## Publications

54 results found

Ferguson Briggs S, Mestel A, 2022, Stability of an inhomogeneous ferrofluid in a channel, subject to a normal field, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 478, ISSN: 1364-5021

The stability of a ferrofluid with a fairly arbitrary non-uniform magnetic susceptibility between two parallel walls, subject to a non-uniform magnetic field acting normal to the walls, is investigated. The susceptibility may depend on position and the field strength, and the stationary state is such that the gradient of the susceptibility with respect to the modulus of the field is negative. Previous work suggests that the configuration may be linearly unstable, as regions of higher susceptibility do not coincide with regions of strongest field, and this is proved here. Adding a constant field in the plane of the layer suppresses parallel instabilities, but has no effect on those orthogonal to it. However, it is demonstrated that stability can be achieved by applying a rapidly rotating field.

Mestel A, Henriques Vaz R, Arslan A,
et al., 2022, Nonlinear laminar ‘dynamos’ linear in one coordinate, *Magnetohydrodynamics*, Vol: 58, Pages: 435-444, ISSN: 0024-998X

We consider two self-similar flow fields, for which the Navier-Stokes equations reduce toODEs. If the magnetic field has a similar structure then the induction equation alsoreduces to ODEs. The coupled system can be regarded as a kinematic dynamo problem,but also the fields grow until they saturate as exact solutions of the nonlinear system.The extent to which these solutions can be regarded as genuine dynamos is discussed.

Ferguson Briggs S, Mestel A, 2022, Linear stability of a ferrofluid centred around a current-carrying wire, *Journal of Fluid Mechanics*, Vol: 942, Pages: 1-29, ISSN: 0022-1120

Investigated first is the linear stability of a Newtonian ferrofluid centred on a rigid wire, surrounded by another ferrofluid with a different magnetic susceptibility. An electric current runs through the wire, generating an azimuthal magnetic field which produces a magnetic stress at the interface of the fluids. Three dimensional disturbances to the system are considered and the linearised Navier-Stokes equations are solved analytically in terms ofan implicit expression for the growth rate of the disturbance. The growth rate is found numerically for arbitrary Reynolds number and given explicitly in the inviscid and Stokes regimes. Investigated next is a ferrofluid, whose magnetic susceptibility varies radially, centred on a rigid wire, subject to an azimuthal, non-uniform field. It is proven that if the gradient of the susceptibility is positive anywhere in the fluid, the system is linearlyunstable. Moreover, it is proven that applying an axial field can stabilise disturbances for both continuous and discontinuous susceptibilities.

Mestel A, Arslan A, Henriques Vaz R,
et al., 2022, Nonlinear laminar 'dynamos' linear in one coordinate, *Magnetohydrodynamics*, Vol: 58, Pages: 435-443, ISSN: 0024-998X

We consider two self-similar flow fields, for which the Navier-Stokes equations reduce to ODEs. If the magnetic field has a similar structure then the induction equation also reduces to ODEs. The coupled system can be regarded as a kinematic dynamo problem, but also the fields grow until they saturate as exact solutions of the nonlinear system. The extent to which these solutions can be regarded as genuine dynamos is discussed.

Arslan A, Mestel A, 2021, Dynamo action between two rotating discs, *Geophysical and Astrophysical Fluid Dynamics*, Vol: 115, Pages: 710-727, ISSN: 0309-1929

Dynamo action is considered in the region between two differentially rotating infinite discs. The boundaries may be insulating, perfectly conducting or ferromagnetic. In the absence of a magnetic field, various well-known self-similar flows arise, generalising that of von Kármán. Magnetic field instabilities with the same similarity structure are sought. The kinematic eigenvalue problem is found to have growing modes for Rem>Rc≃100. The growth rate is real for the perfectly conducting and ferromagnetic cases, but may be complex for insulating boundaries. As Rem→∞ it is shown that the dynamo can be fast or slow, depending on the flow structure. In the slow case, the growth rate is governed by a magnetic boundary layer on one of the discs. The growing field saturates in a solution to the nonlinear dynamo problem. The bifurcation is found to be subcritical and nonlinear dynamos are found for Rem≳0.7Rc. Finally, the flux of magnetic energy to large r is examined, to determine which solutions might generalise to dynamos between finite discs. It is found that the fast dynamos tend to have inward energy flux, and so are unlikely to be realised in practice. Slow dynamos with outward flux are found. It is suggested that the average rotation rate should be non-zero in practice.

