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
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: