Fluid Dynamics of Floating Particles. Fluid dynamics of floating particles (with experiments by Wang, Bai, and Joseph). J. Fluid Mech. Submitted.
Fluid dynamics of floating particles (with experiments by Wang, Bai, and Joseph). J. Fluid Mech. Submitted.
D.D. Joseph, J. Wang, R. Bai and H. Hu. 2003. Particle motion in a liquid film rimming the inside of a rotating cylinder. J. Fluid Mech. 496, 139-163
CONTACT LINE MOVES
Weight of displaced fluids
Generalized Archimedes principle
FORCE BALANCE mg=Fc+Fp
= Floating depth. The more it sinks, the more it is buoyed up.
Fp= ρlgvw + ρagva + (ρl + ρa)gh2A = Pressure Force
Buoyant weight of liquid cylinder above the contact ring
FLOATING DISKS PINNED AT SHARP EDGES
The contact line is fixed and the angle is determined by the force balance; just the opposite.
The floating depth is not determined by wettability.
ψ = 90º
ψ > 90º
The effective contact angle
Equilibrium Contact Angle
n is not defined
ranges over an interval 180º-
θ; 90º at a square corner
The effective angle at a sharp corner is not determined by the Young-Dupré law; it is determined by dynamics.
Cubes can float in different ways. This cube has an interface on a sharp edge and smooth faces.
The depth to which a cube sinks into the lower fluid increases with increasing value of the cylinder density. The contact angle on the plane faces is 120 degrees and the interface at the sharp edges AD and BC is fixed.
(a) Initial state. (b) ρP =1.5, (c) ρP =1.2, and (d) ρP =1.1. Notice that in (c) and (d) the interface near the edges AD and BC rises, as for these cases the particle position is higher than the initial position.
When there are two or more particles hanging in an interface, lateral forces are generated. Usually, these forces are attractive.
The lateral forces arise from pressure imbalance due to the meniscus and from a capillary imbalance.
After Poynting and Thompson 1913.
Neutrally buoyant copolymer spheres d = 1mm cluster in an air/water interface.
DYNAMICS (Gifford and Scriven 1971)
“casual observations… show that floating needles and many other sorts of particles do indeed come together with astonishing acceleration. The unsteady flow fields that are generated challenge analysis by both experiment and theory. They will have to be understood before the common-place ‘capillary attraction’ can be more than a mere label, so far as dynamic processes are concerned.”
Sand in Glycerin
Sand in Water
Frequency = 8 Hz
Light Particles in Water
Heavier-than-Water Particles in Water
Aqueous Triton Mixture
W region occupied by fluids and solids
P(t) region occupied by solids
Equations in W/P(t)
div u = 0 in W
The body force l - a22 l is chosen so that u = U + wr, ss= 0is a rigid motion on P(t) where U(t) and w(t) satisfy
The multiplier field satisfies
normal to n, t plane
uex is in f, normal to t, points inward
uex=an + bn2
t • uex= 0 ,
nf • uex= 0 ,
n • nf = cos a