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 Zhu & Fan (1998, pp. 18–27) and Soo (1990), pertains to a small spherical particle of diameter ${\displaystyle d_{p}}$ having mean density ${\displaystyle \rho _{p}}$ whose center is located at ${\displaystyle {\boldsymbol {X}}_{p}(t)}$. The particle moves with Lagrangian velocity ${\displaystyle {\boldsymbol {U}}_{p}(t)={\text{d}}{\boldsymbol {X}}_{p}/{\text{d}}t}$ in a fluid of density ${\displaystyle \rho _{f}}$, dynamic viscosity ${\displaystyle \mu }$ and Eulerian velocity field ${\displaystyle {\boldsymbol {u}}_{f}({\boldsymbol {x}},t)}$. The fluid velocity field surrounding the particle consists of the undisturbed, local Eulerian velocity field ${\displaystyle {\boldsymbol {u}}_{f}}$ plus a disturbance field – created by the presence of the particle and its motion with respect to the undisturbed field ${\displaystyle {\boldsymbol {u}}_{f}.}$ 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, ${\displaystyle {\boldsymbol {U}}_{f}(t)={\boldsymbol {u}}_{f}({\boldsymbol {X}}_{p}(t),t)}$. 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:

${\displaystyle {\begin{aligned}{\frac {\pi }{6}}\rho _{p}d_{p}^{3}{\frac {{\text{d}}{\boldsymbol {U}}_{p}}{{\text{d}}t}}&=\underbrace {3\pi \mu d_{p}\left({\boldsymbol {U}}_{f}-{\boldsymbol {U}}_{p}\right)} _{\text{term 1}}-\underbrace {{\frac {\pi }{6}}d_{p}^{3}{\boldsymbol {\nabla }}p} _{\text{term 2}}+\underbrace {{\frac {\pi }{12}}\rho _{f}d_{p}^{3}\,{\frac {\text{d}}{{\text{d}}t}}\left({\boldsymbol {U}}_{f}-{\boldsymbol {U}}_{p}\right)} _{\text{term 3}}\\&+\underbrace {{\frac {3}{2}}d_{p}^{2}{\sqrt {\pi \rho _{f}\mu }}\int _{t_{_{0}}}^{t}{\frac {1}{\sqrt {t-\tau }}}\,{\frac {\text{d}}{{\text{d}}\tau }}\left({\boldsymbol {U}}_{f}-{\boldsymbol {U}}_{p}\right)\,{\text{d}}\tau } _{\text{term 4}}+\underbrace {\sum _{k}{\boldsymbol {F}}_{k}} _{\text{term 5}}.\end{aligned}}}$

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:^[1]

1. Stokes' drag,
2. Froude–Krylov force due to the pressure gradient in the undisturbed flow, with ${\displaystyle {\boldsymbol {\nabla }}}$ the gradient operator and ${\displaystyle p({\boldsymbol {x}},t)}$ the undisturbed pressure field,
3. added mass,
4. Basset force and
5. other forces acting on the particle, such as gravity, etc.

The particle Reynolds number ${\displaystyle R_{e}:}$

${\displaystyle R_{e}={\frac {\max \left\{\left|{\boldsymbol {U}}_{p}-{\boldsymbol {U}}_{f}\right|\right\}\,d_{p}}{\mu /\rho _{f}}}}$

has to be less than unity, ${\displaystyle R_{e}<1}$, for the BBO equation to give an adequate representation of the forces on the particle.^[2]

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

${\displaystyle -{\boldsymbol {\nabla }}p=\rho _{f}{\frac {{\text{D}}{\boldsymbol {u}}_{f}}{{\text{D}}t}}-\mu \nabla ^{2}{\boldsymbol {u}}_{f},}$

with ${\displaystyle {\text{D}}{\boldsymbol {u}}_{f}/{\text{D}}t}$ the material derivative of ${\displaystyle {\boldsymbol {u}}_{f}.}$ Note that in the Navier–Stokes equations ${\displaystyle {\boldsymbol {u}}_{f}({\boldsymbol {x}},t)}$ is the fluid velocity field, while, as indicated above, in the BBO equation ${\displaystyle {\boldsymbol {U}}_{f}}$ is the velocity of the undisturbed flow as seen by an observer moving with the particle. Thus, even in steady Eulerian flow ${\displaystyle {\boldsymbol {u}}_{f}}$ depends on time if the Eulerian field is non-uniform.

Notes[]

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