Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface

Terrones, Guillermo; Carrara, Mark D.

doi:10.1063/1.4921648

Title: Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface

Abstract

For a spherical interface of radius R separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration, we perform a linear stability analysis in terms of spherical surface harmonics Y _n to derive the dispersion relation. The instability behavior is investigated by computing the growth rates and the most-unstable modes as a function of the spherical harmonic degree n. This general methodology is applicable to the entire parameter space spanned by the Atwood number, the viscosity ratio, and the dimensionless number B = (α_RΡ²₂/μ²_²)¹^/³ R (where α_R, Ρ₂ and μ₂ are the local radial acceleration at the interface, and the density and viscosity of the denser overlying fluid, respectively). While the mathematical formulation here is general, this paper focuses on instability that arises at a spherical viscous fluid/vacuum interface as there is a great deal to be learned from the effects of one-fluid viscosity and sphericity alone. To quantify and understand the effect that curvature and radial accelerationhave on the Rayleigh-Taylor instability, a comparison of the growth rates, under homologous driving conditions, between the planar and spherical interfaces is performed. The derived dispersion relation for the planar interface accounts for an underlying finitemore »« less

Authors:

^[1];

^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Fri May 01 00:00:00 EDT 2015

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE Office of Science (SC). Basic Energy Sciences (BES); USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1735910

Alternate Identifier(s):: OSTI ID: 1238594

Report Number(s):: LA-UR-20-24207; LA-UR-13-20570
Journal ID: ISSN 1070-6631; TRN: US2205267

Grant/Contract Number:: 89233218CNA000001; AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Physics of Fluids

Additional Journal Information:: Journal Volume: 27; Journal Issue: 5; Journal ID: ISSN 1070-6631

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; viscosity; perturbation theory; real analysis; flow instabilities; dispersion function; covariance and correlation; fluid mechanics; viscous liquid; Newtonian fluids; linear stability analysis

Citation Formats


                    Terrones, Guillermo, and Carrara, Mark D. Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface.  United States: N. p., 2015. 
Web.  doi:10.1063/1.4921648.

Copy to clipboard


                    Terrones, Guillermo, & Carrara, Mark D. Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface.  United States.  https://doi.org/10.1063/1.4921648

Copy to clipboard


                    Terrones, Guillermo, and Carrara, Mark D. Fri .  
"Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface".  United States.  https://doi.org/10.1063/1.4921648.  https://www.osti.gov/servlets/purl/1735910.

Copy to clipboard


                    
@article{osti_1735910,

  title        = {Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface},

  author       = {Terrones, Guillermo and Carrara, Mark D.},

  abstractNote = {For a spherical interface of radius R separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration, we perform a linear stability analysis in terms of spherical surface harmonics Y n to derive the dispersion relation. The instability behavior is investigated by computing the growth rates and the most-unstable modes as a function of the spherical harmonic degree n. This general methodology is applicable to the entire parameter space spanned by the Atwood number, the viscosity ratio, and the dimensionless number B = (αRΡ²2/μ²²)¹/³ R (where αR, Ρ2 and μ2 are the local radial acceleration at the interface, and the density and viscosity of the denser overlying fluid, respectively). While the mathematical formulation here is general, this paper focuses on instability that arises at a spherical viscous fluid/vacuum interface as there is a great deal to be learned from the effects of one-fluid viscosity and sphericity alone. To quantify and understand the effect that curvature and radial accelerationhave on the Rayleigh-Taylor instability, a comparison of the growth rates, under homologous driving conditions, between the planar and spherical interfaces is performed. The derived dispersion relation for the planar interface accounts for an underlying finite fluid region of thickness L and normal acceleration αR. Under certain conditions, the development of the most-unstable modes at a spherical interface can take place via the superposition of two adjacent spherical harmonics Yn and Yn+1. This bimodality in the evolution of disturbances in the linear regime does not have a counterpart in the planar configuration where the most-unstable modes are associated with a unique wave number.},

  doi          = {10.1063/1.4921648},

  journal      = {Physics of Fluids},

  number       = 5,

  volume       = 27,

  place        = {United States},

  year         = {Fri May 01 00:00:00 EDT 2015},

  month        = {Fri May 01 00:00:00 EDT 2015}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1063/1.4921648

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 23 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Bubble Dynamics and Cavitation
journal, January 1977

Plesset, M. S.; Prosperetti, A.
Annual Review of Fluid Mechanics, Vol. 9, Issue 1
DOI: 10.1146/annurev.fl.09.010177.001045

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells
journal, September 1990

Mikaelian, Karnig O.
Physical Review A, Vol. 42, Issue 6
DOI: 10.1103/PhysRevA.42.3400

The Character of the Equilibrium of an Incompressible Fluid Sphere of Variable Density and Viscosity Subject to Radial Acceleration
journal, January 1955

Chandrasekhar, S.
The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 8, Issue 1
DOI: 10.1093/qjmam/8.1.1

