en>Rjwilmsi: Filling in journal cites, using AWB (10395)

2014-08-21T19:07:36Z

Filling in journal cites, using AWB (10395)

← Older revision		Revision as of 21:07, 21 August 2014
Line 1:		Line 1:
	~~{{merge~~ to~~\|Oseen equations\|date=October 2012}}~~		I merely could hardly disappear completely your website prior to suggesting which i basically treasured the usual information anyone offer to your . uniform manufacturersite visitors? Is likely to be yet again frequently in an effort to check out cross-check innovative posts
	~~{{Multiple issues\|refimprove = April 2011\|expert = Physics\|date = April 2011}}~~

	~~{{Use dmy dates\|date=August 2012}}~~		[http://tiotopuniform.com Patella]

	~~In 1910~~ [[Carl Wilhelm Oseen]] proposed what became known as '''Oseen's approximation''', to treat problems of [[fluid dynamics]] in which a flow field involves a small disturbance of a constant [[Volumetric flow rate\|mean flow]], as in a stream of liquid. His work was based on the experiments of [[George Gabriel Stokes\|G.G. Stokes]], who had studied a sphere of radius "<math>a</math>" falling in a fluid of [[viscosity]] <math>\mu\,</math>. Oseen developed a correction term, which included [[inertia]]l factors, for the velocity used in Stokes' calculations, to solve the problem. His approximation lead to an improvement to Stokes' calculations.

	~~==Importance==~~

	The method and formulation for analysis of flow at a very low [[Reynolds number]] is important. The slow motion of small particles in a fluid is common in [[bio-engineering]]. Oseen's drag formulation can be used in connection with flow of fluids under various special conditions, such as: containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens.<ref name="Fung">{{harvtxt\|Fung\|1997}}</ref> The fluid does not even have to be a liquid, and the particles do not need to be solid. It can be used in a number of applications, such as smog formation and [[atomizer nozzle\|atomization]] of liquids.

	~~==Bio-engineering application==~~

	Blood flow in small vessels, such as [[capillary\|capillaries]], is characterized by small [[Reynolds number\|Reynolds]] and [[Womersley number]]s. A vessel of diameter of {{nowrap\|10 µm}} with a flow of {{nowrap\|1 milimetre/second}}, viscosity of {{nowrap\|0.02 poise}} for blood, [[density]] of {{nowrap\|1 g/cm<sup>3</sup>}} and a heart rate of {{nowrap\|2 Hz}}, will have a Reynolds number of 0.005 and a Womersley number of 0.0126. At these small Reynolds and Womersley numbers, the viscous effects of the fluid become predominant. Understanding the movement of these particles is essential for drug delivery and studying [[metastasis]] movements of cancers.

	~~==Calculations==~~

	Oseen considered the sphere to be stationary and the fluid to be flowing with a [[velocity]] (<math>U</math>) at an infinite distance from the sphere. Inertial terms were neglected in Stokes’ calculations.<ref name=Fung/> It is a limiting solution when the Reynolds number tends to zero. When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed. Oseen substituted the following velocity values into the [[Navier-Stokes equations]].

	~~:<math>v_1 = U + v_1^', \qquad v_2 = v_2', \qquad v_3 = v_3'.</math>~~

	~~Inserting these into the Navier-Stokes equations and neglecting the quadratic terms in the primed quantities leads to the derivation of Oseen’s approximation:~~

	~~:<math>U{\partial v_1'\over\partial x_1} = -{1 \over \rho}{\partial p\over\partial x_1} + \nu\nabla^2 v_i' \qquad \left({i=1,2,3}\right).</math>~~

	~~When Stokes' solution was solved on the basis of Oseen's approximation, it showed that the resultant [[hydrodynamic force]] (drag) is given by~~

	~~:<math>F= 6\pi\,\mu\,a U\left(1+{3\over 8} N_R\right),</math>~~

	~~: where:~~
	~~:* <math>N_R</math> is the Reynolds number based on radius of the sphere~~
	~~:* <math>F</math> is the hydrodynamic force~~
	~~:* <math>U</math> is the flow~~
	~~:* <math>a</math> is the radius of the sphere~~
	~~:* <math>\mu\,</math> is the fluid viscosity~~

	~~The force from Oseen's equation differs from that of Stokes by a factor of~~
	~~:<math>1+\left({3\over 8}\right) N_R.</math>~~

	~~==Error in Stokes' solution==~~

	~~The Navier Stokes equations read:<ref>{{harvtxt\|Mei\|2011}}</ref>~~

	~~:<math>\triangledown v' ~ = 0</math>~~

	~~:<math>v \triangledown v' ~ = - \triangledown p + \nu \triangledown ^2 v',</math>~~

	~~but when the velocity ﬁeld is:~~

	~~:<math>v_y = U \cos\theta\left({1 + {a^3 \over 2r^3} - {3a \over 2r}}\right) </math>~~

