Time-dependent gap Hele-Shaw cell with a ferrofluid: Evidence for an interfacial singularity inhibition by a magnetic field

Abstract

We consider the flow of a ferrofluid droplet in a Hele-Shaw cell with a time-dependent gap width. When the surface tension and applied magnetic field are zero, interfacial instabilities develop and the droplet breaks. We execute a mode-coupling approach to the problem and focus on understanding how the development of singularities is affected by the action of an external field. Our analytical results indicate that the introduction of an azimuthal magnetic field profoundly modifies pattern formation, allowing the inhibition of interfacial singularities. We suggest the magnetic field can be used as a controllable parameter to discipline singular behavior.

Magnetic fluid rheology and flows

“…(1) [131]. Other ferrofluid Hele -Shaw cell flows investigated azimuthal magnetic fields from a long straight current-carrying wire along the axis perpendicular to the walls with a timedependent gap [132,133]; labyrinthine instability in miscible magnetic fluids in a horizontal Hele -Shaw cell with a vertical magnetic field [134]; theory and experiments of interacting ferrofluid drops [135 & ]; numerical simulations [136] and measurements [137] of viscous fingering labyrinth instabilities; fingering instability of an expanding air bubble in a horizontal Hele -Shaw cell [138] and of a rising bubble in a vertical Hele-Shaw cell [139]; fingering instabilities of a miscible magnetic fluid droplet in a rotating Hele -Shaw cell [140]; theory and experiments of the Rayleigh -Taylor instability [141 & ]; and natural convection of magnetic fluid in a bottom-heated square Hele-Shaw cell with two insulated side walls with heat transfer measurements and liquid crystal thermography [142].…”

Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

“…From equation (2.1), we understand that a magnetic body force ∝ |M|∇|H| acts on the thin film, where M is the magnetization vector of the ferrofluid. For the purposes of studying the interface [29,[32][33][34], we assume that the ferrofluid is uniformly magnetized, and the magnetization is collinear with the external field, i.e. M = χ H, where χ is the constant magnetic susceptibility.…”

“…(Although it is possible to also derive arbitrary-amplitude long-wave equations [22,37], Homsy [21] argued that the distinguished limit of 1 leads to the model equations capturing the essential physics.) Note that ε 1 is determined by the geometric configuration; specifically, R 0 is chosen sufficiently large to allow the Cartesian approximation, but small enough to ensure that ∇|H| is still the dominant term in the magnetic body force [29,32,33]. Note that demagnetization can still be neglected because it can be made arbitrarily small via the thickness b [34].…”

Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves

“…A long wire through the origin, carrying an electric current I, produces the azimuthal component. The combined magnetic field H forms an angle with the initially undisturbed interface [16]. The droplet experiences a body force ∝ |M|∇|H|, where M is the magnetization.…”

“…facial dynamics (e.g., [12,15,16]). The droplet interface is written as h(θ, t) = R + ξ(θ, t), where ξ(θ, t) = +∞ k=−∞ ξ k (t)e ikθ represents the perturbation of the initially circular interface, with complex Fourier amplitudes ξ k (t) ∈ C and azimuthal wavenumbers k ∈ Z.…”