Mannix P, Mestel A, 2021, Bistability and hysteresis of axisymmetric thermal convection between differentially rotating spheres, *Journal of Fluid Mechanics*, Vol: A12, ISSN: 0022-1120

Heating a quiescent fluid from below gives rise to cellular convective motion as the temperature gradient becomes sufficiently steep. Typically, this transition increases heat transfer. Differentially rotating spherical shells also generate a state of cellular motion, which in this case transports angular momentum. When both effects are present, it is often assumed that the fluid adopts a configuration which maximises the transfer of angular momentum and heat. Depending on how the equilibrium is reached, however, this maximisation may not always be achieved, with two different stable equilibria often co-existing for the same heating and rotation strengths. We want to understand why the fluid motion in a spherical shell is bistable, and how this scenario might arise. We consider a deep, highly viscous fluid layer, of relevance to the ice shells of Saturn's and Jupiter's moons. We find that bistability depends largely on the relative strength of heating and differential rotation, as characterised by the Rayleigh number Ra and inner sphere Reynolds number Re1, and that the nature of the transition between bistable states depends strongly on the ratio of momentum diffusivity ν to thermal diffusivity κ defined by the Prandtl number Pr=ν/κ. In particular, we find that the transition between solutions at large Pr, depends on the strength of thin thermal layers and can occur either due to the destabilisation of an equatorial jet by buoyancy forces, or alternatively of a polar thermal plume by differential rotation. Our results demonstrate that, although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance.Keywords

Sumner L, Mestel A, Reichenbach J, 2021, Steady streaming as a method for drug delivery tothe inner ear, *Scientific Reports*, Vol: 11, Pages: 1-12, ISSN: 2045-2322

The inner ear, or cochlea, is a fluid-filled organ housing the mechanosensitive hair cells. Sound stimulation is relayed to the hair cells through waves that propagate on the elastic basilar membrane. Sensorineural hearing loss occurs from damage to the hair cells and cannot currently be cured. Although drugs have been proposed to prevent damage or restore functionality to hair cells, a difficulty with such treatments is ensuring adequate drug delivery to the cells. Because the cochlea is encased in the temporal bone, it can only be accessed from its basal end. However, the hair cells that are responsible for detecting speech-frequency sounds reside at the opposite, apical end. In this paper we show that steady streaming can be used to transport drugs along the cochlea. Steady streaming is a nonlinear process that accompanies many fluctuating fluid motions, including the sound-evoked waves in the inner ear. We combine an analytical approximation for the waves in the cochlea with computational fluid dynamic simulations to demonstrate that the combined steady streaming effects of several different frequencies can transport drugs from the base of the cochlea further towards the apex. Our results therefore show that multi-frequency sound stimulation can serve as a non-invasive method to transport drugs efficiently along the cochlea.

Mestel A, Mannix P, 2019, Weakly nonlinear mode-interactions in spherical Rayleigh-Benard convection, *Journal of Fluid Mechanics*, Vol: 874, Pages: 359-390, ISSN: 0022-1120

In an annular spherical domain with separation d, the onset of convective motion occursat a critical Rayleigh number Ra = Rac. Solving the linear stability problem, it is shownthat degenerate points (d = dc; Rac) exist where two modes simultaneously becomeunstable. Considering the weakly nonlinear evolution of these modes, it is found thatspatial resonances play a crucial role in determining the preferred convection pattern forneighbouring modes (` : ` 1) and non-neighbouring even modes (` : ` 2). Derivingcoupled amplitude equations relevant at all degeneracies we outline the inuence ofchanges in d; Ra and Prandtl number Pr. A particular conclusion is that only evenmodes have pure mode solutions, and that odd modes exist only as a component of mixedmode solutions. The mode-dependent inuence of Pr on the saturation of mixed modesolutions is shown to be markedly di erent in the limits Pr ! 0 and Pr ! 1. Usingdirect numerical simulation (DNS) to verify all results, time periodic solutions are alsooutlined for small Pr. The 2 : 1 periodic signature observed to be general of oscillationsin a spherical annulus, is explained using the structure of the equations derived.

Mestel A, Henriques Vaz R, 2019, Some similarity solutions for 3-D boundary layers, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 475, Pages: 1-11, ISSN: 1364-5021

A similarity solution of a 3−dimensional boundarylayer is investigated. The outer flow is given by U =(−xz, −yz, z2), corresponding to an axisymmetricpoloidal circulation with constant potential vorticity.This flow is an exact solution of the Navier–Stokes. Awall is introduced at y = 0 along which a boundarylayer develops towards x = 0. We show that asimilarity reduction to a system of ODEs is possible.Two distinct solutions are found, one of them throughnumerical path-continuation, and their stability isinvestigated. A second 3-D solution is also identifiedfor 2-D outer flow. The solutions are generalised forouter flows scaling with different powers of z andsimilar results are found. This behaviour is related tothe non-uniqueness of the Falkner-Skan flows in a 3-Dsense, with a transverse wall-jet.

