The BBO equation, in the formulation as given by and , pertains to a small spherical particle of diameter d_p having mean
density \rho_p whose center is located at \boldsymbol{X}_p(t). The particle moves with
Lagrangian velocity \boldsymbol{U}_p(t)=\text{d} \boldsymbol{X}_p / \text{d}t in a fluid of density \rho_f,
dynamic viscosity \mu and
Eulerian velocity field \boldsymbol{u}_f(\boldsymbol{x},t). The fluid velocity field surrounding the particle consists of the undisturbed, local Eulerian velocity field \boldsymbol{u}_f plus a disturbance field – created by the presence of the particle and its motion with respect to the undisturbed field \boldsymbol{u}_f. 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, \boldsymbol{U}_f(t)=\boldsymbol{u}_f(\boldsymbol{X}_p(t),t). 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: : \begin{align} \frac{\pi}{6} \rho_p d_p^3 \frac{\text{d} \boldsymbol{U}_p}{\text{d} t} &= \underbrace{3 \pi \mu d_p \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 1}} - \underbrace{\frac{\pi}{6} d_p^3 \boldsymbol{\nabla} p}_{\text{term 2}} + \underbrace{\frac{\pi}{12} \rho_f d_p^3\, \frac{\text{d}}{\text{d} t} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 3}} \\ & + \underbrace{\frac{3}{2} d_p^2 \sqrt{\pi \rho_f \mu} \int_{t_{_0}}^t \frac{1}{\sqrt{t-\tau}}\, \frac{\text{d}}{\text{d} \tau} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)\, \text{d} \tau}_{\text{term 4}} + \underbrace{\sum_k \boldsymbol{F}_k}_{\text{term 5}} . \end{align} 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: • Stokes' drag, •
Froude–Krylov force due to the
pressure gradient in the undisturbed flow, with \boldsymbol{\nabla} the
gradient operator and p(\boldsymbol{x},t) the undisturbed pressure field, • added mass, • Basset force and • other forces acting on the particle, such as
gravity, etc. The particle Reynolds number R_e: :R_e = \frac{\max\left\{ \left| \boldsymbol{U}_p - \boldsymbol{U}_f \right| \right\}\, d_p}{\mu/\rho_f} has to be less than unity, R_e , 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: : -\boldsymbol{\nabla} p = \rho_f \frac{\text{D} \boldsymbol{u}_f}{\text{D} t} - \mu \nabla^2 \boldsymbol{u}_f, with \text{D} \boldsymbol{u}_f / \text{D} t the
material derivative of \boldsymbol{u}_f. Note that in the Navier–Stokes equations \boldsymbol{u}_f(\boldsymbol{x},t) is the fluid velocity field, while, as indicated above, in the BBO equation \boldsymbol{U}_f is the velocity of the undisturbed flow as seen by an observer moving with the particle. Thus, even in steady Eulerian flow \boldsymbol{u}_f depends on time if the Eulerian field is non-uniform. ==Notes==