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In a body submerged in a [[fluid]], unsteady forces due to [[acceleration]] of that body with respect to the fluid, can be divided into two parts: the [[virtual mass effect]] and the '''Basset force'''.
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The Basset force term describes the force due to the lagging [[boundary layer]] development with changing relative velocity (acceleration) of bodies moving through a fluid.<ref>C. Crowe et al., Multiphase flows with droplets and particles, CRC Press, 1998, ISBN 0-8493-9469-4, p. 81</ref>
The Basset term accounts for [[viscosity|viscous]] effects and addresses the temporal delay in boundary layer development as the relative velocity changes with time. It is also known as the "history" term.
The Basset force is difficult to implement and is commonly neglected for practical reasons; however, it can be substantially large when the body is accelerated at a high rate.<ref>R.W. Johnson, The handbook of fluid dynamics, CRC Press, 1998, ISBN 0-8493-2509-9, pp.&nbsp;18–3</ref>
 
This force in an accelerating [[Stokes flow]] has been proposed by [[Joseph Valentin Boussinesq]] in 1885 and [[Alfred Barnard Basset]] in 1888. Consequently, it is also referred to as the '''Boussinesq–Basset force'''.<ref name=CAS2004>{{citation | author1 = F. Candelier | author2 = J. R. Angilella | author3 = M. Souhar | title = On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex | year = 2004 | journal = Physics of Fluids | volume = 16 | issue = 5 | pages = 1765–1776 | doi = 10.1063/1.1689970 |bibcode = 2004PhFl...16.1765C }}</ref><ref name=Michaelides2003>{{ citation | author = E. E. Michaelides | title = Hydrodynamic force and heat/mass transfer from particles, bubbles, and drops—The Freeman Scholar Lecture | year = 2003 | journal = Journal of Fluids Engineering | volume = 125 | issue = 2 | pages = 209–238 | doi = 10.1115/1.1537258 }}</ref>
 
==Acceleration of a flat plate==
 
Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to ''u<sub>0</sub>''—in the direction of the plate–fluid interface plane.
 
The equation of motion for the fluid—[[Stokes flow]] at low [[Reynolds number]]—is
 
:<math>\frac{\partial u}{\partial t}=\nu_c\,\frac{\partial^2u}{\partial y^2},</math>
 
where ''u''(''y'',''t'') is the velocity of the fluid, at some time ''t'', parallel to the plate, at a distance ''y'' from the plate, and ''ν<sub>c</sub>'' is the [[kinematic viscosity]] of the fluid (c~continuous phase).
The solution to this equation is,<ref>{{citation | author=F. M. White | title=Viscous fluid flow | at=Chapter 3 | origyear=2006 | publisher=McGraw Hill | publication-place=New York | year=2006 }}</ref>
 
:<math>u=u_0 - u_0\, \operatorname{erf}\left(\frac{y}{2\sqrt{\nu_ct}}\right) = u_0 \operatorname{erfc}\left(\frac{y}{2\sqrt{\nu_ct}}\right),</math>
 
where erf and erfc denote the [[error function]] and the [[complementary error function]], respectively.
 
Assuming that an acceleration of the plate can be broken up into a series of such step changes in the velocity, it can be shown that the cumulative effect on the shear stress on the plate is
 
:<math>\tau=\sqrt{\frac{\rho_c\mu_c}{\pi}}\int\limits_0^t\frac{\frac{\partial u_p}{\partial t'}}{\sqrt{t-t'}} \, dt', </math>
 
where ''u<sub>p</sub>(t)'' is the velocity of the plate, ''ρ<sub>c</sub>'' is the [[mass density]] of the fluid, and ''μ<sub>c</sub>'' is the [[viscosity]] of the fluid.
 
==Acceleration of a spherical particle==
[[Joseph Valentin Boussinesq|Boussinesq]] (1885) and Basset (1888) found that the force '''''F''''' on an accelerating spherical particle in a viscous fluid is<ref name=CAS2004/><ref name=Michaelides2003/><ref>{{citation | author=J. V. Boussinesq |title=Sur la résistance qu’oppose un fluide indéfini au repos, sans pesanteur, au mouvement varié d’une sphère solide qu’il mouille sur toute sa surface, quand les vitesses restent bien continues et assez faibles pour que leurs carrés et produits soient négligeables | journal=Comptes Rendu de l'Academie des Sciences | volume=100 | pages=935–937 | year=1885 }}</ref><ref>{{citation | author=A. B. Basset | title=Treatise on hydrodynamics | at=Chapter 22 | volume=2 | origyear=1888 | publisher=Deighton, Bell and Co. | publication-place=Cambridge | year=1961 }}</ref>
 
:<math>\mathbf{F}=\frac{3}{2}D^2\sqrt{\pi\rho_c\mu_c}\int\limits_0^t\frac{\frac{D\mathbf{u}}{Dt'}-\frac{\partial\mathbf{v}}{\partial t'}}{\sqrt{t-t'}}dt',</math>
 
where ''D'' is the particle diameter, {{frac|D|Dt'}} is the [[material derivative]], and '''''u''''' and '''''v''''' are the fluid and particle velocity vectors, respectively.
 
==See also==
 
* [[Basset–Boussinesq–Oseen equation]]
* [[Stokes boundary layer]]
 
==References==
 
<references/>
 
{{DEFAULTSORT:Basset Force}}
[[Category:Fluid dynamics]]

Revision as of 15:01, 1 March 2014

A collection of suggestions on how to start traveling smarter and safer makes the best beginning point for a beginner to commence planning their own trips a bit simpler. Below is just such a collection that will hopefully help the eager novice into sooner or later becoming a pro when it comes to traveling.

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