Added mass

In fluid mechanics, added mass is the inertia added to a system because an accelerating or decelerating body must move some volume of surrounding fluid as it moves through it, since the object and fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees.

The concept of added mass can be thought of as a classical physics analogue of the quantum mechanical concept of quasiparticles. It is, however, not to be confused with relativistic mass increase.

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History

Friedrich Bessel proposed the concept of added mass in 1828 to describe the motion of a pendulum in a fluid. The period of such a pendulum increased relative to its period in a vacuum (even after accounting for buoyancy effects), indicating that the surrounding fluid increased the effective mass of the system. (See: G. G. Stokes, Trans. Camb. Philos. Soc. 9, 8, 1851.)

Applications

The added mass can be incorporated into most physics equations by considering an effective mass as the sum of the mass and added mass. This sum is commonly known as the "virtual mass".

A simple formulation of the added mass for a spherical body permits Newton's classical second law to be written in the form

F = ma becomes F = (m + m_added)a.

One can show that the added mass for a sphere (of radius r) is 2 / 3πr³ρ_fluid. For a general body, the added mass becomes a tensor (referred to as the induced mass tensor), with components depending on the direction of motion of the body. It should be noted that not all elements in the added mass tensor will have dimension mass, some will be mass*length and some will be mass*length².

All bodies accelerating in a fluid will be affected by added mass, but since the added mass is dependent on the density of the fluid, the effect is often neglected for dense bodies falling in much less dense fluids. For situations where the density of the fluid is comparable to or greater than the density of the body, the added mass can often be greater than the mass of the body and neglecting it can introduce significant errors into a calculation.

For example, a spherical air bubble rising in water has a mass of 4 / 3πr³ρ_air but an added mass of 2 / 3πr³ρ_water. Since water is approximately 800 times denser than air (at RTP), the added mass in this case is approximately 400 times the mass of the bubble.

See also

• Response Amplitude Operator for the use of added mass in ship design.

References

• MIT OpenCourse Ware
• Naval Civil Engineering Laboratory