MathMagic - Revision history

en>Chivalry: Fix link to iOS page.

2015-01-02T22:32:43Z

Fix link to iOS page.

← Older revision		Revision as of 00:32, 3 January 2015
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	~~{{Cleanup\|date=April 2011}}~~		Hello! My name is Miriam. I smile that I could join to the entire globe. I live in Great Britain, in the south region. I dream to go to the various countries, to obtain familiarized with interesting people.


	In [[fluid mechanics]], '''dynamic similarity''' refers to the phenomenon that when there are two geometrically similar vessels (same shape, different sizes) with the same boundary conditions (ex. No-slip, center-line velocity) and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.<ref>Jones, Robert T. "Blood Flow," ''Annual Review of Fluid Mechanics'', 1(1969)223:244.</ref>

	~~==Derivation==~~
	~~[[Reynolds number]] and [[Womersley number]] are the only two physical parameters necessary to solve an incompressible fluid flow problem. Reynolds Number is given by:~~
	~~<math>N_R = {VL{\rho}\over {\mu}}\,\!</math>~~

	~~The terms of the equation itself represent the following:~~
	~~:<math>N_R = {\text{Convective Inertial Force}\over \text{Shear Force}}\,\~~!~~</math>.~~
	~~When Reynolds Number is large, it shows that the flow is dominated by convective inertial effects; When Reynolds Number is small, it shows that the flow~~ is ~~dominated by shear effects~~.
	~~Womersley Number is given by:~~
	~~:<math>N_W = L\sqrt{{\omega}{\rho}\over {\mu}}=\sqrt{N_S}\,\!</math>,~~
	~~which is simply the square-root of Stokes Number, the terms of the equation itself represents the following:~~
	~~:<math style="vertical-align:-100%;"> N_W = {\text{Transient Inertial Force}\over \text{Shear Force}}\,\!</math>.~~
	~~When Womersley Number is large(around 10 or greater), it shows that the flow is dominated by oscillatory inertial forces and~~ that ~~the velocity profile is flat. When the Womersley parameter is low, viscous forces tend~~ to
	~~dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with~~ the ~~driving pressure gradient.<ref>Ku, David N~~. ~~"Blood Flow~~ in ~~Arteries," ''Annual Review of Fluid Mechanics'', 1(1969)223:44.</ref>~~

	~~Starting with [[Navier–Stokes equation]] for Cartesian flow:~~

	: <math>{\rho} \left (\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v\frac{{\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right )= {\rho}g-\frac{{\partial}P}{{\partial}x}+{\mu}\left( \frac{{\partial^2}u}{{\partial}x^2}+\frac{{\partial^2}v}{{\partial}y^2}+\frac{{\partial^2}w}{{\partial}z^2}\right)\,\!</math>.

	~~The terms of the equation itself represent the following:~~

	: <math>\text{transient inertial forces + convective inertial forces}=\text{gravitational force + Pressure force + viscous forces}\,\!</math> <ref>Fung, Yuan-cheng. "Biomechanics: Circulation," ''Dynamic Similarity'', "New York: Springer", 2(2008)130:134.</ref>

	~~Ignoring gravitational forces and dividing the equation by density (<math>\rho</math>) yields:~~

	: <math>\left (\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v\frac{{\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right )= -\frac{1}{\rho}\frac{{\partial}P}{{\partial}x}+{\nu}\left( \frac{{\partial^2}u}{{\partial}x^2}+\frac{{\partial^2}v}{{\partial}y^2}+\frac{{\partial^2}w}{{\partial}z^2}\right)\,\!</math>,

	where <math>\nu={\mu}/{\rho}</math> is the kinematic viscosity. Since both Reynolds and Womersley numbers are dimensionless, Navier-Stokes must be represented as a dimensionless expression as well. Choosing <math>V</math>, <math>\omega</math>, and <math>L</math> as a characteristic velocity, frequency, and length respectively yields dimensionless variables:
	~~Dimensionless Length Term (same for y' and z'):<math>x' = {x / L}\~~,~~\!</math>,~~
	~~Dimensionless Velocity Term (same for v' and w'): <math>u' = {u / V}\,\!</math>,~~
	~~Dimensionless Pressure Term: <math>P' = {P / {{\rho}V^2}}\,\!</math>,~~
	~~Dimensionless Time Term: <math>t' = t{\omega}\,\!</math>.~~
	~~Dividing the Navier-Stokes equation by <math>\frac{V^2}{L}\,\!</math> (Convective Inertial Force term) gives:~~