The stability of the surface of a cavitation bubble
journal, January 1953

Binnie, A. M.
Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 49, Issue 1
DOI: 10.1017/S0305004100028152

On the Bell–Plesset effects: The effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability
journal, November 2004

Epstein, R.
Physics of Plasmas, Vol. 11, Issue 11
DOI: 10.1063/1.1790496

An overview of Rayleigh-Taylor instability
journal, July 1984

Sharp, D. H.
Physica D: Nonlinear Phenomena, Vol. 12, Issue 1-3
DOI: 10.1016/0167-2789(84)90510-4

Unstable normal mode for Rayleigh–Taylor instability in viscous fluids
journal, January 1977

Menikoff, R.; Mjolsness, R. C.; Sharp, D. H.
Physics of Fluids, Vol. 20, Issue 12
DOI: 10.1063/1.861831

Chemical and hydrodynamical analysis of stability of a spherical interface
journal, August 1976

Sørensen, Torben Smith; Hennenberg, Marcel; Steinchen, Annie
Journal of Colloid and Interface Science, Vol. 56, Issue 2
DOI: 10.1016/0021-9797(76)90243-5

Stability and mix in spherical geometry
journal, August 1990

Mikaelian, Karnig O.
Physical Review Letters, Vol. 65, Issue 8
DOI: 10.1103/PhysRevLett.65.992

On the Stability of Fluid Flows with Spherical Symmetry
journal, January 1954

Plesset, M. S.
Journal of Applied Physics, Vol. 25, Issue 1
DOI: 10.1063/1.1721529

Eigenvalues and degeneracies for n ‐dimensional tensor spherical harmonics
journal, October 1984

Rubin, Mark A.; Ordóñez, Carlos R.
Journal of Mathematical Physics, Vol. 25, Issue 10
DOI: 10.1063/1.526034

The Non-Linear Field Theories of Mechanics
book, January 1965

Truesdell, C.; Noll, W.
The Non-Linear Field Theories of Mechanics / Die Nicht-Linearen Feldtheorien der Mechanik
DOI: 10.1007/978-3-642-46015-9_1

Fastest growing linear Rayleigh-Taylor modes at solid∕fluid and solid∕solid interfaces
journal, March 2005

Terrones, Guillermo
Physical Review E, Vol. 71, Issue 3
DOI: 10.1103/PhysRevE.71.036306

Stability of a compressed gas bubble in a viscous fluid
journal, June 1989

Iooss, G.; Laure, P.; Rossi, M.
Physics of Fluids A: Fluid Dynamics, Vol. 1, Issue 6
DOI: 10.1063/1.857402

Theory of the Rayleigh-Taylor instability
journal, August 1991

Kull, H. J.
Physics Reports, Vol. 206, Issue 5
DOI: 10.1016/0370-1573(91)90153-D

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shells
journal, September 2005

Mikaelian, Karnig O.
Physics of Fluids, Vol. 17, Issue 9
DOI: 10.1063/1.2046712

Works referencing / citing this record:

Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration
journal, October 2018

Li, Yikai; Zhang, Peng; Kang, Ning
Physics of Fluids, Vol. 30, Issue 10
DOI: 10.1063/1.5050517

Similar Records in DOE PAGES and OSTI.GOV collections:

Finite boundary effects on the spherical Rayleigh–Taylor instability between viscous fluids

Journal Article Oren, Garrett H. ; Terrones, Guillermo - AIP Advances

For the Rayleigh–Taylor unstable arrangement of a viscous fluid sphere embedded in a finite viscous fluid spherical shell with a rigid boundary and a radially directed acceleration, a dispersion relation is developed from a linear stability analysis using the method of normal modes. [Formula: see text] is the radially directed acceleration at the interface. ρ_i denotes the density, μ_i is the viscosity, and R_i is the radius, where i = 1 is the inner sphere and i = 2 is the outer sphere. The dispersion relation is a function of the following dimensionless variables: viscosity ratio [Formula: see text], densitymore »« less
https://doi.org/10.1063/5.0090277

Full Text Available
TAYLOR INSTABILITY OF A DEBRIS INTERFACE UNDER VARIABLE ACCELERATION. Chap. 8 of THEORETICAL STUDY OF HYDROMAGNETIC STABILITY AND TURBULENCE. INVESTIGATIONS AND DETAILED RESULTS. Annual Report, Period: January 1-December 31, 1961

Technical Report Jarem, J

S> The Taylor instability of the interface separating an expanding plasma from the hydromagnetic region is studied. The plasma is treated as incompressible and the bubble interface which is assumed to be under variable acceleration is treated as a surface of discontinuity. The motion of the surface of discontinuity is prescribed and the flows determined so as to be compatible. The fluids on either side of the interface are viewed as being ideal hydromagnetic fluids and the boundary conditions appropriate to a moving interface are satisfied. The problem of small motions about such time dependent flows is considered by meansmore »« less
Stability of a compressed gas bubble in a viscous fluid

