In fluid dynamics, the Basset–Boussinesq–Oseen equation (BBO equation) describes the motion of – and forces on – a small particle in unsteady flow at low Reynolds numbers.
The equation is named after Joseph Valentin Boussinesq, Alfred Barnard Basset and Carl Wilhelm Oseen.
The BBO equation, in the formulation as given by Zhu & Fan (1998, pp.
18–27) and Soo (1990), pertains to a small spherical particle of diameter
The particle moves with Lagrangian velocity
and Eulerian velocity field
The fluid velocity field surrounding the particle consists of the undisturbed, local Eulerian velocity field
plus a disturbance field – created by the presence of the particle and its motion with respect to the undisturbed field
For very small particle diameter the latter is locally a constant whose value is given by the undisturbed Eulerian field evaluated at the location of the particle center,
The small particle size also implies that the disturbed flow can be found in the limit of very small Reynolds number, leading to a drag force given by Stokes' drag.
Unsteadiness of the flow relative to the particle results in force contributions by added mass and the Basset force.
The BBO equation states: This is Newton's second law, in which the left-hand side is the rate of change of the particle's linear momentum, and the right-hand side is the summation of forces acting on the particle.
The terms on the right-hand side are, respectively, the:[1] The particle Reynolds number
, for the BBO equation to give an adequate representation of the forces on the particle.
18–27) suggest to estimate the pressure gradient from the Navier–Stokes equations: with
is the fluid velocity field, while, as indicated above, in the BBO equation
is the velocity of the undisturbed flow as seen by an observer moving with the particle.
Thus, even in steady Eulerian flow
depends on time if the Eulerian field is non-uniform.