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]

Stokes' drag,
pressure gradient, with $\boldsymbol{\nabla}$ the gradient operator,
added mass,
Basset force and
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

↑ With Zhu & Fan (1998, pp. 18–27) referring to Soo (1990)
↑ Zhu & Fan (1998, pp. 18–27)
↑ 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.