Capillarity and Archimedes' Principle of Floatation

25 Pages·2009·0.25 MB·English

Capillarity and Archimedes’ Principle of Floatation
John McCuan and Ray Treinen
January 15, 2009
Abstract
Weconsidersomeofthecomplications thatariseinattemptingtogeneralize
a version of Archimedes’ principle of floatation to account for capillary effects.
The main result provides a means to relate the floating position (depth in
the liquid) of a symmetrically floating sphere in terms of readily observable
geometric quantities.
A similar result is obtained for an idealized case corresponding to a sym-
metrically floating infinite cylinder. Certain possibilities are also outlined in
the event symmetry is relaxed in this latter problem.
Central to all of these results is a specialized variational formula for floating
bodies which was derived in a special case earlier [Pac. J. Math. 231 (2007)
pp.167–191] and is here generalized to account for gravitational forces.
1 Introduction
We wish to consider the following version of Archimedes’ principle:
An object, when deposited into a bath of liquid, displaces a volume of liquid
having mass equal to the effective mass of the object.
This principle takes no account of the effects of surface tension or surface energies
associated with wetting. Indeed, simple experiments show that it is possible, under
certain circumstances, for a convex object with density greater than that of a given
liquidbathtofloat(only)partiallysubmergedonthesurfaceofthebath. Finn[Fin08]
has recently given the first rigorous proof of this fact, at least in an idealized situation
which we describe in §4 below.
1
Eveninthecaseofaconvexobjectwithconstantdensityρsmallerthanthedensity
ρ of the liquid, how to interpret and generalize Archimedes’ principle to include
0
surface tension and surface energies is not entirely obvious. In order to illustrate this,
let us briefly consider the notion of displaced liquid referred to in the principle above.
The principle states, after all, that
ρ V = ρV , (1)
0 d m
where V is the volume of the object and V is the volume of the displaced liquid.
m d
According to the account of Vetruvius the displaced volume V of liquid is determined
d
by assuming the bathis filled precisely to the rimof the vessel so that when the object
is deposited, the displaced volume is simply that which spills onto the floor, or into
a larger vessel if one wishes to avoid the mess. One presumes that the interface
is assumed planar in this discussion, and hence for a floating object the displaced
volume is also that portion of the volume of the object below the surface of the liquid
while floating.
With surface tension, a bath may often be filled slightly above the rim of the
vessel (or far above the rim of the vessel [Mie02] if the gravity is small); the interface
is also required to meet the floating object at a prescribed contact angle depending on
materials and not necessarily compatible with the condition that a planar interface
meet the object in a manner appropriate for the depth of floatation. These factors
and others may conspire to render the interface decidedly nonplanar in general and
suggest we look elsewhere for the displaced volume of liquid.
Onemight alsouseavessel withhighersides. Infact, weshallrestrict attention, in
the physical three-dimensional case, to a cylindrical vessel with circular cross section
of radius R and having initial depth large enough to completely immerse the object,
even if it floats. See Figure 1. Under the foregoing Archimedean assumptions, the
object of density ρ < ρ will still float in a geometric position (i.e., attitude with
0
respect to the interface) congruent to that obtained when the liquid spilled over the
rim, but the planar interface and the object will be higher; the entire level of the
interface configuration will rise, being translated upward through a vertical distance
V /(πR2) as if the displaced volume had been injected at the bottom of the vessel.
d
It should be noted that the attendant volume V of raised liquid (i.e., the volume of
r
liquid above the level of the original interface) is necessarily less than the displaced
volume, being determined by the relation
V +V = V
r c d
where V is the volume of a certain cavity swept out by the submerged portion of the
c
bottom of the object as it is vertically translated through a distance V /(πR2). Thus,
d
2
Figure 1: The displaced liquid of Archimedean floatation
there is a somewhat delicate relation between the displaced volume, the rise in the
liquid interface, and the manner in which the object floats, even in the Archimedean
case.
When surface tension and wetting energies are taken into account, the interface
itselfmustalsobeexpected tochangeinaglobalwayuponintroductionofthefloating
object. It is clear that making sense of the displaced volume in terms of the liquid
rise in this case would be difficult at best. See Figure 2.
