Linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian fluid

Abstract

We performed a linear stability analysis of a Newtonian ferrofluid cylinder surrounded by a Newtonian non-magnetic fluid in an azimuthal magnetic field. A wire is used at the centre of the ferrofluid cylinder to create this magnetic field. Isothermal conditions are considered and gravity is ignored. An axisymmetric perturbation is imposed at the interface between the two fluids and a dispersion relation is obtained allowing us to predict whether the flow is stable or unstable with respect to this perturbation. This relation is dependent on the Ohnesorge number of the ferrofluid, the dynamic viscosity ratio, the density ratio, the magnetic Bond number, the relative magnetic permeability and the dimensionless wire radius. Solutions to this dispersion relation are compared with experimental data from Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and, more recently, Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502). A better agreement than the inviscid theory and the theory that only takes into account the viscosity of the ferrofluid is shown with the data of Arkhipenko et al. (Fluid Dyn., vol. 15, issue 4, 1981, pp. 477–481) and those of Bourdin et al. (Phys. Rev. Lett., vol. 104, issue 9, 2010, 094502) for small wavenumbers.

Linear stability of a ferrofluid centred around a current-carrying wire

“…Both works consider axisymmetric disturbances, and the results obtained by Canu & Renoult (2021) show that accounting for viscosity agrees better with the experimental results of Arkhipenko et al (1980) and Bourdin et al (2010) than the inviscid system. Moreover, Canu & Renoult (2021) highlight the importance of having a surrounding liquid to mirror experimental conditions and for drug targeting applications. Blyth & Parau (2014) and Doak & Vanden-Broeck (2019) also consider the effect of a non-magnetic fluid surrounding the jet, but in the inviscid limit.…”

“…(1980) and Bourdin, Barci & Falcon (2010) suggests that viscous effects are important in the development of the instability. Cornish (2018) considers the highly viscous limit, and Canu & Renoult (2021) consider a Newtonian ferrofluid jet, surrounded by a Newtonian non-magnetic fluid. Both works consider axisymmetric disturbances, and the results obtained by Canu & Renoult (2021) show that accounting for viscosity agrees better with the experimental results of Arkhipenko et al.…”

“…(2010) than the inviscid system. Moreover, Canu & Renoult (2021) highlight the importance of having a surrounding liquid to mirror experimental conditions and for drug targeting applications. Blyth & Parau (2014) and Doak & Vanden-Broeck (2019) also consider the effect of a non-magnetic fluid surrounding the jet, but in the inviscid limit.…”

“…Of the previous works, Bashtovoi & Krakov (1978) and Rannacher & Engel (2006) consider non-axisymmetric disturbances, where they assume an inviscid irrotational system, but the experimental work performed by Arkhipenko et al (1980) and Bourdin, Barci & Falcon (2010) suggests that viscous effects are important in the development of the instability. Cornish (2018) considers the highly viscous limit, and Canu & Renoult (2021) consider a Newtonian ferrofluid jet, surrounded by a Newtonian non-magnetic fluid. Both works consider axisymmetric disturbances, and the results obtained by Canu & Renoult (2021) show that accounting for viscosity agrees better with the experimental results of Arkhipenko et al (1980) and Bourdin et al (2010) than the inviscid system.…”

“…Cornish (2018) considers the highly viscous limit, and Canu & Renoult (2021) consider a Newtonian ferrofluid jet, surrounded by a Newtonian non-magnetic fluid. Both works consider axisymmetric disturbances, and the results obtained by Canu & Renoult (2021) show that accounting for viscosity agrees better with the experimental results of Arkhipenko et al (1980) and Bourdin et al (2010) than the inviscid system. Moreover, Canu & Renoult (2021) highlight the importance of having a surrounding liquid to mirror experimental conditions and for drug targeting applications.…”