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In [[fluid dynamics]], '''Faxén's laws''' relate a sphere's velocity <math>\mathbf{U}</math> and angular velocity <math>\mathbf{\Omega}</math> to the forces, torque, stresslet and flow it experiences under low Reynolds number (creeping flow) conditions.
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==First law==
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Faxen's first law was introduced in 1922 by Swedish physicist [[Hilding Faxén]], who at the time was active at [[Uppsala University]], and is given by<ref name="Chen">http://dx.doi.org/10.1006/jcis.1999.6552 Chen, Shing Bor, and Xiangnan Ye. "Faxen's laws of a composite sphere under creeping flow conditions." ''Journal of colloid and interface science'' 221.1 (2000): 50-57</ref><ref name="Durlofsky">http://dx.doi.org/10.1017/S002211208700171X Durlofsky, Louis, John F. Brady, and Georges Bossis. "Dynamic simulation of hydrodynamically interacting particles." Journal of fluid mechanics 180.1 (1987): 21-49, equations (2.15a, b, c). Note sign change.</ref>
:<math>
\mathbf{F} = 6 \pi \mu a \left[ \left( 1 + \frac{a^2}{6} \nabla^2\right) \mathbf{u}' - (\mathbf{U} - \mathbf{u}^\infty) \right],
</math>
where
* <math>\mathbf{F}</math> is the force exerted by the fluid on the sphere
* <math>\mu</math> is the Newtonian viscosity of the solvent in which the sphere is placed
* <math>a</math> is the sphere's radius
* <math>\mathbf{U}</math> is the (translational) velocity of the sphere
* <math>\mathbf{u}'</math> is the disturbance velocity caused by the other spheres in suspension (not by the background impressed flow), evaluated at the sphere centre
* <math>\mathbf{u}^\infty</math> is the background impressed flow, evaluated at the sphere centre (set to zero in some references).
 
It can also be written in the form
:<math>
\mathbf{U} - \mathbf{u}^\infty = \mathbf{u}' + b_0 \mathbf{F} + \frac{a^2}{6} \nabla^2 \mathbf{u}',
</math>
where <math>b_0 = -\frac{1}{6 \pi \mu a}</math> is the mobility.
 
In the case that the pressure gradient is small compared with the length scale of the sphere's diameter, and when there is no external force, the last two terms of this form may be neglected.  In this case the external fluid flow simply advects the sphere.
 
==Second law==
Faxen's second law is given by<ref name="Chen" /><ref name="Durlofsky" />
:<math>
\mathbf{T} = 8 \pi \mu a^3 \left[ \frac{1}{2} \left( \boldsymbol{\nabla} \times \mathbf{u}' \right) - (\mathbf{\Omega} - \mathbf{\Omega}^\infty) \right],
</math>
where
*<math>\mathbf{T}</math> is the torque exerted by the fluid on the sphere
*<math>\mathbf{\Omega}</math> is the angular velocity of the sphere
*<math>\mathbf{\Omega}^\infty</math> is the angular velocity of the background flow, evaluated at the sphere centre (set to zero in some references).
 
=='Third law'==
Batchelor and Green<ref>http://dx.doi.org/10.1017/S0022112072002435 Batchelor, G. K., and J. T. Green. "The hydrodynamic interaction of two small freely-moving spheres in a linear flow field." ''J. Fluid Mech'' 56.2 (1972): 375-400</ref> derived an equation for the stresslet, given by<ref name="Chen" /><ref name="Durlofsky" />
:<math>
\boldsymbol{\mathsf{S}} = \frac{10}{3} \pi \mu a^3 \left[ -2\boldsymbol{\mathsf{E}}^\infty + \left(1 + \frac{1}{10} a^2 \nabla^2\right) \left(\boldsymbol{\nabla} \mathbf{u}' + (\boldsymbol{\nabla} \mathbf{u}')^\mathrm{T}\right)\right],
</math>
where
*<math>\boldsymbol{\mathsf{S}}</math> is the stresslet (symmetric part of the first moment of force) exerted by the fluid on the sphere,
*<math>\boldsymbol{\nabla}\mathbf{u}'</math> is the velocity gradient tensor; <math>{}^\mathrm{T}</math> represents transpose; and so <math>\frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{u}' + (\boldsymbol{\nabla} \mathbf{u}')^\mathrm{T}\right]</math> is the rate of strain, or deformation, tensor.
*<math>\boldsymbol{\mathsf{E}}^\infty = \frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{u}^\infty + (\boldsymbol{\nabla} \mathbf{u}^\infty)^\mathrm{T}\right]</math> is the rate of strain of the background flow, evaluated at the sphere centre (set to zero in some references).
 
