The coalescence of water drops I. A theoretical model of approaching drops |
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| Authors: | Nitza Arbel Zev Levin |
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| Institution: | (1) Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel;(2) National Center for Atmospheric Research, 80307 Boulder, Colorado, USA |
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| Abstract: | The approach of two water drops in the absence of air flow around them is theoretically investigated. By assuming deformation criteria it is possible to solve the equation of motion of the drops under the influence of a variety of forces. These forces include the viscous force exerted by the air between the two deformed surfaces, the London-Van Der Waals forces and the force of gravity. It is found that the viscous forces dominate over the whole distance of the interaction. The equations have analytical solutions when a head-on approach is considered and when the deformation of the drops is assumed constant during the interaction. The equations were solved numerically for other deformation criteria and for non head-on approaches.The results of the present model are used in the following paper to compute the coalescence efficiencies of water drops. The model is primarily applicable to situations in which the large drop is stationary and the small one approaches it from below. However, it could also be used for interaction between freely falling drops as long as their relative velocities exceed about 13 cm/sec.Appendix: List of symbols
C
constant of the motion
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D
distance between the deformed surfaces of the drops
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D
o
initial value ofD
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D
m
the value at which the viscous force is maximum
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D
N
normalized distance
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D
s
the distance at which the velocity of approach vanishes
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F
c
centrifugal force
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F
g
force due to gravity
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F
N
normalized viscous force
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F
LV
force due to London-Van der Waals effect
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F
R
radial component of the force
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F
V
viscous force
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F
t
tangential component of the force
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g
acceleration due to gravity
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M
L
mass of large drop
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m
s
mass of small drop
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p
ratio of radii of interacting drops
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R
radius of an arbitrary drop
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r
distance between the centers of mass of the two drops
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R
D
radius of deformation
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R
L
radius of larger drop
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R
s
radius of smaller drop
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t
time
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u
defined in equation 20 — has the meaning of kinetic energy
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v
relative velocity of the deformed surfaces
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v
0
initial value ofv
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V
0
initial relative velocity of the centers of the drops
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V
c
critical impact velocity
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V
i
impact velocity
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V
N
,v
n
normalized velocity
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V
t
tangential component of the velocity
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W
i
velocity of the small drop at infinity for it to reach the pointD
0 at velocityV
0
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x
instantaneous impact distance
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average critical impact distance for coalescence
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x
0
initial value of the impact distance
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x
c
critical impact distance for coalescence
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coefficient of deformation
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i
impact angle according toWhelpdale andList (1971)
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coefficient of deformation
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viscosity
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surface tension
-  F
s
sum of forces acting on the small drop
- F
L
sum of forces acting on the large drop
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time constant
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R
Rayleigh's oscillation period
On sabbatical leave (1976–77) from the Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel.The National Center for Atmospheric Research is sponsored by the National Science Foundation. |
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| Keywords: | Cloud drop coalescence |
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