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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 number]]s. The equation is named after [[Joseph Valentin Boussinesq]], [[Alfred Barnard Basset]] and [[Carl Wilhelm Oseen]].
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| ==Formulation==
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| One formulation of the BBO equation is the one given by {{harvtxt|Zhu|Fan|1998|pp=18–27}}, for a spherical particle of diameter <math>d_p</math>, position <math>\boldsymbol{x}=\boldsymbol{X}_p(t)</math> and mean [[density]] <math>\rho_p</math> moving with particle velocity <math>\boldsymbol{U}_p=\text{d} \boldsymbol{X}_p / \text{d}t</math> – in a fluid of density <math>\rho_f</math>, [[dynamic viscosity]] <math>\mu</math> and with ambient (undisturbed local) [[flow velocity]] <math>\boldsymbol{U}_f:</math><ref>With {{harvtxt|Zhu|Fan|1998|pp=18–27}} referring to {{harvtxt|Soo|1990}}</ref>
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| :<math>
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| \begin{align}
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| \frac{\pi}{6} \rho_p d_p^3 \frac{\text{d} \boldsymbol{U}_p}{\text{d} t}
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| &= \underbrace{3 \pi \mu d_p \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 1}}
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| - \underbrace{\frac{\pi}{6} d_p^3 \boldsymbol{\nabla} p}_{\text{term 2}}
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| + \underbrace{\frac{\pi}{12} \rho_f d_p^3\,
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| \frac{\text{d}}{\text{d} t} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 3}}
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| \\ &
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| + \underbrace{\frac{3}{2} d_p^2 \sqrt{\pi \rho_f \mu}
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| \int_{t_{_0}}^t \frac{1}{\sqrt{t-\tau}}\, \frac{\text{d}}{\text{d} \tau} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)\,
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| \text{d} \tau}_{\text{term 4}}
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| + \underbrace{\sum_k \boldsymbol{F}_k}_{\text{term 5}} .
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| \end{align}
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| </math>
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| This is [[Newton's second law]], with in the [[left-hand side]] the particle's [[time derivative|rate of change]] of [[linear momentum]], and on the [[right-hand side]] the [[force]]s acting on the particle. The terms on the right-hand side are respectively due to the:<ref>{{harvtxt|Zhu|Fan|1998|pp=18–27}}</ref>
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| # [[Stokes' drag]],
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| # [[pressure gradient]], with <math>\boldsymbol{\nabla}</math> the [[gradient]] operator,
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| # [[added mass]],
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| # [[Basset force]] and
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| # other forces on the particle, such as due to [[gravity]], etc.
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| The particle Reynolds number <math>R_e:</math>
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| :<math>R_e = \frac{\max\left\{ \left| \boldsymbol{U}_p - \boldsymbol{U}_f \right| \right\}\, d_p}{\mu/\rho_f}</math>
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| has to be small, <math>R_e<1</math>, for the BBO equation to give an adequate representation of the forces on the particle.<ref>{{Cite book | publisher = Springer | isbn = 9780792333760 | last = Green | first = Sheldon I. | title = Fluid Vortices | year = 1995 | page = 831 }}</ref>
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| Also {{harvtxt|Zhu|Fan|1998|pp=18–27}} suggest to estimate the pressure gradient from the [[Navier–Stokes equations]]: | |
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| :<math> | |
| -\boldsymbol{\nabla} p
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| = \rho_f \frac{\text{D} \boldsymbol{u}_f}{\text{D} t}
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| - \mu \boldsymbol{\nabla}\!\cdot\!\boldsymbol{\nabla} \boldsymbol{u}_f,
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| </math>
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| with <math>\text{D} \boldsymbol{u}_f / \text{D} t</math> the [[material derivative]] of <math>\boldsymbol{u}_f.</math> Note that in the Navier–Stokes equations <math>\boldsymbol{u}_f(\boldsymbol{x},t)</math> is the fluid velocity field, while in the BBO equation <math>\boldsymbol{U}_f</math> is the undisturbed fluid velocity at the particle position: <math>\boldsymbol{U}_f(t)=\boldsymbol{u}_f(\boldsymbol{X}_p(t),t).</math>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| *{{Cite book
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| | publisher = Springer
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| | isbn = 9783540646129
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| | editor-last = Johnson
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| | editor-first = Richard W.
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| | title = The Handbook of Fluid Dynamics
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| | year = 1998
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| | chapter = Chapter 18 – Multiphase flow: Gas/Solid
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| | last1 = Zhu
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| | first1 = Chao
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| | last2 = Fan
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| | first2 = Liang-Shi
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| }}
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| *{{Cite book
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| | publisher = Ashgate Publishing
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| | isbn = 9780566090332
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| | last = Soo
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| | first = Shao L.
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| | title = Multiphase Fluid Dynamics
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| | year = 1990
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| }}
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| {{refend}}
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| {{DEFAULTSORT:Basset-Boussinesq-Oseen equation}}
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| [[Category:Equations of fluid dynamics]]
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I'm Lanny (30) from Oberdorfl, Austria.
I'm learning Swedish literature at a local high school and I'm just about to graduate.
I have a part time job in a the office.
Also visit my homepage :: cheapdomains (go to this site)