Note there is no rate of strain on the sphere (no <math>\boldsymbol{\mathsf{E}}</math>) since the spheres are assumed to be rigid.
 
Faxén's law is a correction to [[Stokes' law]] for the friction on spherical objects in a [[viscosity|viscous]] [[fluid]], valid where the object moves close to a wall of the container.<ref>[http://www2.bioch.ox.ac.uk/~oubsu/ebjknight/gloss.html Single molecule measurements and biological motors - Glossary], accessed on May 12, 2009</ref>
 
== See also ==
* [[Immersed boundary method]]
 
== Notes ==
{{Reflist}}
 
== References ==
*{{citation | doi=10.1002/andp.19223731003 | first=H. | last=Faxén |title=Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Flüssigkeit, die zwischen zwei parallelen ebenen Wänden eingeschlossen ist | journal=Annalen der Physik | year=1922  | volume=373 | issue=10 | pages=89–119 |bibcode = 1922AnP...373...89F }}
*{{citation | first1=J. | last1=Happel | first2=H. | last2=Brenner | title=Low Reynolds Number Hydrodynamics | publisher=Kluwer | location=Dordrecht | year=1991}}
 
{{DEFAULTSORT:Faxen's law}}
[[Category:Equations of fluid dynamics]]

Latest revision as of 20:25, 5 November 2014

To go nude or never to go nude? That is the issue that many vacationers wind up asking themselves. It"s likely that you"ll have to choose, if you are thinking about spending your summer vacation at an international beach. Several offshore beaches are classified as being nude beaches.

Bare beaches are perfect for adults who are touring with a crowd. Whether you are vacationing overseas with your pals or you"re vacationing with your spouse or spouse, many people find bare shores desirable. In ways, many couples find nude beaches to be exotic; ergo, making a intimate setting. Get more on How To Get Into Modeling Even If You Dont Reside In A Huge City | LG LED by visiting our wonderful encyclopedia.

In america, nudity is really a controversial topic. If seeing nude men and women frolicking on the beach does not bother you, you might want to consider selecting a nude beach as the next summer holiday destination. When it comes to choosing the nude beach to go to, you have numerous possibilities. Below can be an overview of three common nudist shores that as your next summer vacation destination you may wish to select.

The Montalivet Beach is found in France. It"s called one of the most popular nude beaches in the world. Italy is frequently credited for the growth and increased popularity of nude beaches. The beach is found on the Atlantic coast and it includes several over night accommodations, including bungalow leases and campsites. The environment at Montalivet Beach is usually reported to be safe for individuals of ages.

Samurai Beach, positioned in Australia is yet another, one of many, top rated nude beaches. Multiple resorts along the seashore, including Bardots Clothing Optional Resort, let nudity. Actually, nudity at the Samurai Beach is really common that it has get to be the home of the Nude Olympics, which are placed on an annual basis.

Greece is home to several nude beaches, such as the common Red Beach. As well as visiting on the beach nude, Greece is well-know for its history. There are a amount of famous landmarks over the Red Beach. Red Beach is fabled for its insufficient limits on nudity. The only restriction, at this popular nude beach, is that clothing must be worn in the dining area. A large number of nude beaches restrict when and where you can be in the nude, but freedom is offered by Red Beach to any or all.

As previously mentioned, nude beaches are perfect for adult crowds. If you"re visiting along with your family and other young kids, you might want to reexamine your decision to vacation at a nude beach. Nudity may produce some confusion, particularly with younger kids, while numerous nude beaches are classified to be family pleasant. If you are still enthusiastic about visiting a beach overseas, below is a synopsis of three common beaches, where clothing is required.

Located in Thailand may be the area of Phi Phi Leh. Discover more on this related use with - Browse this URL: webcam modeling. The most well-known beach could be the region may be the Maya Beach. The area was broken in the tsunami a few years back, but, for probably the most part, a rebirth have been seen by the area. As well as traditional beach activities, new facilities, services, and activities have already been included for visitors.

Cannes is located in the French Riviera. Cannes is most famous for its consistent star appointments. While Cannes is frequently seen as a costly holiday destination, perhaps you are able take pleasure in the beaches and the scenery on a budget.

The Bondi Beach, situated in Australia, is a popular destination. Shores line the coast of the large island. Due to its recognition, Bondi Beach is frequently overcrowded. Despite significant crowds, visitors frequently enjoy the numerous on and off shore activities bought at Bondi Beach.

Along with the aforementioned beaches, there are a many additional overseas trips. Whether you decide to vacation at a nude beach or a beach where clothing is necessary, you"re sure to obtain the great summer vacation destination.

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Word Count 654.

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