# 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

One formulation of the BBO equation is the one given by Zhu & Fan (1998, pp. 18–27), for a spherical particle of diameter $d_p$, position $\boldsymbol{x}=\boldsymbol{X}_p(t)$ and mean density $\rho_p$ moving with particle velocity $\boldsymbol{U}_p=\text{d} \boldsymbol{X}_p / \text{d}t$ – in a fluid of density $\rho_f$, dynamic viscosity $\mu$ and with ambient (undisturbed local) flow velocity $\boldsymbol{U}_f:$[1]

\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, with in the left-hand side the particle's rate of change of linear momentum, and on the right-hand side the forces acting on the particle. The terms on the right-hand side are respectively due to the:[2]

1. Stokes' drag,
2. pressure gradient, with $\boldsymbol{\nabla}$ the gradient operator,
4. Basset force and
5. other forces on the particle, such as due to 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 small, $R_e<1$, for the BBO equation to give an adequate representation of the forces on the particle.[3]

Also Zhu & Fan (1998, pp. 18–27) suggest to estimate the pressure gradient from the Navier–Stokes equations:

$-\boldsymbol{\nabla} p = \rho_f \frac{\text{D} \boldsymbol{u}_f}{\text{D} t} - \mu \boldsymbol{\nabla}\!\cdot\!\boldsymbol{\nabla} \boldsymbol{u}_f,$

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 in the BBO equation $\boldsymbol{U}_f$ is the undisturbed fluid velocity at the particle position: $\boldsymbol{U}_f(t)=\boldsymbol{u}_f(\boldsymbol{X}_p(t),t).$

## Notes

1. ^ With Zhu & Fan (1998, pp. 18–27) referring to Soo (1990)
2. ^ Zhu & Fan (1998, pp. 18–27)
3. ^ Green, Sheldon I. (1995). Fluid Vortices. Springer. p. 831. ISBN 9780792333760.

## References

• Zhu, Chao; Fan, Liang-Shi (1998). "Chapter 18 – Multiphase flow: Gas/Solid". In Johnson, Richard W. The Handbook of Fluid Dynamics. Springer. ISBN 9783540646129.
• Soo, Shao L. (1990). Multiphase Fluid Dynamics. Ashgate Publishing. ISBN 9780566090332.