In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
, the Cheerios effect is a colloquial name for the phenomenon of floating objects appearing to either attract or repel one another. The example which gives the effect its name is the observation that pieces of breakfast cereal (for example,
Cheerios
Cheerios is a brand of cereal manufactured by General Mills in the United States, consisting of pulverized oats in the shape of a solid torus. In some countries, including the United Kingdom, Cheerios is marketed by Cereal Partners under the Ne ...
) floating on the surface of a bowl will tend to clump together, or appear to stick to the side of the bowl.
Description
The effect is observed in small objects which are supported by the surface of a liquid. There are two types of such objects: objects which are sufficiently
buoyant
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
that they will always float on the surface (for example, Cheerios in milk), and objects which are heavy enough to sink when immersed, but not so heavy as to overcome the
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
of the liquid (for example, steel pins on water). Objects of the same type will appear to attract one another, and objects of opposite types will appear to repel one another.
In addition, the same attractive or repulsive effect can be observed between objects and the wall of the container. Once again there are two possibilities: the interface between the liquid and the container wall is either a concave or a convex
meniscus
Meniscus may refer to:
*Meniscus (anatomy), crescent-shaped fibrocartilaginous structure that partly divides a joint cavity
*Meniscus (liquid)
The meniscus (plural: ''menisci'', from the Greek for "crescent") is the curve in the upper surface ...
. Buoyant objects will be attracted in the case of a concave meniscus and repelled for convex. Non-buoyant floating objects will do the opposite.
Explanation
All objects in a fluid experience two opposed forces in the vertical direction: gravity (determined by the mass of the object) and buoyancy (determined by the density of the fluid and the volume of liquid displaced by the object). If the buoyant force is greater than the force of gravity acting on an object, it will rise to the top of the liquid. On the other hand, an object immersed in a liquid which experiences a gravitational force greater than its buoyant force will sink.
At the surface of the liquid, a third effect comes into play -
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
. This effect is due to the fact that molecules of the liquid are more strongly attracted to each other than they are to the air above the liquid. As such,
non-wetting objects on the surface of the liquid will experience an upward force due to surface tension. If the upward force is sufficient to balance the force of gravity on the object, it will float on the surface of the liquid, while deforming the surface down. By contrast, objects with a net positive buoyancy will deform the water surface upward around them as they press against the surface.
This deformation of the liquid surface, combined with the net upwards or downwards force experienced by each object, is the cause of the Cheerios effect. Objects experiencing a net upward force will follow the surface of the liquid as it curves upward. Therefore two objects with an upward deformation will move toward each other as each follows the surface of the liquid upward. Similarly, objects with a net downward force will follow the curve of the liquid surface in the downward direction, and will move horizontally together as they do so.
The same principle holds at the side of the container, where the surface of the liquid is deformed by the
meniscus
Meniscus may refer to:
*Meniscus (anatomy), crescent-shaped fibrocartilaginous structure that partly divides a joint cavity
*Meniscus (liquid)
The meniscus (plural: ''menisci'', from the Greek for "crescent") is the curve in the upper surface ...
effect. If the container is wetting with respect to the liquid, the meniscus will slope upwards at the wall of the container, and buoyant objects will move towards the wall as a result of travelling upward along the surface. By contrast, non-buoyant floating objects will move away from the walls of such a container for the same reason.
More complex behavior resulting from the same principles can be observed in shapes which do not have simple concave or convex meniscus behavior. When such objects come close to each other they rotate in the plane of the water surface until they find an optimum relative orientation then move toward each other.
Simplified calculation
Writing in the
American Journal of Physics
The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current Fre ...
,
Dominic Vella and
L. Mahadevan of
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...
discuss the Cheerios effect and suggest that it may be useful in the study of the
self-assembly
Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...
of small structures. They calculate the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
between two
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s of
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and radius
floating distance
apart in liquid of density
as
:
where
is the surface tension,
is a modified
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first kind,
is the
Bond number
Bond or bonds may refer to:
Common meanings
* Bond (finance), a type of debt security
* Bail bond, a commercial third-party guarantor of surety bonds in the United States
* Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
, and
:
is a nondimensional factor in terms of the
contact angle
The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liq ...
.
Here
is a convenient meniscus length scale.
References
{{reflist
Fluid mechanics