Mannix P, Mestel A, 2019, Weakly nonlinear mode interactions in spherical Rayleigh-Bénard convection, *Journal of Fluid Mechanics*, ISSN: 0022-1120

In an annular spherical domain with separation d, the onset of convective motion occurs at a critical Rayleigh number Ra = Rac. Solving the axisymmetric linear stability problem, shows that degenerate points (d = dc, Rac) exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes (`, `±1) and non-neighbouring even modes (`, `±2). Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in d, Ra and Prandtl number Pr . Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small Pr . The 2 : 1 periodic signature observed to be general for oscillations in a spherical annulus, is explained using the structure of the equations. The relevance of all solutions presented isdetermined by computing their stability with respect to non-axisymmetric perturbations at large Pr .

Sargent C, Mestel A, 2019, Trapped modes of the Helmholtz equation in infinite waveguides with wall indentations and circular obstacles, *IMA Journal of Applied Mathematics*, Vol: 84, Pages: 312-344, ISSN: 0272-4960

Trapped modes of the Helmholtz equation are investigated in infinite, two-dimensional acoustic waveg-uides with Neumann or Dirichlet walls. A robust boundary element scheme is used to study modes bothinside and outside the continuous spectrum of propagating modes. An effective method for distinguishingbetween genuine trapped modes and spurious solutions induced by the domain truncation is presented.The method is also suitable for the detection and study of “nearly trapped modes” (NTM). These areof great practical importance as they display many features of trapped modes but do not require perfectgeometry.An infinite, two-dimensional channel is considered with one or two discs on its centreline. The walls mayhave rectangular, triangular or smooth cavities. The combination of a circular obstacle and a rectangularcavity, in both Neumann and Dirichlet guides is studied, illustrating the possible use of a movable disc todetect wall irregularities.The numerical method is validated against known results and many new modes are identified, both insideand outside the continuous spectrum. Results obtained suggest that at least one symmetry line is animportant condition for the formation of trapped mode type resonances. The addition of a symmetry-preserving geometric parameter to a problem which has a discrete embedded trapped mode solution fora specific geometry, tends to lead to a continuous set of trapped modes.

Henriques Vaz R, Tettamanti F, Mestel AJ, 2018, 'Unforced' Navier-Stokes solutions derived from convection in a curved channel, *Journal of Fluid Mechanics*, Vol: 848, Pages: 676-695, ISSN: 0022-1120

Steady Boussinesq flow in a weakly curved channel driven by a horizontal temperature gradient is considered. Linear variation in the transverse direction is assumed so that the problem reduces to a system of ordinary differential equations. A series expansion in G , a parameter proportional to the Grashof number and the square root of the curvature, reveals a real singularity and anticipates hysteresis. Numerical solutions are found using path continuation and the bifurcation diagrams for different parameter values are obtained. Multivalued solutions are observed as G and the Prandtl number vary. Often fields with the imposed structure that satisfy all the governing equations are insensitive to the boundary conditions and can be regarded as perturbations of the homogeneous (or ‘unforced’) problem. Four such unforced solutions are found. In two of these the velocity remains coupled with temperature which, formally, scales as 1/G as G→0 . The other two are purely hydrodynamic. The existence of such solutions is due to the unbounded nature of the domain. It is shown that these occur not only for the Dean equations, but constitute previously unreported solutions of the full Navier–Stokes equations in an annulus of arbitrary curvature. Two additional unforced solutions are found for large curvature.

Henriques Vaz R, Tettamanti Boshier FA, Mestel AJ, 2018, Dynamos in an annulus with fields linear in the axial coordinate, *Geophysical and Astrophysical Fluid Dynamics*, Vol: 112, Pages: 222-234, ISSN: 0309-1929

Dynamo action is considered in a conducting cylindrical annulus surrounded by an insulator. The drivingvelocity field is assumed to be linear in the axial coordinate and to satisfy the incompressible Navier-Stokesequations. Such flows have recently been shown to exist with no forcing other than the similarity structure.Magnetic field instabilities with the same spatial structure are sought. The kinematic eigenvalue problem isfound to have two growing modes for moderate values of the magnetic Reynolds number,Rm. AsRm→∞it is shown that the modes are governed by layers on the outer wall. The growing field saturates in a solutionto the nonlinear dynamo problem. Three distinct steady solution families are found and the complicatedbifurcation structure is investigated.

