Geometrical optics stability analysis of rotating visco-diffusive flows

Oleg Kirillov

Geometrical optics stability analysis of rotating visco-diffusive flows

Oleg Kirillov^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Abstract

The geometrical optics stability analysis has proved successful in solving problems of ideal hydrodynamics and magnetohydrodynamics related to the stability of 3D flows of both compressible and incompressible fluids [1-3]. The method is based on perturbations of a background flow in a small parameter, representing a short wavelength. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. Thus, detecting instabilities localized near a particular fluid element location, moving with the flow, reduces to solving a system of ODEs for the wave vector and amplitude, along particle paths in the underlying flow, with coefficients depending on the unperturbed velocity field [1-3]. Inclusion of viscosity or diffusivity in the analysis was long-time believed to be only stabilizing. However, starting with the works [4,5] that considered both viscosity and magnetic diffusivity in the geometrical optics analysis of helical magnetorotational instability it became possible to extend the method to visco-diffusive [6] and multiple-diffusive [7] flows for a wide range of Prandtl, Schmidt and magnetic Prandtl numbers. In the talk we review these works and discuss new applications.

References:
[1] K.S. Eckhoff (1981) J. Differ. Equ. 40 (1), 94–115.
[2] A. Lifschitz, E. Hameiri (1991) Phys. Fluids 3, 2644–2651.
[3] S. Friedlander, M.M. Vishik (1991) Phys. Rev. Lett. 66, 2204–2206.
[4] O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) JFM, 760: 591- 633.
[5] O.N. Kirillov (2017) Proc. A, 473(2205): 20170344.
[6] O.N. Kirillov, I. Mutabazi (2017) JFM, 818: 319-343.
[7] J. Labarbe, O.N. Kirillov (2021). Phys. Fluids, 33(10): 104108

Original language	English
Title of host publication	Book of abstracts
Subtitle of host publication	The 65th British Applied Mathematics Colloquium
Place of Publication	Newcastle Upon Tyne
Publisher	Newcastle University
Pages	96-96
Number of pages	1
Publication status	Published - 9 Apr 2024
Event	British Applied Mathematics Colloquium - Newcastle, United Kingdom Duration: 9 Apr 2024 → 11 Apr 2024 https://conferences.ncl.ac.uk/bamc2024/

Conference

Conference	British Applied Mathematics Colloquium
Country/Territory	United Kingdom
City	Newcastle
Period	9/04/24 → 11/04/24
Internet address	https://conferences.ncl.ac.uk/bamc2024/

Keywords

Rotating flows
magnetohydrodynamcs (MHD)
stratified flows
lenticular vortices
thermal gradient
instabilities
WKB

Access to Document

BAMC_Book_of_AbstractsFinal published version, 2.06 MB

https://conferences.ncl.ac.uk/media/sites/conferencewebsites/bamc2024/BAMC%20Book%20of%20Abstracts.pdf

Cite this

@inproceedings{32a3fa2bfd474b479a7ee90c3be6a85a,

title = "Geometrical optics stability analysis of rotating visco-diffusive flows",

abstract = "The geometrical optics stability analysis has proved successful in solving problems of ideal hydrodynamics and magnetohydrodynamics related to the stability of 3D flows of both compressible and incompressible fluids [1-3]. The method is based on perturbations of a background flow in a small parameter, representing a short wavelength. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. Thus, detecting instabilities localized near a particular fluid element location, moving with the flow, reduces to solving a system of ODEs for the wave vector and amplitude, along particle paths in the underlying flow, with coefficients depending on the unperturbed velocity field [1-3]. Inclusion of viscosity or diffusivity in the analysis was long-time believed to be only stabilizing. However, starting with the works [4,5] that considered both viscosity and magnetic diffusivity in the geometrical optics analysis of helical magnetorotational instability it became possible to extend the method to visco-diffusive [6] and multiple-diffusive [7] flows for a wide range of Prandtl, Schmidt and magnetic Prandtl numbers. In the talk we review these works and discuss new applications. References: [1] K.S. Eckhoff (1981) J. Differ. Equ. 40 (1), 94–115. [2] A. Lifschitz, E. Hameiri (1991) Phys. Fluids 3, 2644–2651. [3] S. Friedlander, M.M. Vishik (1991) Phys. Rev. Lett. 66, 2204–2206. [4] O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) JFM, 760: 591- 633. [5] O.N. Kirillov (2017) Proc. A, 473(2205): 20170344. [6] O.N. Kirillov, I. Mutabazi (2017) JFM, 818: 319-343. [7] J. Labarbe, O.N. Kirillov (2021). Phys. Fluids, 33(10): 104108",

keywords = "Rotating flows, magnetohydrodynamcs (MHD), stratified flows, lenticular vortices, thermal gradient, instabilities, WKB",