Journal Article Iooss, G ; Laure, P ; Rossi, M - Phys. Fluids A; (United States)

The stability of the spherical shape of the free surface of a gas bubble compressed by an incompressible fluid as it appears in the inertial confinement fusion problem is considered. (i) The equations derived by Prosperetti (Accad. Naz. Lincei 62, 196 (1977)) generalizing the Plesset equation are recovered in cases when the outer fluid is nonviscous, the flow being not potential, and it is shown that vorticity may change drastically the results of the potential case. (ii) In the case of viscous external fluid, the equations derived by Prosperetti (Q. Appl. Math. 1, 399 (1977)) and other external conditions onmore »« less
https://doi.org/10.1063/1.857402
Effects of initial radius of the interface and Atwood number on nonlinear saturation amplitudes in cylindrical Rayleigh-Taylor instability

Journal Article Liu, Wanhai ; LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190 ; Yu, Changping ; ... - Physics of Plasmas

Nonlinear saturation amplitudes (NSAs) of the first two harmonics in classical Rayleigh-Taylor instability (RTI) in cylindrical geometry for arbitrary Atwood numbers have been analytically investigated considering nonlinear corrections up to the fourth-order. The NSA of the fundamental mode is defined as the linear (purely exponential) growth amplitude of the fundamental mode at the saturation time when the growth of the fundamental mode (first harmonic) is reduced by 10% in comparison to its corresponding linear growth, and the NSA of the second harmonic can be obtained in the same way. The analytic results indicate that the effects of the initial radiusmore »« less
https://doi.org/10.1063/1.4901088
Viscous effects on the Rayleigh-Taylor instability with background temperature gradient

Journal Article Gerashchenko, S. ; Livescu, D., E-mail: livescu@lanl.gov - Physics of Plasmas

The growth rate of the compressible Rayleigh-Taylor instability is studied in the presence of a background temperature gradient, Θ, using a normal mode analysis. The effect of Θ variation is examined for three interface types corresponding to the combinations of the viscous properties of the fluids (inviscid-inviscid, viscous-viscous, and viscous-inviscid) at different Atwood numbers, At, and when at least one of the fluids' viscosity is non-zero, as a function of the Grashof number. For the general case, the resulting ordinary differential equations are solved numerically; however, dispersion relations for the growth rate are presented for several limiting cases. An analyticalmore »« less
https://doi.org/10.1063/1.4959810

Similar Records

Title: Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface

Abstract

Citation Formats

Bubble Dynamics and Cavitation journal, January 1977

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells journal, September 1990

The Character of the Equilibrium of an Incompressible Fluid Sphere of Variable Density and Viscosity Subject to Radial Acceleration journal, January 1955

The stability of the surface of a cavitation bubble journal, January 1953

On the Bell–Plesset effects: The effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability journal, November 2004

An overview of Rayleigh-Taylor instability journal, July 1984

Unstable normal mode for Rayleigh–Taylor instability in viscous fluids journal, January 1977

Chemical and hydrodynamical analysis of stability of a spherical interface journal, August 1976

Stability and mix in spherical geometry journal, August 1990

On the Stability of Fluid Flows with Spherical Symmetry journal, January 1954

Eigenvalues and degeneracies for n ‐dimensional tensor spherical harmonics journal, October 1984

The Non-Linear Field Theories of Mechanics book, January 1965

Fastest growing linear Rayleigh-Taylor modes at solid∕fluid and solid∕solid interfaces journal, March 2005

Stability of a compressed gas bubble in a viscous fluid journal, June 1989

Theory of the Rayleigh-Taylor instability journal, August 1991

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shells journal, September 2005

Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration journal, October 2018

Bubble Dynamics and Cavitation
journal, January 1977

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells
journal, September 1990

The Character of the Equilibrium of an Incompressible Fluid Sphere of Variable Density and Viscosity Subject to Radial Acceleration
journal, January 1955

The stability of the surface of a cavitation bubble
journal, January 1953

On the Bell–Plesset effects: The effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability
journal, November 2004

An overview of Rayleigh-Taylor instability
journal, July 1984

Unstable normal mode for Rayleigh–Taylor instability in viscous fluids
journal, January 1977

Chemical and hydrodynamical analysis of stability of a spherical interface
journal, August 1976

Stability and mix in spherical geometry
journal, August 1990

On the Stability of Fluid Flows with Spherical Symmetry
journal, January 1954

Eigenvalues and degeneracies for n ‐dimensional tensor spherical harmonics
journal, October 1984

The Non-Linear Field Theories of Mechanics
book, January 1965

Fastest growing linear Rayleigh-Taylor modes at solid∕fluid and solid∕solid interfaces
journal, March 2005

Stability of a compressed gas bubble in a viscous fluid
journal, June 1989

Theory of the Rayleigh-Taylor instability
journal, August 1991

Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shells
journal, September 2005

Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration
journal, October 2018