https://www.findaphd.com/search/ProjectDetails.aspx?PJID=92456

# Double-diffusive instabilities in rotating flows (Advert Reference: RDF18/MPE/KIRILLOV)

## Project Description

While common sense tends to assign to dissipation the role of a vibration damper, as early as 1879 Kelvin and Tait predicted viscosity-driven instability of Maclaurin’s spheroids, thus presenting a class of Hamiltonian equilibria, which, although stable in the absence of dissipation, become unstable due to the action of dissipative forces. The universality of the dissipation-induced instabilities yields their manifestation in many fields of physics such as solid- and fluid mechanics, magnetohydrodynamics, plasma physics, nonlinear waves, to name few. For instance, a rotating and self-gravitating mass of inviscid fluid (imagine a rotating sun, for example) tends to acquire a shape that is different from a perfect sphere. The simplest possible shape has been found to be an oblate spheroid by Colin Maclaurin in 1742. This shape is stable at small and moderate rotating speeds. In 1963 Roberts and Stewartson proved that viscosity of the fluid makes the interval of stable rotation speeds smaller than in the case of the inviscid fluid thus proving the prediction of Kelvin and Tait. In 1970 Chandrasekhar found that radiation of gravitational waves has a similar destabilizing effect. In 1977 Lindblom and Detweiler discovered that acting together, viscous dissipation and radiative losses can inhibit destabilizing action of each other at some particular proportion of their strengths. However, a single instability criterion for the Maclaurin spheroids in the presence of the two dissipation mechanisms is still an open question. The aim of the PhD project is to find such a criterion by solving a boundary value problem with a variety of boundary conditions. The non-self-adjoint boundary value problem will require a perturbative, asymptotic and numerical treatment and will involve such concepts as negative energy waves, sensitivity analysis of boundary eigenvalue problems, dissipation-induced instabilities, and Hamiltonian mechanics. A particular interest is to explore a connection between the radiative Chandrasekhar-Friedman-Schutz instability of the Maclaurin spheroids and the anomalous Doppler effect. The theory is mathematically challenging but highly demanded by both astrophysical and geophysical communities as it is linked to many modern applications including tidal effects in planets and stars. As a second step, the developed theory and its methodology will be extended to double-diffusive instabilities in differentially rotating flows of thermo- and electrically-conducting fluids in magnetic fields that are related with the modern experimental works on the Taylor-Couette flow with the quasi-Keplerian shear profile.

## Eligibility and How to Apply:

Please note eligibility requirement:

• Academic excellence of the proposed student i.e. 2:1 (or equivalent GPA from non-UK universities [preference for 1st class honours]); or a Masters (preference for Merit or above); or APEL evidence of substantial practitioner achievement.

• Appropriate IELTS score, if required.

• Applicants cannot apply for this funding if currently engaged in Doctoral study at Northumbria or elsewhere.

For further details of how to apply, entry requirements and the application form, see:

https://www.northumbria.ac.uk/research/postgraduate-research-degrees/how-to-apply/

Please note: Applications that do not include a research proposal of approximately 1,000 words (not a copy of the advert), or that do not include the advert reference (e.g. RDF18/…) will not be considered.

Deadline for applications: 28 January 2018

Start Date: 1 October 2018

Northumbria University takes pride in, and values, the quality and diversity of our staff. We welcome applications from all members of the community. The University holds an Athena SWAN Bronze award in recognition of our commitment to improving employment practices for the advancement of gender equality and is a member of the Euraxess network, which delivers information and support to professional researchers.

## Funding Notes

The studentship includes a full stipend, paid for three years at RCUK rates (for 2017/18, this is £14,553 pa) and fees.

## References

O.N. Kirillov (2017) Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics. Proc. R. Soc. London A, 473(2205): 20170344

O.N. Kirillov, I. Mutabazi (2017) Short wavelength local instabilities of a circular Couette flow with radial temperature gradient. Journal of Fluid Mechanics, 818: 319-343.

F. Stefani, T. Albrecht, R. Arlt, M. Christen, A. Gailitis, M. Gellert, A. Giesecke, O. Goepfert, J. Herault, O. N. Kirillov, G. Mamatsashvili, J. Priede, G. Rudiger, M. Seilmayer, A. Tilgner, T. Vogt (2017) Magnetic field dynamos and magnetically triggered flow instabilities. IoP Conference Series, Materials Science and Engineering, 228: 012002

F. Stefani, O.N. Kirillov (2015) Destabilization of rotating flows with positive shear by azimuthal magnetic fields. Phys. Rev. E, 92: 051001(R)

O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) Local instabilities in magnetized rotational flows: A short-wavelength approach. Journal of Fluid Mechanics, 760: 591- 633

O.N. Kirillov, F. Stefani (2013) Extending the range of the inductionless magnetorotational instability. Physical Review Letters, 111(6): 061103

Kirillov ON. (2013) Nonconservative stability problems of modern physics, De Gruyter Studies in Mathematical Physics 14. Berlin, Boston: De Gruyter.

O.N. Kirillov, F. Stefani, Y. Fukumoto (2012) A unifying picture of helical and azimuthal MRI, and the universal significance of the Liu limit. The Astrophysical Journal, 756(83): 6pp

O.N. Kirillov, D.E. Pelinovsky, G. Schneider (2011) Paradoxical transitions to instabilities in hydromagnetic Couette-Taylor flows. Physical Review E, 84(6): 065301(R)

O.N. Kirillov, F. Stefani (2010) On the relation of standard and helical magnetorotational instability. The Astrophysical Journal, 712: 52-68

O.N. Kirillov (2010) Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices. Zeitschrift fur angewandte Mathemtik und Physik-ZAMP, 61(2): 221-234