Increasing Flow Complexity in Time-Dependent Modulated Ferrofluidic Couette Flow

Abstract

⎯We present numerical simulations of ferrofluidic Couette flow in between counter-rotating cylinders with a spatially homogeneous magnetic field but subject to time-periodic modulation. Such a modulation can lead to a significant inner Reynolds number () enhancement for primary bifurcating solutions, for either helical or toroidal flow structures. Moreover, the external introduced modulation frequency renders the different solutions to step up one level in the hierarchy of complexity. Fixed point solutions become limit cycles and limit cycles become 2-tori. Moreover, for sufficiently strong modulation amplitudes of the magnetic field stability can be exchanged between primary bifurcating spiral and ribbon solutions, both appearing at a common threshold. In addition, a stable bifurcation branch with direct connection of ribbon and Taylor vortices via wavy Taylor vortices is found.

Ferrofluidic wavy Taylor vortices under alternating magnetic field

“…The magnetic field H and the magnetization M are conveniently normalized by the quantity √ ρ/μ 0 ν/d, with free space permeability μ 0 . By using a modified Niklas approach [23,24,48,49], the effect of the magnetic field and the magnetic properties of the ferrofluid on the velocity field can be characterized by a single (time-dependent) function, the modulation frequency, respectively. See appendix A for further details.…”

“…is the Niklas coefficient [23], μ is the dynamic viscosity, Φ is the volume fraction of the magnetic material, S is the symmetric component of the velocity gradient tensor [50,51] and λ 2 is the material-dependent transport coefficient [50], which we choose to be λ 2 = 4/5 [50,51,54]. Using (3.2), the magnetization in (3.1) can be eliminated to obtain the following ferrohydrodynamical equation of motion [29,[48][49][50][51]:…”

“…The magnetic field H and the magnetization M are conveniently normalized by the quantity ρ/μ0ν/d, with free space permeability μ0. By using a modified Niklas approach [23,24,48,49], the effect of the magnetic field and the magnetic properties of the ferrofluid on the velocity field can be characterized by a single (time-dependent) function, szfalse[xfalse]false(tfalse)=sS,zfalse[xfalse]+sM,zfalse[xfalse]sin⁡(ΩHz[x]t), with sS,zfalse[xfalse] being the static contribution of the driving, sM,zfalse[xfalse] the modulation amplitude and ΩHz,false[xfalse] the modulation frequency, respectively. See appendix A for further details.…”

Forced Vortex Merging and Splitting Events in Ferrofluidic Couette Flow

“…The ferrofluid in the annulus is assumed isothermal, and incompressible with kinematic viscosity 𝜈. Using the gap width, 𝑑 = 𝑅 𝑜 − 𝑅 𝑖 , as the length scale and the radial diffusion time, 𝑑 2 ∕𝜈, as the time scale, and normalizing the pressure with 𝜌𝜈 2 ∕𝑑 2 , the non-dimensionalized ferro-hydrodynamical equations of motion [29,[35][36][37] are given by:…”

“…The magnetic field 𝐇 and the magnetization 𝐌 are conveniently normalized by the quantity √ 𝜌∕𝜇 0 𝜈∕𝑑, with free space permeability 𝜇 0 . By using a modified Niklas approach [28][29][30]37,38], the effect of the magnetic field and the magnetic properties of the ferrofluid on the velocity field can be characterized by a single (time-dependent) parameter…”