Rectilinear magnetophoresis of a single oil droplet in a paramagnetic rare-earth solution

We studied the transient motion of single oil droplet dispersed in paramagnetic rare-earth dysprosium salt solution in a strong magnetic pressure gradient <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>(</a:mo><a:mrow><a:mi mathvariant="bold">∇</a:mi></a:mrow><a:msup><a:mi>B</a:mi><a:mn>2</a:mn></a:msup><a:mo>)</a:mo></a:mrow></a:math>. For that purpose, we used a concentric ring-rod NdFeB permanent magnetic assembly and oriented its axis parallel to gravity. The resulting <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mrow><c:mi mathvariant="bold">∇</c:mi></c:mrow><c:msup><c:mi>B</c:mi><c:mn>2</c:mn></c:msup></c:mrow></c:math> field stratifies monotonically along its symmetry axis, reaching <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mo>≈</e:mo><e:mn>600</e:mn><e:mspace width="0.16em"/><e:msup><e:mrow><e:mi mathvariant="normal">T</e:mi></e:mrow><e:mn>2</e:mn></e:msup><e:mo>/</e:mo><e:mi mathvariant="normal">m</e:mi></e:mrow></e:math> at the magnet surface. Initially, an oil droplet rises in a stagnant aqueous solution driven by buoyancy force. As the droplet approaches the magnet, the Kelvin force acting on the droplet becomes dominating. Consequently, the droplet starts to oscillate vertically rectilinearly. Dynamically, the trajectory of the oscillating droplet is resembled perfectly with a damped linear harmonic oscillation system. The phenomenon is robust for all four experimented organic phases despite that their momentum dissipation time scales differ in two orders of magnitudes. Correlating hydrodynamic forces with damping terms in the linear harmonic oscillation system, an experimentally quantifiable constraint is derived for the memory kernel of the droplet's Basset-Boussinesq force. The droplet's memory kernel, different from that of spherical gas and particle that decays at the dimensionless time following <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mo>∝</i:mo><i:msup><i:mi>t</i:mi><i:mrow><i:mo>′</i:mo><i:mo>−</i:mo><i:mn>1</i:mn><i:mo>/</i:mo><i:mn>2</i:mn></i:mrow></i:msup></i:mrow></i:math>, decays exponentially <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msup><j:mi>e</j:mi><j:msup><j:mi>t</j:mi><j:mo>′</j:mo></j:msup></j:msup></j:math>. This is found valid for the highest experimented droplet Reynolds number up to 45. Statistics from experiments show that the droplet memory kernel is correlated with droplet diameter instead of the viscosity ratio between the two phases. After hydrodynamic forces have dissipated kinetic energy, the droplet is magnetically levitated statically, where the Kelvin force is counterbalancing buoyancy. By incorporating magnetic pressure into the Young-Laplace equation, we calculate the interfacial tension between the two phases based on the deformation of droplet which agrees well with measurements validated by a tensiometer.