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-{{merge to|Oseen equations|date=October 2012}}
+#REDIRECT [[Oseen equations]] {{R from merge}}
-{{Multiple issues|
-{{refimprove|date=April 2011}}
-{{expert-subject|Physics|date=April 2011}}
-}}
-{{Use dmy dates|date=August 2012}}
-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 [[logarithm]]ically 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]]