Magnetic field driven micro-convection in the Hele-Shaw cell: the Brinkman model and its comparison with experiment

Abstract

The micro-convection caused by the ponderomotive forces of the self-magnetic field in a magnetic fluid is studied here both numerically and experimentally. The theoretical approach based on the general Brinkman model substantially improves the description with respect to the previously proposed Darcy model. The predictions of both models are here compared to finely controlled experiments. The Brinkman model, in contrast to the Darcy model, allows us to describe the formation of mushrooms on the plumes of the micro-convective flow and the width of the fingers. In the Brinkman approach, excellent quantitative agreement is also obtained for the finger velocity dynamics and the velocity maximal values as a function of the magnetic Rayleigh number. The diffusion coefficient of particles of the water-based magnetic colloid determined by the threshold field strength value of the micro-convection is significantly larger than the diffusion coefficient of individual particles. This result is confirmed by independent measurements of the diffusion coefficient at the smearing of the diffusion front.

Rivalry of diffusion, external field and gravity in micro-convection of magnetic colloids

“…The experimental conditions of previous experiments [13] at which we have observed the characteristic fingering pattern of the magnetic micro-convection, as visible in Fig.1(a), had a cell thickness ℎ = 130 m and fluid density difference Δ 0 = 0.148 g/cm 3 . According to the model of gravitational influence, described in §2.2, this corresponds to a gravitational Rayleigh number = 13 ′ 500.…”

“…At time = 0 we assume a straight interface with a concentration step, where = 1 corresponds to magnetic fluid and = 0 to non-magnetic fluid. The process can be theoretically mod-elled with a system of Brinkman, continuity and convectiondiffusion equations, as described in [13]. Without going in details, we note that the phenomenon is characterized with a dimensionless magnetic Rayleigh number = ( ) 2 ℎ 2 12 , where is the susceptibility of magnetic fluid, is magnetic field, ℎ is the cell thickness, is the fluid viscosity, assumed to be equal across the fluid, and is the diffusion coefficient of magnetic nanoparticles.…”

“…Without going in details, we note that the phenomenon is characterized with a dimensionless magnetic Rayleigh number = ( ) 2 ℎ 2 12 , where is the susceptibility of magnetic fluid, is magnetic field, ℎ is the cell thickness, is the fluid viscosity, assumed to be equal across the fluid, and is the diffusion coefficient of magnetic nanoparticles. For the instability to appear a critical field is necessary and it corresponds to a critical magnetic Rayleigh number crit ≈ 6, while finger size is approximately equal to cell thickness ℎ and does not depend on [13]. If no magnetic field is applied, the magnetic fluid should slowly mix with non-magnetic fluid via diffusion.…”

“…Holes are made in one of the glass slide to glue in tubing connectors. For micro-convection experiments a channel shape that allows to merge two droplets is used, as explained in [13]. For no-field experiments we exploit a continuous microfluidics channel, as described in [14].…”

“…Changing the thickness of the cell ℎ allows to expand the verification of the magnetic micro-convection theoretical model [13], introduced in §2.1. One of the predictions is the change of finger size.…”