# What does the net force do with this droplet in magnetic field?

Consider a spherical droplet with radious $R_d$ which is coated with an immiscible ferrofluid with volume equal to $2v_f$. While the droplet is in rest on a surface, if I approach two magnets in opposite directions, the ferrofluid will gather in two poles ($v_f$ in each pole and I may denote its corresponding radios with $R_f$) and different forces will be applied to them including magnetic force, surface tension force, interfacial forces between liquids and air, droplet weight, etc.

Question: I'm trying to find that under what conditions (for base droplet and ferrofluid properties, field strength, $R_d$, $v_f$, etc), the magnetic field would be capable of exerting enough force to biforcate the droplet (which is maybe equivalent to start narrowing from A-A)

I think that force applied to droplet on A-A axis would be:

$F_m=$force applied to ferrofluid cap by magnetic field

$F_T=2\pi r\gamma$

is surface tension force applied to the interface of air and water droplet at A-A

$F_L=\Delta P.A=\gamma(1/r_1+1/r_2)\pi r^2$

is Laplace pressure in which $r_1$ and $r_2$ are radii of curvature which could be considered equal together.

$F_c=K\pi r^2$ force taht should be applied at A-A to overcome cohesion between water molecules in which $K$ should be cohesion coefficient between water molecules.

Also, $F_a$ is adhesion force between main droplet and ferrofluid coating and should be larger than the force required to be applied to A-A for bifurcation:

$F_m>F_T\ cos\theta-F_L+F_c$

• I would be shocked if the solution to this problem didn't involve lagrangian analysis of the problem, and looking for some sort of configuration that minimzed the energy – Jerry Schirmer Feb 15 '15 at 2:04
• Beware that the familiar "magnetic attraction" between two dipoles, or between a dipole and a polarizable object, requires that the magnetic field be non-uniform; in general polarizable materials are drawn towards regions of strong field. For the two ends of the droplet to be drawn apart, as you've sketched, you'd have to have a field strength minimum in the middle of the droplet. The whole scenario sounds very fiddly and unrealistic to me. However if you imagine a charged fluid layer atop an insulating droplet, there might be a charge density where the charged fluid wants to fragment. – rob Feb 15 '15 at 4:35
• @rob Thanks for comments. Just 3 points to know your opinion about: 1) I think that in case I use two permanent magnets and put the base droplet between them, magnetic field would be maximum in two poles and minimum in the middle point (droplet center) as you mentioned. 2) What do you mean by: "there might be a charge density where the charged fluid wants to fragment" I want to perform some experiments for this to see whether it will happen. Do you think that it is necessary to modify any condition etc in above question? 3) Do you have any idea for bifurcation a droplet in a more feasible way? – vorujak Feb 15 '15 at 5:58
• (1) If the field gradient is strong enough to stretch the droplet, it would be hard to get the droplet to the center — it'll tend to be attracted to one magnet or the other as your hand wobbles while you steer the droplet to the interaction region. (2) If your conducting fluid separates into two regions, they'll be repelled from each other and want to stay apart. (3) You might try to use angular momentum to split the droplet, with a centrifuge table; perhaps you can image with a strobe. (4) Is your droplet on a surface? floating in space? suspended in another medium? – rob Feb 15 '15 at 6:12
• @rob Thanks for comments. My droplet has a ferrofluid oily coating and I prefer to use this property for bifurcating it. I liked your idea for using angular momentum, in this case, instead of using nano-sized particles and in order to have enough momentum for droplet rupture, maybe it's better to put larger magnet(s) into droplet. Do you think it would be Ok? Regarding (4), I planned to perform experiments on a superhydrophobic plate to minimize surface effects. – vorujak Feb 15 '15 at 12:47