Basset–Boussinesq–Oseen equation
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. Formulation. The BBO equation, in the formulation as given by and , pertains to a small spherical particle of diameter formula_1 having mean density formula_2 whose center is located at formula_3. The particle moves with Lagrangian velocity formula_4 in a fluid of density formula_5, dynamic viscosity formula_6 and Eulerian velocity field formula_7. The fluid velocity  field surrounding the particle consists of the undisturbed, local Eulerian velocity field formula_8 plus a disturbance field – created by the presence of the particle and its motion with respect to the undisturbed field formula_9 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, formula_10. 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: The particle Reynolds number formula_14 has to be less than unity, formula_16, for the BBO equation to give an adequate representation of the forces on the particle. Also suggest to estimate the pressure gradient from the Navier –Stokes equations: with formula_18 the material derivative of formula_9 Note that in the Navier–Stokes equations formula_7 is the fluid velocity field, while, as indicated above, in the BBO equation formula_21 is the velocity of the undisturbed flow as seen by an observer moving with the particle. Thus, even in steady Eulerian flow formula_8 depends on time if the Eulerian field is non-uniform.