In fluid mechanics

Fluid mechanics

Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

, the Cheerios effect is the tendency for small wettable

Wetting

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...

 floating objects to attract one another. An example of the Cheerios effect is the phenomenon whereby breakfast cereal

Breakfast cereal

A breakfast cereal is a food made from processed grains that is often, but not always, eaten with the first meal of the day. It is often eaten cold, usually mixed with milk , water, or yogurt, and sometimes fruit but sometimes eaten dry. Some cereals, such as oatmeal, may be served hot as porridge...

 tends to clump together or cling to the sides of a bowl of milk

Milk

Milk is a white liquid produced by the mammary glands of mammals. It is the primary source of nutrition for young mammals before they are able to digest other types of food. Early-lactation milk contains colostrum, which carries the mother's antibodies to the baby and can reduce the risk of many...

. It is named for the breakfast cereal Cheerios

Cheerios

Cheerios is a brand of breakfast cereal by General Mills introduced on May 1, 1941 as the first oat-based, ready-to-eat cold cereal. Originally named CheeriOats, the name was changed to Cheerios in 1945 because of a trade name dispute with Quaker Oats. The name fit the "O" shape of the cereal pieces...

 and is due to surface tension

Surface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

 and buoyancy

Buoyancy

In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

. The same effect governs the behavior of bubbles on the surface of soft drink

Soft drink

A soft drink is a non-alcoholic beverage that typically contains water , a sweetener, and a flavoring agent...

s.

Description

This clumping behavior applies to any small macroscopic object that floats or clings to the surface of a liquid. This can include a multitude of things, including hair particles in shaving cream and fizzy beer bubbles. The effect is not noticeable for boats and other large floating objects because the force of surface tension is relatively small at that scale. (The Casimir effect

Casimir effect

In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...

, with a similar result, occurs at nanoscopic scale, and boats and other large objects floating in a choppy sea are subject to its classical equivalent; both are caused by waves, not surface tension.)

Explanation

The quality of surface tension

Surface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

 allows the surface of a liquid

Liquid

Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

 to act like a flexible membrane. A variety of weak forces act between liquid molecules to cause this effect.

At the interface between water and air, water molecules at the surface are pulled forcefully by water molecules beneath them but experience only a weak outward pull from the air molecules above. Therefore, the surface of the water caves in slightly, forming a curve known as a meniscus

Meniscus

The meniscus is the curve in the upper surface of a liquid close to the surface of the container or another object, caused by surface tension. It can be either convex or concave. A convex meniscus occurs when the molecules have a stronger attraction to each other than to the material of the...

.

Water adjacent to the side of a container curves either upward or downward, depending on whether the liquid is attracted to or repulsed by the material of the wall. For example, since water is attracted to glass, the water surface in a glass container will curve upwards near the container walls, as this shape increases the contact area between the water and the glass. A floating object which is less dense than water, seeking the highest point, will thus find its way to the edges of the container. A similar argument explains why bubbles on surfaces attract each other: a single bubble raises the water level locally, causing other bubbles in the area to be attracted to the first. Conversely, dense objects like paper clips can rest on liquid surfaces due to surface tension. These objects deform the liquid surface downward. Other dense objects, seeking to move downward but constrained to the surface by surface tension, will be attracted to the first. Objects denser than water will repel objects less dense than water: dense objects deform the water surface downward, and less dense objects tend to move upward, away from the dense object. Objects with an irregular meniscus also deform the water surface, forming "capillary multipoles". When such objects come close to each other, they first rotate in the plane of the water surface until they find an optimum relative orientation. Subsequently they are attracted to each other by surface tension.

Writing in the American Journal of Physics

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 is Jan Tobochnik of Kalamazoo College.-Aims and scope:...

, Dominic Vella and L. Mahadevan

L. Mahadevan

Lakshminarayanan Mahadevan is an Indian American mathematician, and Lola England de Valpine Professor of Applied Mathematics, at Harvard University...

 of Harvard University

Harvard University

Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...

 discuss the cheerios effect and suggest that it may be useful in the study of self assembly of small structures. They calculate the force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

 between two sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s of density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

  and radius floating distance apart in liquid of density as


where is the surface tension, is a modified Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

 of the first kind, is the Bond number, and

is a nondimensional factor in terms of the contact angle

Contact angle

The contact angle is the angle at which a liquid/vapor interface meets a solid surface. The contact angle is specific for any given system and is determined by the interactions across the three interfaces. Most often the concept is illustrated with a small liquid droplet resting on a flat...

.
Here is a convenient meniscus length scale.