Surface Tension Supported Floating of Heavy Objects: Why
Elongated Bodies Float Better?
Edward Bormashenko
1
Ariel University, Physics Faculty, Ariel, POB 3, 40700, Israel
E-mail address: [email protected]
Floating of bodies heavier than the supporting liquid is discussed. Floating of
cylindrical, ellipsoidal bodies and rectangular plates is discussed. It is demonstrated
that more elongated bodies of a fixed volume are better supported by capillary forces,
due to the increase in the perimeter of the triple line. Thus, floating of metallic
needles obtains reasonable explanation.
I.
INTRODUCTION
Floating of bodies heavier than the supporting liquid is a fascinating physical
phenomenon1-5, giving rise to a variety of effects, including the ability of water
striders to walk on water and self-assembly of floating particles.6-8 An understanding
of floating is crucial for a variety of biological and engineering problems, including
the behavior of colloidal particles attached to a liquid surface, the formation of lipid
droplets and liquid lenses, etc.9-12 The phenomenon of floating of heavy bodies results
from the complex interplay of gravity and surface tension.1-5 The characteristic
dimension describing the interrelation between these factors is called the capillary
length and it is defined by the expression: lca   /  l g (where γ and ρl are the
surface tension and density of the liquid respectively).13-14 When the characteristic
dimension of floating particles is much less than the capillary length, gravity is
negligible, and floating is totally prescribed by the surface tension.11 When the
characteristic dimension of a floating body is on the order of magnitude of the
capillary length, both gravity and surface tension related effects should be
considered.1-5 It is noteworthy that the capillary length is on the same order of
magnitude of several millimeters for all liquids.13 We focus in our study on the
floating of heavy bodies comparable with the capillary length. It is well known that a
metallic needle may float on a water surface, when carefully placed on it.14-15
However, a metallic ball of the same mass will sink; the question is why? The paper
4-15
proposes a simple qualitative model clarifying the impact of the shape on the floating
ability of heavy objects.
II.
FLOATING OF HEAVY NEEDLES
It is well known that a steel needle may be carefully placed on the surface of
water. The needle will float as shown in Fig. 1. At the same time, a steel ball of the
same mass will sink. The floating ability of a needle results from the interplay of
gravity and surface tension. For the sake of simplicity consider the situation where a
liquid wets a cylindrical needle along a line dividing the floating body into equal parts
(ABCD is the medial longitudinal cross-section of the needle, bounded by the triple
line), as shown in Fig. 2. The line, at which solid, liquid and gaseous phases meet is
called the triple (or three phase) line.13-14 The gravity effects, including the buoyancy,
result in the force f grav roughly estimated by:
f grav  gV   l g

V 
 gd 2 l (   l ) ,
2 4
2
(1)
where ρ, V, l and d are the density, volume, length and diameter of the needle
respectively (for the accurate calculation of the buoyancy see Ref. 5). The capillary
force f cap supporting floating may be very roughly estimated as:
f cap    2 (d  l ) ,
(2)
where  is the perimeter of the cross-section ABCD. The interrelation between the
capillarity and gravity-induced effects will be described by the dimensionless
number:

f cap
f grav
.
(3)
Substituting Exp. (1) and (2) into (3) yields:

where  
8
g (  
l
2
f cap
f grav

d l
,
d 2l
(4)
. It should be mentioned that the dimensionless number  is
)
actually the inverse of the well-known Eötvös (or Bond) number.13,14 Now consider
floating of various cylindrical needles of the same volume. Thus, Exp. (5) takes place:

4
d 2 l  V  const .
4-16
(5)
Hence, the length of the needle may be expressed as l 
4V
. Substituting this
d 2
expression into Formula 4, and considering the constancy of volume (5) yields for  :
( d ) 

V
(d 
4V
) .
d 2
(6)
The function (d ) is schematically depicted in Figure 3. It possesses a minimum,
when d  d *  23
V

. Considering V 

4
d 2 l gives rise to d *  2l . It means that the
influence of capillary forces supporting floating is minimal for needles possessing
close longitudinal and lateral dimensions, whereas this influence is maximal for very
long ( d *  2l ) and very short ( d *  l ) needles. Somewhat curiously, very short
needles could also be treated as oblong (elongated) objects, and the inequality   1
takes place. We conclude that the relative influence of capillary forces is maximal for
elongated objects (in turn, the relative influence of gravity for these objects is
minimal). This consideration qualitatively explains the floating ability of heavy
needles.
III. FLOATING OF RECTANGULAR PLATES
Floating of heavy (    l ) rectangular plates may be treated in a similar way.
Consider floating a rectangular plate with the thickness of h and lateral dimensions
axb. When a liquid wets the plate along the line ABCD, dividing it into equal parts
(see Fig. 4), the gravity related effects are approximately given by:
f grav  abhg (  
l
).
(7)
f cap    2 (a  b) .
(8)
2
The capillary force is estimated as:
Consider rectangular plates of the same thickness h and same volume V=abh, but
differently shaped. We have for the dimensionless number  :

where   
2
gV (  
l
2
f cap
f grav

2 (a  b)
gV (  
l
2
  ( a  b) ,
)
. Exp. (9) could be rewritten as:
)
4-17
(9)
(a )   (a 
V
).
ah
(10)
Function (a ) is qualitatively similar to that depicted in Fig 3. It possesses a
minimum, when a  a * 
V
.
h
Considering V  abh , yields a *  b . Thus, the influence of capillary forces is
minimal for square plates of a fixed thickness. Again, the influence of capillarity
supporting the floating is maximal for elongated rectangular plates.
IV. FLOATING OF ELLIPSOIDAL OBJECTS
Treatment of ellipsoidal objects is more complicated. Consider a heavy ellipsoidal
body possessing the semi-principal axes of length a, b, c, floating as shown in Fig. 5.
Under assumptions made in Sections III and IV the gravitational force is given by:
f grav 