We are not so ambitious to propose here a general definition of the displaced
volume of liquid for an arbitrary floating object. If we assume, however, that the
contact line on the floating object is the intersection of the boundary of the object
with a horizontal plane, then we have recourse to the alternative mentioned above.
Namely, V is simply the volume of the object “below the surface of the liquid,” that
d
is to say below the horizontal plane containing the contact line. This is not only
a convenient definition, but at least in the case of a sphere leads to an interesting
Figure 2: The difficulty of finding the displaced liquid when surface tension and
wetting energies are acting
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analogueofArchimedes’ principle and, surprisingly, onethat canbeviewed asadirect
generalization of it.
Before stating this relation, let us recast Archimedes’ principle in the special case
we have described. Under Archimedean assumptions, a sphere of radius a floating in
the center of a cylindrical vessel should have circular contact line determined by an
¯
azimuthal angle φ. See Figure 3. The volume of the sphere below the planar interface
Figure3: Azimuthal anglesdetermined byahorizontalcontact line(left)anddiffering
azimuthal angles in the two-dimensional case (right)
is
1
V = πa3 sin2φ¯cosφ¯+2+2cosφ¯ .
d
3
Using (1), we obtain the following(cid:0) (cid:1)
Theorem 1 According to Archimedes’ principle, a homogeneous sphere of density
ρ > ρ will sink to the bottom of a bath of density ρ , and a homogeneous sphere of
0 0
density ρ < ρ will float at a level determined by
0
2ρ
cos3φ¯−3cosφ¯= 2 1− . (2)
ρ
(cid:18) 0(cid:19)
It is easily checked that the function F(φ¯) = cos3φ¯− 3cosφ¯ is increasing from −2
to 2 on [0,π] with zero derivative at the endpoints and strictly positive derivative
interior to the interval. Thus, for each positive value 0 ≤ ρ ≤ ρ , the condition (2)
0
determines a unique azimuthal angle. See Figure 4.
We obtain the following result under assumptions described in detail below.
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Theorem 2 A sphere that floats in a centrally symmetric position as described above
under the effects of surface tension and adhesion effects of an axially symmetric bath
¯
must float at a level determined by the azimuthal angle φ satisfying
6 cosγ 3sinγ 2ρ
cos3φ¯−3cosφ¯+ H¯ + sin2φ¯− sin(2φ¯) = 2 1− , (3)
κa a κa2 ρ
(cid:18) 0(cid:19)
h i
where κ = ρg/σ is the capillary constant determined by the gravitational acceleration
¯
g and the surface tension σ, H is the mean curvature of the liquid interface at the
contact line, and γ is the contact angle of the liquid interface with the floating sphere.
¯
The function F(φ) appearing on the left in equation (3) also takes the values −2 and
¯ ¯
2 at the endpoints φ = 0 and π respectively. However, F is decreasing at φ = 0 and
decreases to a unique local interior minimum, thus allowing for values of ρ > ρ and
0
determining explicitly a unique maximum density ρ = ρ (a,γ,κ,H¯) for which
max max
ρ > ρ implies no floatation is possible. From the unique interior minimum, the
max
function strictly increases to a unique (interior) maximum value greater than 2 from
¯
which it strictly decreases to the value 2 at the other endpoint φ = π. It will be noted
¯
from this description that a unique azimuthal angle φ is determined for all values of
0 < ρ < ρ , and that two values are possible for certain values of ρ ≥ ρ (as long as
0 0
ρ is not too large). We presume by continuity that the physically relevant value for
heavy floating spheres is the larger one determined by (3); see the list of conjectures
and open problems at the end for further comments.
2
4 4
1 3 3
2 2
1 1
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
-1 -1 -1
-2 -2
-2
Figure 4: The azimuthal angles determined by Theorems 1 (left) and 2 (middle);
plotted together on the right
It could be legitimately objected that the quantity H¯ appearing in our formula
is somewhat unnatural due, first of all, to the fact that there is nothing of the sort
appearing in Archimedes’ principle which we wish to generalize. Upon reflection,
however, it becomes clear that some globally determined quantities must appear;
the situation when capillarity is taken account of is necessarily more complicated
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than Archimedean floatation. On the other hand, we agree that the value H¯ should,
in principle, be determined by other parameters not appearing in (3), namely the
volume of liquid in the cylinder before the floating object is introduced, and the
contact angle γ between the free surface interface and the wall of the cylinder. In
out
summary, intuition suggests:
An accurate description of how an object floats under the influence of
capillarity should require the inclusion of global information on the config-
uration including but not limited to the volume of liquid and the contact
angle between the liquid interface and relatively distant structures such the
surface of the containing vessel.