	~~:<math>v_z = - U \sin\theta \left({1 - {a^3 \over 4r^3} - {3a \over 4r}}\right).</math>~~

	~~In the far ﬁeld <math>{r\over a}</math> >> 1, the viscous stress is dominated by the last term. That is:~~

	~~:<math>\triangledown^2 v' = O\left({a^3\over r^3}\right).</math>~~

	~~The inertia term is dominated by the term:~~

	~~:<math>U{\partial v'\over\partial z_1} \sim O\left({a^2 \over r^2}\right).</math>~~

	~~The error is then given by the ratio:~~

	~~:<math>U {{\partial v'\over \partial z_1} \over {\nu \triangledown^2 v'}} = O \left({r \over a} \right).</math>~~

	~~This becomes unbounded for <math>{r \over a}</math> >> 1, therefore the inertia cannot be ignored in the far ﬁeld.~~
	~~By taking the curl, Stokes equation gives~~
	~~<math>\triangledown ^2 \zeta\,= 0.</math>~~
	Since the body is a source of [[vorticity]],<math>\zeta\,</math> would become unbounded [[logarithmic]]ally for large <math>{r \over a}.</math> This is certainly unphysical and is known as [[Stokes' paradox]].

	~~==Modifications to Oseen's approximation==~~

	One may question, however, whether the correction term was chosen by chance, because in a frame of reference moving with the sphere, the fluid near the sphere is almost at rest, and in that region inertial force is negligible and Stokes' equation is well justified.<ref name=Fung/> Far away from the sphere, the flow velocity approaches ''U'' and Oseen's approximation is more accurate.<ref name=Fung/> But Oseen's equation was obtained applying the equation for the entire flow field. This question was answered by Proudman and Pearson in 1957,<ref>{{harvtxt\|Proudman\|Pearson\|1957}}</ref> who solved the Navier-Stokes equations and gave an improved Stokes' solution in the neighborhood of the sphere and an improved Oseen’s solution at infinity, and matched the two solutions in a supposed common region of their validity. They obtained:

	~~:<math>F= 6\pi\,\mu\,a U\left( 1 + {3 \over 8} N_R + {9 \over 40} N_R^2 \ln N_R + \mathcal{O}( N_R^2) \right).</math>~~

	~~==References==~~
	~~; Notes~~
	~~{{reflist}}~~
	~~; Sources~~
	* {{citation \| last=Fung \|first=Yuan-cheng \|title=Biomechanics: Circulation \|edition=2nd \|location=New York, NY \|publisher=Springer-Verlag \|year=1997 }}
	* {{citation \|first=C.C. \|last=Mei \|author-link=Chiang C. Mei \|work=Advanced Environmental Fluid Mechanics \|publisher=Web.Mit.edu \|date=4 April 2011 \|url=http://~~web.mit.edu/fluids-modules/www/low_speed_flows/2-6oseen~~.~~pdf \|format=pdf \|title=Oseen's improvement for slow flow past a body \|accessdate=2013-02-28 }}~~
	* {{Citation
	~~\| doi = 10.1017/S0022112057000105~~
	~~\| volume = 2~~
	~~\| issue = 3~~
	~~\| pages = 237–262~~
	~~\| last1 = Proudman~~
	~~\| first1 = I.~~
	~~\| first2 = J.R.A.~~
	~~\| last2 = Pearson~~
	~~\| title = Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder~~
	~~\| journal = Journal of Fluid Mechanics~~
	~~\| year = 1957~~
	~~\|bibcode = 1957JFM.....2..237P }}~~

	~~[[Category:Fluid mechanics]~~]

en>Double sharp at 17:09, 8 August 2013

2013-08-08T17:09:21Z

Show changes

en>Double sharp: /* Oxides */ ?!

2012-08-24T09:32:07Z

Oxides: ?!

New page

It involves expertise and knowledge of various tools and technologies used for creating websites. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. Several claim that Wordpress just isn't an preferred tool to utilise when developing a professional site. You can add new functionalities and edit the existing ones to suit your changing business needs. For a Wordpress website, you don't need a powerful web hosting account to host your site. The least difficult and very best way to do this is by acquiring a Word - Press site. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. all the necessary planning and steps of conversion is carried out in this phase, such as splitting, slicing, CSS code, adding images, header footer etc. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. Additionally Word - Press add a default theme named Twenty Fourteen. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. The company gains commission from the customers' payment. This includes enriching the content with proper key words, tactfully defining the tags and URL. More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. If you enjoyed this information and you would such as to obtain additional facts relating to [http://GET7.pw/backup_plugin_229245 wordpress backup] kindly browse through the website. Automated deal feed integration option to populate your blog with relevant deals. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.

Compounds of berkelium - Revision history

en>Rjwilmsi: Filling in journal cites, using AWB (10395)

en>Double sharp at 17:09, 8 August 2013

en>Double sharp: /* Oxides */ ?!