Boshier FAT, Mestel AJ, 2017, Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers, *Journal of Fluid Mechanics*, Vol: 818, Pages: 241-259, ISSN: 1469-7645

Steady, incompressible flow down a slowly-curving circular pipe is considered, analyti-cally and numerically. Both real and complex solutions are investigated. Using high-orderHermite–Pad ́e approximants, the Dean series solution is analytically continued outside itscircle of convergence where it predicts a complex solution branch for real, positive Deannumber,K. This is confirmed by numerical solution. It is shown that other previouslyunknown solution branches exist for allK >0, which are related to an unforced com-plex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokesequations in most geometries. By means of path continuation, numerical solutions arefollowed around the complexK-plane. The standard Dean two-vortex solution is shownto lie on the same hypersurface as the eigensolution and the four-vortex solutions foundin the literature.Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case.There is an imaginary singularity limiting convergence of the Dean series, an unforcedsolution atK= 0 and nonuniqueness forK >0, culminating in a real bifurcation.

Boshier FAT, Mestel AJ, 2014, Extended series solutions and bifurcations of the Dean equations, *JOURNAL OF FLUID MECHANICS*, Vol: 739, Pages: 179-195, ISSN: 0022-1120

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Setchi A, Mestel AJ, Siggers JH,
et al., 2013, Mathematical model of flow through the patent ductus arteriosus, *JOURNAL OF MATHEMATICAL BIOLOGY*, Vol: 67, Pages: 1487-1506, ISSN: 0303-6812

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- Citations: 5

Setchi A, Mestel AJ, Parker KH,
et al., 2013, Low-Reynolds-number flow through two-dimensional shunts, *JOURNAL OF FLUID MECHANICS*, Vol: 723, Pages: 21-39, ISSN: 0022-1120

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- Citations: 4

Rosin MS, Mestel AJ, 2012, Quasi-global galactic magnetorotational instability with Braginskii viscosity, *MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY*, Vol: 425, Pages: 74-86, ISSN: 0035-8711

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- Citations: 4

Mestel AJ, Zabielski L, 2012, Laminar instability of pressure-driven dynamos in multiple helical pipes, *GEOPHYSICAL AND ASTROPHYSICAL FLUID DYNAMICS*, Vol: 106, Pages: 493-507, ISSN: 0309-1929

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- Citations: 1

Setchi AT, Siggers JH, Parker KH, et al., 2010, Mathematical Modeling of Two-Dimensional Flow through Patent Ductus Arteriosus in an Adult, 6th World Congress of Biomechanics (WCB 2010), Publisher: SPRINGER, Pages: 386-+, ISSN: 1680-0737

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- Citations: 2

Lock RM, Mestel AJ, 2008, On annular self-similar solutions in resistive magnetogasdynamics, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 464, Pages: 2535-2547, ISSN: 1364-5021

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- Citations: 1

Lock RM, Mestel AJ, 2008, Annular self-similar solutions in ideal magnetogasdynamics, *JOURNAL OF PLASMA PHYSICS*, Vol: 74, Pages: 531-554, ISSN: 0022-3778

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- Citations: 28

Dubash N, Mestel AJ, 2007, Behavior near critical etc, *Physics of Fluids*, Vol: 19

Dubash N, Mestel AJ, 2007, Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field, *PHYSICS OF FLUIDS*, Vol: 19, ISSN: 1070-6631

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- Citations: 34

Dubash N, Mestel AJ, 2007, Behavior near critical for a conducting drop in an electric field, *PHYSICS OF FLUIDS*, Vol: 19, ISSN: 1070-6631

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- Citations: 3

Zabielski L, Mestel AJ, 2007, A double-helix laminar dynamo, *JOURNAL OF FLUID MECHANICS*, Vol: 573, Pages: 237-246, ISSN: 0022-1120

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- Citations: 2

Dubash N, Mestel AJ, 2007, Behaviour of a conducting drop in a highly viscous fluid subject to an electric field, *Journal of Fluid Mechanics*, Vol: 581, Pages: 469-493

Zabielski L, Mestel AJ, 2006, Nonlinear dynamos in laminar, helical pipe flow, *PHYSICS OF FLUIDS*, Vol: 18, ISSN: 1070-6631

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- Citations: 2

Zabiekski L, Mestel AJ, 2005, Kinematic dynamo action in a helical pipe, *JOURNAL OF FLUID MECHANICS*, Vol: 535, Pages: 347-367, ISSN: 0022-1120

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- Citations: 5

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