We believe this intuition is correct, but a theorem giving the exact relation among
all the relevant quantities requires an understanding of the family of solutions of
the ordinary differential equation governing the interface which is beyond what we
currently have. The structure of this family of solutions is notoriously complicated,
and there are many basic questions even about particular special solutions which are
still open. For a survey of some of the recent results, see [Fin86, Vog82, Sie06, Sie80,
Nic02, EKT04, Tur80, JP68, Tre08].
On the other hand, it simply requires a change of perspective to designate H¯
as a locally measurable independent parameter, and the form of Theorem 2 gives
quantitative content to a competing intuition which says:
The depth at which an object floats in a liquid bath (relative to the “level”
of the bath) should only depend on the relative densities, the contact angle
γ of the interface with the surface of the floating object, and quantities
measured locally near the object.
The choice and appearance of the particular quantity H¯ is further explained in §5 in
reference to Conjecture 4. The result above may also be defended on precisely the
ground that it is, like Archimedes’ result, essentially algebraic and beautifully simple
in that sense.
2 Variational Formulation
The general assumptions of our model are outlined in [McC07] though the derivation
given there was aimed at the zero gravity case in which buoyancy plays no role, and
the effects of gravity were not properly considered. For the sake of making this paper
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somewhat more self-contained we include a short review/summary of the model and
amend the deficiencies in the former derivation.
Quite generally, we consider a solid structure Σ = Σ ∪ Σ consisting of a sta-
s m
tionary part Σ and a movable, or floating, part Σ . In addition, we hypothesize an
s m
equilibrium liquid interface Λ with corresponding wetted region W = W ∪ W , so
s m
that the liquid volume V satisfies ∂V = Λ∪W and the contact line/triple interface
is given by ∂Λ = ∂W. Under these assumptions, we consider the variational problem
associated with
E = σ|Λ|−σβ|W|+G (4)
where G = G and G is a position dependent function representing field forces
V∪Σm
such as gravity.1
R
Under rather general hypotheses, as described in [McC07] a family of variations
leaving Σ fixed leads to the following (standard) variational formulas
m
|V˙| = − 2HX˙ ·N + X˙ ·~n
ZΛ Z∂Λ
where H is the mean curvature defined on Λ, X˙ is the variation vector, N is the unit
normal pointing out of the liquid volume V, and ~n is the unit conormal to N and ∂Λ
pointing out of Λ.
|W˙ | = X˙ ·~ν,
Z∂Λ
where ~ν is the unit conormal to NW and ∂W pointing out of W; note that NW
denotes the unit normal to W pointing out of V and may also be denoted by N on
the interior of W where no ambiguity arises.
G˙ = GX˙ ·N and |V˙| = X˙ ·N.
ZΛ ZΛ
These last two formulas apparently require an interesting and somewhat delicate
applicationofmoregeneralmathematicalprinciples offluidmechanics, andweoutline
their derivation under more general assumptions below.
For now, we assemble E˙/σ −λ|V˙| from the constituent parts above where λ is a
Lagrange multiplier associated with the volume constraint:
E˙/σ −λ|V˙| = (−2H +G/σ −λ)X˙ ·N + (X˙ ·~n−βX˙ ·~ν).
ZΛ Z∂Λ
1We included only G in [McC07].
V
R
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The vanishing of this quantity for all variation vectors X˙ results in the well known
geometric boundary value problem
2H = G/σ −λ on Λ
(5)

cosγ = β on ∂Λ

since~n = (~n·NW)NW+cosγ~ν. In the special case under consideration in this paper,
G represents the limiting value ρ gz taken as a limit from inside the liquid, so that
0
2H = κz −λ
where κ = ρ g/σ is a capillary constant for the problem.
0
A more general variation allowing rigid motion of Σ takes the form
m
X = X(p;t,h) : M ×(−ǫ,ǫ)×(−δ,δ) → R3
where M = Σ∪V is considered as an abstract manifold; see Figure 5.