author = "Oleg Kirillov",

year = "2024",

month = apr,

day = "9",

language = "English",

pages = "96--96",

booktitle = "Book of abstracts",

publisher = "Newcastle University",

note = "British Applied Mathematics Colloquium ; Conference date: 09-04-2024 Through 11-04-2024",

url = "https://conferences.ncl.ac.uk/bamc2024/",

}

Kirillov, O 2024, Geometrical optics stability analysis of rotating visco-diffusive flows. in Book of abstracts: The 65th British Applied Mathematics Colloquium., 52, Newcastle University, Newcastle Upon Tyne, pp. 96-96, British Applied Mathematics Colloquium, Newcastle, United Kingdom, 9/04/24. <https://conferences.ncl.ac.uk/media/sites/conferencewebsites/bamc2024/BAMC%20Book%20of%20Abstracts.pdf>

TY - GEN

T1 - Geometrical optics stability analysis of rotating visco-diffusive flows

AU - Kirillov, Oleg

PY - 2024/4/9

Y1 - 2024/4/9

N2 - The geometrical optics stability analysis has proved successful in solving problems of ideal hydrodynamics and magnetohydrodynamics related to the stability of 3D flows of both compressible and incompressible fluids [1-3]. The method is based on perturbations of a background flow in a small parameter, representing a short wavelength. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. Thus, detecting instabilities localized near a particular fluid element location, moving with the flow, reduces to solving a system of ODEs for the wave vector and amplitude, along particle paths in the underlying flow, with coefficients depending on the unperturbed velocity field [1-3]. Inclusion of viscosity or diffusivity in the analysis was long-time believed to be only stabilizing. However, starting with the works [4,5] that considered both viscosity and magnetic diffusivity in the geometrical optics analysis of helical magnetorotational instability it became possible to extend the method to visco-diffusive [6] and multiple-diffusive [7] flows for a wide range of Prandtl, Schmidt and magnetic Prandtl numbers. In the talk we review these works and discuss new applications. References: [1] K.S. Eckhoff (1981) J. Differ. Equ. 40 (1), 94–115. [2] A. Lifschitz, E. Hameiri (1991) Phys. Fluids 3, 2644–2651. [3] S. Friedlander, M.M. Vishik (1991) Phys. Rev. Lett. 66, 2204–2206. [4] O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) JFM, 760: 591- 633. [5] O.N. Kirillov (2017) Proc. A, 473(2205): 20170344. [6] O.N. Kirillov, I. Mutabazi (2017) JFM, 818: 319-343. [7] J. Labarbe, O.N. Kirillov (2021). Phys. Fluids, 33(10): 104108

AB - The geometrical optics stability analysis has proved successful in solving problems of ideal hydrodynamics and magnetohydrodynamics related to the stability of 3D flows of both compressible and incompressible fluids [1-3]. The method is based on perturbations of a background flow in a small parameter, representing a short wavelength. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. Thus, detecting instabilities localized near a particular fluid element location, moving with the flow, reduces to solving a system of ODEs for the wave vector and amplitude, along particle paths in the underlying flow, with coefficients depending on the unperturbed velocity field [1-3]. Inclusion of viscosity or diffusivity in the analysis was long-time believed to be only stabilizing. However, starting with the works [4,5] that considered both viscosity and magnetic diffusivity in the geometrical optics analysis of helical magnetorotational instability it became possible to extend the method to visco-diffusive [6] and multiple-diffusive [7] flows for a wide range of Prandtl, Schmidt and magnetic Prandtl numbers. In the talk we review these works and discuss new applications. References: [1] K.S. Eckhoff (1981) J. Differ. Equ. 40 (1), 94–115. [2] A. Lifschitz, E. Hameiri (1991) Phys. Fluids 3, 2644–2651. [3] S. Friedlander, M.M. Vishik (1991) Phys. Rev. Lett. 66, 2204–2206. [4] O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) JFM, 760: 591- 633. [5] O.N. Kirillov (2017) Proc. A, 473(2205): 20170344. [6] O.N. Kirillov, I. Mutabazi (2017) JFM, 818: 319-343. [7] J. Labarbe, O.N. Kirillov (2021). Phys. Fluids, 33(10): 104108

KW - Rotating flows

KW - magnetohydrodynamcs (MHD)

KW - stratified flows

KW - lenticular vortices

KW - thermal gradient

KW - instabilities

KW - WKB

M3 - Conference contribution

SP - 96

EP - 96

BT - Book of abstracts

PB - Newcastle University

CY - Newcastle Upon Tyne

T2 - British Applied Mathematics Colloquium

Y2 - 9 April 2024 through 11 April 2024

ER -