Basset–Boussinesq–Oseen equation

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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[edit]

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,
  3. added mass,
  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[edit]

  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[edit]

  • 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.