Figure 5: The variation map and its notation
It is assumed here, as indicated in the figure that h parameterizes a family of rigid
motions w = w(x;h) to which Σ is subject. Denoting derivatives with respect to h
m
by an acute accent, we find
|Λ´| = − 2HX´ ·N + X´ ·~n, (6)
ZΛ Z∂Λ
|W´ | = − 2HWX´ ·N + X´ ·~ν, (7)
ZWm Z∂Wm
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G´ = GX´ ·N + GX´ ·NW + G X´ ·Nm. (8)
m
ZΛ ZWm Z∂Σm
This last term requires some explanation. The quantity G denotes the value of the
m
volumetric forcefieldpotentialtakenasalimit frominsidethemovablesolidstructure
Σ . In the special case of a floating object of density ρ, we typically take G = ρgz.
m m
Also in this last identity Nm denotes the unit normal to the boundary ∂Σ of the
m
movable/floating solid structure and points out of Σ , so that Nm = −NW on their
m
common domain of definition W . Finally, we include a brief derivation.
m
Up until this point, we have stated all variational formulae in their final form,
that is to say with the parameters of the variation set to zero so that X˙ represents
d
X(p;t)
dt
t=0
(cid:12)
where X = X(p;t) : M ×(−ǫ,ǫ). For this calculation, we must temporarily assume
(cid:12)
the parameters t and h are not evaluated at zero. Notationally, this is conveniently
indicated by a tilde so that Σ˜ = X(Σ ) = X(Σ ;t;h), and we will evaluate at
m m m
t = h = 0 at the end.
Consideration of the second term should suffice. Setting
G = G ,
m m
ZΣ˜m
we have
G = G ◦Xdet DX,
m m
ZΣm
where X represents the restriction of the variation to Σ and the derivative is taken
m
in M ⊂ R3 with respect to p. Euler’s kinematical formula tells us how a material
integral changes with the flow of a region of fluid. We can cast our present situation
into this framework starting with the preliminary identity
∂
det DX = (divR3 v)◦X det DX
∂h
where v(x;h) = X´(X−1(x;h);h) is the spatial velocity associated with the flow X =
X(p;h) and we have simply suppressed the t dependence. It might be expected (or
hoped) that in our situation the motion/flow associated with the variation should be
particularly simple, at least on the solid movable object Σ , and that we might have,
m
for example, X(p;h) ≡ w(p;h) there. However, taking into account the motion of
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the liquid and that of the contact line of the liquid interface Λ in particular, it is
clear that this would violate the continuity assumption on the variation X : M ×
(−ǫ,ǫ)×(−δ,δ) → R3. Having made this concession and subjected ourselves to the
added complication that other authors seem to have avoided, it is some consolation,
aspointedoutbyFinn[Fin05], thattheinternalmotionoftheliquid underavariation
of the free surface interface could be very complicated, and we are taking account of
such possibilities.
In any case, we continue to obtain
G´m = DGm ·v+divR3 v
ZΣ˜m
= divR3(Gmv)
ZΣ˜m
= G v·Nm,
m
Z∂Σ˜m
so that
G´ = G X´ ·Nm.
m m
h=0 Z∂Σm
A similar argument applies to the integral over V appearing in G and also yields
(cid:12)
(cid:12)
|V´| = X´ ·N + X´ ·N
ZΛ ZWm
where we have returned to the general assumption on evaluation, that t = h = 0.
Combining this with (6-8), we have
´ ´ ´ ´ ´
E/σ −λ|V| = (−2H +G/σ −λ)X ·N + (X ·~n−βX ·~ν) +
ZΛ Z∂Λ
β 2HWX´ ·N + (G/σ −λ)X´ ·N + (G /σ)X´ ·Nm
m
ZWm ZWm Z∂Σm
= X´ ·~n−cosγ X´ ·~ν +
Z∂Wm Z∂Wm
cosγ 2HWX´ ·N + (G/σ −λ)X´ ·N + (G /σ)X´ ·Nm.
m
ZWm ZWm Z∂Σm
Next we refer to a calculation from [McC07] which uses the fact that w−1(X;h) ∈ Σ
m
when X = X(p;h) ∈ w(Σ ;h) to show that
m
X´ −w´ ∈ T Σ .
X m
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