4
abcg (   l ) ,
3
2
(11)
4
where abc  V is the volume of the ellipsoid. The capillary force is estimated
3
f cap   , where  is the perimeter (circumference) of the ellipse, bounded by the
triple line and depicted in red in Fig. 5. The circumference of the ellipse is the
complete elliptic integral of the second kind. It may be reasonably evaluated with the
approximate Ramanujan formula:    [3(a  b)  10ab  3(a 2  b 2 ) ] . Thus, the
dimensionless number  equals:

where   

gV (  
l
2
f cap
f grav
  [3(a  b)  10ab  3(a 2  b 2 ) ] ,
(12)
. Now we fix the values of the volume V and the vertical
)
semi-axis c. Thus, we obtain:
3V

3V
 ; 
.
4ac a
4c
b
(13)
Substituting Exp. 13 into Exp. (12) yields:
(a )   [3(a 

a
)  10  3(a 2 
2
a2
)].
Differentiation of Exp. (14) yields the somewhat cumbersome expression:
4-18
(14)
d(a )

 3 (1  2 )(1 
da
a
However,
1
it
a (1 
could

a2
be
easily
a (1 
2
10  3( a 
2
a2
always
)
)
10  3( a 
demonstrated
positive;
a2
2
)
is

hence
2
a2
that
).
(15)
)
the
expression
d(a )
 0 takes
da
place
when a    . Hence, the function (a ) has a physically meaningful minimum,
when a   
3V
 ab . Finally, this results in a=b. We conclude that the role
4c
of capillary forces is minimal in the degenerated case, when horizontal semi-axes of
an ellipsoid are the same, namely the cross-section bounded by the triple line is a
circle. This result is expected intuitively from the variational considerations. Indeed,
we fixed the volume V and the vertical semi-axis c of the floating ellipsoid. In this
situation, the area of the cross-section bounded by the triple line S  ab turns out to
be fixed (recall, that V 
4
4
abc  Sc  const ). Thus, we are actually seeking the
3
3
minimal circumference of the figure possessing the given area, thereby supplying the
minimum to the capillary force. It is well-known from the variational analysis that a
circle has the minimum possible perimeter for a given area. Curiously, the use of the
approximated Ramanujan formula for the calculation of the ellipse perimeter leads to
an accurate solution. We come to the conclusion that the more elongated ellipsoidal
body is better supported by capillary forces. The approach presented in the paper is
qualitative; we did not calculate carefully neither gravity nor surface tension induced
effects, as it performed in Ref. 1, 4, 5, 15. However this qualitative approach, based
on very crude approximations, explains the fascinating floating ability of heavy
elongated objects.
V. DISCUSSION
When the characteristic dimensions of floating bodies are comparable with the
capillary length, both gravity and capillary forces are responsible for the phenomenon
of floating. The capillary force is proportional to the perimeter of the cross-section of
the body circumscribed by the triple line. This perimeter increases with the elongation
4-19
for a given volume of a body. This conclusion is true for different symmetries of
floating objects, including cylindrical and ellipsoidal bodies and rectangular plates
(for rectangular plates and ellipsoidal objects vertical dimensions are also fixed).
These arguments qualitatively explain the floating of metallic needles (whereas a
metallic ball of the same mass will sink).
Acknowledgements
I am thankful to Mrs. Ye. Bormashenko for her kind help in preparing this
manuscript. I am grateful to Professor G. Whyman for inspiring discussions. I am
thankful to Dr. R. Grynyov for experiments demonstrating floating of heavy objects.
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Figures legends
Fig. 1. Steel needle floating in water.
Fig. 2. Scheme of a floating needle supported by the surface tension. ABCD is the
cross-section bounded by the triple (three-phase) line.
Fig. 3. The dimensionless number  depicted schematically as a function of the
diameter of the needle.
Fig. 4. Floating rectangular plate with the dimensions of axbxh.
Fig. 5. Floating ellipsoidal object; a,b,c are semi-axes. The cross-section
circumscribed by the triple line is shown in red.
Fig. 1. Steel needle floating in water.
l
C
D
B
d
A
water
Fig. 2. Scheme of a floating needle supported by the surface tension. ABCD is the
cross-section bounded by the triple (three-phase) line.
4-21
Ψ
d
d*=2l
Fig. 3. The dimensionless number  depicted schematically as a function of the
diameter of the needle.
C
b
B
d
a
D
h
A
water
Fig. 4. Floating rectangular plate with the dimensions of axbxh.
4-22
c
a
b
b
a
c
water
Fig. 5. Floating ellipsoidal object; a,b, c are semi-axes. The cross-section
circumscribed by the triple line is shown in red.
4-23