Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may also be a parcel of fluid). The pressure of a fluid increases with depth below the fluid's surface because of the weight of the fluid above. Thus, the pressure at the bottom of a container of fluid is greater than at the top of the container. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. This pressure difference results in a net upward force on the object. The magnitude of the force is proportional to the pressure difference, and, as explained by Archimedes' principle, is equivalent to the weight of the fluid that would otherwise occupy the submerged volume of the object, i.e. the displaced fluid.
An object under the influence of gravityNote: More generally, in a non-inertial reference frame, which either has a gravitational field or is accelerating due to a force other than gravity defining a "downward" direction. with an average density greater than the surrounding fluid tends to sink because its weight is greater than the weight of the fluid it displaces. If the object is less dense, buoyancy can keep the object afloat.In the absence of surface tension, the mass of fluid displaced is equal to the submerged volume multiplied by the fluid density. High repulsive surface tension will cause the body to float higher than expected, though the same total volume will be displaced, but at a greater distance from the object. Where there is doubt about the meaning of "volume of fluid displaced", this should be interpreted as the overflow from a full container when the object is floated in it, or as the volume of the object below the average level of the fluid.
Buoyancy also applies to fluid mixtures, and is the most common driving force of convection currents. In these cases, the mathematical modelling is altered to apply to continua, but the principles remain the same. Examples of buoyancy driven flows include the spontaneous separation of air and water or oil and water.
Buoyancy is a function of the force of gravity or other source of acceleration on objects of different densities, and for that reason is considered an apparent force, in the same way that centrifugal force is an apparent force as a function of inertia. Buoyancy can exist in environments without gravity, but without some source of acceleration creating a non-inertial reference frame, buoyancy does not exist.
The center of buoyancy of an object is the center of gravity of the displaced volume of fluid.
Archimedes' principle
Archimedes' principle is named after Archimedes of Syracuse, who first discovered this law in 212 BC.. For objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:
—with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.
Mathematically we note.
Where is the local gravitational acceleration, the density of the fluid, and the displaced volume; the negative sign arises since the buoyant force acts in the opposite direction as the object's weight. Archimedes' principle does not consider the surface tension (capillarity) acting on the body, but this additional force modifies only the amount of fluid displaced and the spatial distribution of the displacement, so the principle remains valid.
It's important to note that the density of an object is defined to be its mass per unit volume.
If an object is fully submerged and we assume that the net force acting upon the object in the vertical direction is zero. If fully submerged the displaced volume is simply the volume of the object.
This implies that objects of greater density than the fluid will sink, and objects of lesser density will float. Example: If a person drops wood into water, buoyancy will keep it afloat.
Applications A common application of Archimedes' principle is hydrostatic weighing. Suppose we can measure the tension of a hanging mass by a force probe. Assuming Archimedes' principle, when the mass is submerged in the fluid and the net force is zero,
Recall that the definition of density states
Thus, the density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes. Below we can denote the ratio of densities.
Forces and equilibrium
The equation to calculate the pressure inside a fluid in equilibrium is:
where f is the force density exerted by some outer field on the fluid, and σ is the Cauchy stress tensor. In this case the stress tensor is proportional to the identity tensor:
Here δij is the Kronecker delta. Using this the above equation becomes:
Assuming the outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function:
Then:
Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field. Let the z-axis point downward. In this case the field is gravity, so Φ = −ρfgz where g is the gravitational acceleration, ρf is the mass density of the fluid. Taking the pressure as zero at the surface, where z is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is
So pressure increases with depth below the surface of a liquid, as z denotes the distance from the surface of the liquid into it. Any object with a non-zero vertical depth will have different pressures on its top and bottom, with the pressure on the bottom being greater. This difference in pressure causes the upward buoyancy force.
The buoyancy force exerted on a body can now be calculated easily, since the internal pressure of the fluid is known. The force exerted on the body can be calculated by integrating the stress tensor over the surface of the body which is in contact with the fluid:
The surface integral can be transformed into a volume integral with the help of the Gauss theorem:
where V is the measure of the volume in contact with the fluid, that is the volume of the submerged part of the body, since the fluid does not exert force on the part of the body which is outside of it.