	: <math>\frac{{N_W}^2}{N_R} \left (\frac{{\partial}u'}{{\partial}t'}\right)+\left (u'\frac{{\partial}u'}{{\partial}x'}+v'\frac{{\partial}u'}{{\partial}y'}+w'\frac{{\partial}u'}{{\partial}z'}\right )= -\frac{{\partial}P'}{{\partial}x'}+\frac{1}{N_R}\left( \frac{{\partial^2}u'}{{\partial}x'^2}+\frac{{\partial^2}v'}{{\partial}y'^2}+\frac{{\partial^2}w'}{{\partial}z'^2}\right)\,\!</math>,

	With the addition of the dimensionless continuity equation (seen below) in any incompressible fluid flow problem the Reynolds and Womersley Numbers are the only two physical parameters that are in the two equations
	~~:<math>\frac{{\partial}u'}{{\partial}x'}+\frac{{\partial}v'}{{\partial}y'}+\frac{{\partial}w'}{{\partial}z'}=0 \,\!</math>, <ref>van de Vosse, Frans M. "Pulse Wave Propagation~~ in the ~~Arterial Tree~~.~~," ''Annual Review of Fluid Mechanics'', 43(2011)467:499.</ref>~~

	~~==Boundary layer thickness==~~
	~~Reynolds and Womersley Numbers are also used~~ to ~~calculate the thicknesses of the [[boundary layers]] that can form from the fluid flow’s viscous effects. Reynolds number is used~~ to calculate the convective inertial boundary layer thickness that can form, and Womersley number is used to calculate the transient inertial boundary thickness that can form. From the Womersley Number it can be shown that the transient inertia force is represented by <math>{\rho}{\omega}V\,\!</math>, and from the last term in the ~~non-modified Navier-Stokes equation that viscous force is represented by <math>{\mu}{V}\over {{\delta}_1}^2\~~,~~\!</math> (subscript one indicates that the boundary layer thickness is that of the transient boundary layer). Setting the two forces equal~~ to ~~each other yields:~~
	~~<math>{\rho}{\omega}V={{\mu}{V}\over {{\delta}_1}^2}\,\!</math>~~
	~~Solving for <math>{\delta}_1\,\!</math> yields:~~
	~~<math>{\delta}_1=\sqrt{{\mu}\over {\rho}{\omega}}\,\!</math>~~
	~~Adding a characteristic length (L) to both sides gives the ratio:~~
	~~<math>{L\over{\delta}_1}=L\sqrt{{\rho}{\omega}\over {\mu}} =L\sqrt{{\omega}\over {\nu}}=N_W \,\!</math>~~
	Therefore it can be seen that when the flow has a high Womersley Number the transient boundary layer thickness is very small, when compared to the characteristic length, which for circular vessels is the radius. As shown earlier the convective inertial force is represented by the term <math>{\rho}{V^2}\over L\,\!</math>, equating that to the viscous force term yields:
	~~<math style="vertical-align:-100%;"> {{\rho}{V^2}\over L}={{\mu}{V}\over {{{\delta}_2}^2}}\,\!</math>~~
	~~Solving for the convective boundary layer thickness yields:~~
	~~<math>{\delta}_2=\sqrt{{\mu}L\over {\rho}V}\,\!</math>~~
	~~Factoring in a characteristic length gives the ratio:~~
	~~<math>{L\over{\delta}_2}=\sqrt{{\rho}VL\over {\mu}} =\sqrt{{VL\over\nu}}=\sqrt{N_R} \,\!</math>~~
	~~From the equation it is shown that for a flow~~ with a large Reynolds Number there will be a correspondingly small convective boundary layer compared to the vessel’s characteristic length.<ref>Skalak, Richard. "Biofluid Mechanics," ''Annual Review Fluid Mechanics'', 21(1989)167:204.</ref> By knowing the Reynolds and Womersley Numbers for a given flow it is able to calculate both transient and convective boundary layer thicknesses, and relate them to a flow in another system. Boundary layer thickness is also useful in knowing when the fluid can be treated as an ideal fluid. This is at a distance that is larger than both boundary layer thicknesses.<ref>Taylor,M G. ~~"Hemodynamics," ''Annual Review of Physiology'', 35(1973)87:116.</ref>~~

	~~== See also ==~~
	* [[Similitude (model)]]
	* [[Dimensionless Number]]
	* [[Reynolds Number]]
	* [[Womersley number]]
	* [[Boundary Layer]]
	* [[Navier-Stokes equation]]

