice, Linear Rayleigh–Taylor instability for viscous, compressible fluids, SIAM J. Math. Anal., 42 (2010), 1688-1720.  doi: 10.1137/090777438.  Google Scholar

[19]

Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal., 207 (2013), 459-531.  doi: 10.1007/s00205-012-0570-z.  Google Scholar

[20]

Y. Guo and I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6 (2013), 1429-1533.  doi: 10.2140/apde.2013.6.1429.  Google Scholar

[21]

D. M. Herbert, On the stability of visco-elastic liquids in heated plane Couette flow, J. Fluid Mech., 17 (1963), 353-359.  doi: 10.1017/S0022112063001397.  Google Scholar

[22]

L. N. Howard, Hydrodynamic and hydromagnetic stability, J. Fluid Mech., 13 (1962), 158-160.   Google Scholar

[23]

X. Hu, Global existence of weak solutions to two dimensional compressible viscoelastic flows, J. Differential Equations, 265 (2018), 3130-3167.  doi: 10.1016/j.jde.2018.05.001.  Google Scholar

[24]

X. Hu and F. Lin, Global solution to two dimensional incompressible viscoelastic fluid with discontinuous data, Comm. Pure Appl. Math., 69 (2016), 372-404.  doi: 10.1002/cpa.21561.  Google Scholar

[25]

X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.  doi: 10.1016/j.jde.2010.10.017.  Google Scholar

[26]

X. Hu and D. Wang, The initial-boundary value problem for the compressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 917-934.  doi: 10.3934/dcds.2015.35.917.  Google Scholar

[27]

X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833.  doi: 10.1137/120892350.  Google Scholar

[28]

J. JangI. Tice and Y. Wang, The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit, Commun. Math. Phys., 343 (2016), 1039-1113.  doi: 10.1007/s00220-016-2603-1.  Google Scholar

[29]

F. Jiang and S. Jiang, On linear instability and stability of the Rayleigh–Taylor problem in magnetohydrodynamics, J. Math. Fluid Mech., 17 (2015), 639-668.  doi: 10.1007/s00021-015-0221-x.  Google Scholar

[30]

F. Jiang and S. Jiang, On the stabilizing effect of the magnetic fields in the magnetic Rayleigh-Taylor problem, SIAM J. Math. Anal., 50 (2018), 491-540.  doi: 10.1137/16M1069584.  Google Scholar

[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.  Google Scholar

[32]

F. Jiang and S. Jiang, On magnetic inhibition theory in non-resistive magnetohydrodynamic fluids, Arch. Ration. Mech. Anal., 233 (2019), 749-798.  doi: 10.1007/s00205-019-01367-8.  Google Scholar

[33]

F. Jiang and S. Jiang, Strong solutions of the equations for viscoelastic fluids in some classes of large data, J. Differential Equations, 282 (2021), 148-183.  doi: 10.1016/j.jde.2021.02.020.  Google Scholar

[34]

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