The magnitude of buoyancy force may be appreciated a bit more from the following argument. Consider any object of arbitrary shape and volume V surrounded by a liquid. The force the liquid exerts on an object within the liquid is equal to the weight of the liquid with a volume equal to that of the object. This force is applied in a direction opposite to gravitational force, that is of magnitude:
where ρf is the density of the fluid, Vdisp is the volume of the displaced body of liquid, and g is the gravitational acceleration at the location in question.
If this volume of liquid is replaced by a solid body of exactly the same shape, the force the liquid exerts on it must be exactly the same as above. In other words, the "buoyancy force" on a submerged body is directed in the opposite direction to gravity and is equal in magnitude to
Though the above derivation of Archimedes principle is correct, a paper by the Brazilian physicist Fabio M. S. Lima in 2012 proposed a more general approach for the evaluation of the buoyant force exerted by any fluid (even non-homogeneous) on a body with arbitrary shape based on the surface integral of the pressure force applied by the fluid over the entire body's surface. This method leads to the prediction that the buoyant force exerted on a rectangular block touching the bottom of a container points downward, which has been confirmed experimentally.
The net force on the object must be zero if it is to be a situation of fluid statics such that Archimedes principle is applicable, and is thus the sum of the buoyancy force and the object's weight
If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise. An object whose weight exceeds its buoyancy tends to sink. Calculation of the upwards force on a submerged object during its accelerating period cannot be done by the Archimedes principle alone; it is necessary to consider dynamics of an object involving buoyancy. Once it fully sinks to the floor of the fluid or rises to the surface and settles, Archimedes principle can be applied alone. For a floating object, only the submerged volume displaces water. For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor.
In order for Archimedes' principle to be used alone, the object in question must be in equilibrium (the sum of the forces on the object must be zero), therefore;
and therefore
showing that the depth to which a floating object will sink, and the volume of fluid it will displace, is independent of the gravitational field regardless of geographic location. (Note: If the fluid in question is seawater, it will not have the same density (ρ) at every location, since the density depends on temperature and salinity. For this reason, a ship may display a Plimsoll line.)
It can be the case that forces other than just buoyancy and gravity come into play. This is the case if the object is restrained or if the object sinks to the solid floor. An object which tends to float requires a tension restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a normal force of constraint N exerted upon it by the solid floor. The constraint force can be tension in a spring scale measuring its weight in the fluid, and is how apparent weight is defined.
If the object would otherwise float, the tension to restrain it fully submerged is:
When a sinking object settles on the solid floor, it experiences a normal force of:
Another possible formula for calculating buoyancy of an object is by finding the apparent weight of that particular object in the air (calculated in Newtons), and apparent weight of that object in the water (in Newtons). To find the force of buoyancy acting on the object when in air, using this particular information, this formula applies:
Buoyancy force = weight of object in empty space − weight of object immersed in fluid
The final result would be measured in Newtons.
Air's density is very small compared to most solids and liquids. For this reason, the weight of an object in air is approximately the same as its true weight in a vacuum. The buoyancy of air is neglected for most objects during a measurement in air because the error is usually insignificant (typically less than 0.1% except for objects of very low average density such as a balloon or light foam).
Simplified model
A simplified explanation for the integration of the pressure over the contact area may be stated as follows:
Consider a cube immersed in a fluid with the upper surface horizontal.
The sides are identical in area, and have the same depth distribution, therefore they also have the same pressure distribution, and consequently the same total force resulting from hydrostatic pressure, exerted perpendicular to the plane of the surface of each side.
There are two pairs of opposing sides, therefore the resultant horizontal forces balance in both orthogonal directions, and the resultant force is zero.
The upward force on the cube is the pressure on the bottom surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal bottom surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the bottom surface.