	~~== References ==~~
	~~<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->~~
	~~{{Reflist}}~~

	~~[[Category:Dimensionless numbers]]~~
	~~[[Category:Fluid dynamics]]~~
	~~[[Category:Biomechanics]]~~

en>A.amitkumar: Reverted edits by 139.55.134.61 (talk) to last revision by 94.246.33.171 (HG)

2013-10-02T18:00:56Z

Reverted edits by 139.55.134.61 (talk) to last revision by 94.246.33.171 (HG)

← Older revision		Revision as of 20:00, 2 October 2013
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	Hi there! :) ~~My name~~ is ~~Zulma~~, I'm a ~~student studying Computer Science~~ from ~~Raversbeuren~~, ~~Germany~~.<br><br>~~Here~~ is ~~my web page~~ :: ~~[http~~://~~tinyurl~~.~~com~~/~~mqsp9rh~~ http://~~tinyurl~~.~~com~~/~~mqsp9rh~~]		{{Cleanup\|date=April 2011}}


			In [[fluid mechanics]], '''dynamic similarity''' refers to the phenomenon that when there are two geometrically similar vessels (same shape, different sizes) with the same boundary conditions (ex. No-slip, center-line velocity) and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.<ref>Jones, Robert T. "Blood Flow," ''Annual Review of Fluid Mechanics'', 1(1969)223:244.</ref>

			==Derivation==
			[[Reynolds number]] and [[Womersley number]] are the only two physical parameters necessary to solve an incompressible fluid flow problem. Reynolds Number is given by:
			<math>N_R = {VL{\rho}\over {\mu}}\,\!</math>

			The terms of the equation itself represent the following:
			:<math>N_R = {\text{Convective Inertial Force}\over \text{Shear Force}}\,\!</math>.
			When Reynolds Number is large, it shows that the flow is dominated by convective inertial effects; When Reynolds Number is small, it shows that the flow is dominated by shear effects.
			Womersley Number is given by:
			:<math>N_W = L\sqrt{{\omega}{\rho}\over {\mu}}=\sqrt{N_S}\,\!</math>,
			which is simply the square-root of Stokes Number, the terms of the equation itself represents the following:
			:<math style="vertical-align:-100%;"> N_W = {\text{Transient Inertial Force}\over \text{Shear Force}}\,\!</math>.
			When Womersley Number is large(around 10 or greater), it shows that the flow is dominated by oscillatory inertial forces and that the velocity profile is flat. When the Womersley parameter is low, viscous forces tend to
			dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with the driving pressure gradient.<ref>Ku, David N. "Blood Flow in Arteries," ''Annual Review of Fluid Mechanics'', 1(1969)223:44.</ref>

			Starting with [[Navier–Stokes equation]] for Cartesian flow:

			: <math>{\rho} \left (\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v\frac{{\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right )= {\rho}g-\frac{{\partial}P}{{\partial}x}+{\mu}\left( \frac{{\partial^2}u}{{\partial}x^2}+\frac{{\partial^2}v}{{\partial}y^2}+\frac{{\partial^2}w}{{\partial}z^2}\right)\,\!</math>.

			The terms of the equation itself represent the following:

			: <math>\text{transient inertial forces + convective inertial forces}=\text{gravitational force + Pressure force + viscous forces}\,\!</math> <ref>Fung, Yuan-cheng. "Biomechanics: Circulation," ''Dynamic Similarity'', "New York: Springer", 2(2008)130:134.</ref>

			Ignoring gravitational forces and dividing the equation by density (<math>\rho</math>) yields:

			: <math>\left (\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v\frac{{\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right )= -\frac{1}{\rho}\frac{{\partial}P}{{\partial}x}+{\nu}\left( \frac{{\partial^2}u}{{\partial}x^2}+\frac{{\partial^2}v}{{\partial}y^2}+\frac{{\partial^2}w}{{\partial}z^2}\right)\,\!</math>,

			where <math>\nu={\mu}/{\rho}</math> is the kinematic viscosity. Since both Reynolds and Womersley numbers are dimensionless, Navier-Stokes must be represented as a dimensionless expression as well. Choosing <math>V</math>, <math>\omega</math>, and <math>L</math> as a characteristic velocity, frequency, and length respectively yields dimensionless variables:
			Dimensionless Length Term (same for y' and z'):<math>x' = {x / L}\,\!</math>,
			Dimensionless Velocity Term (same for v' and w'): <math>u' = {u / V}\,\!</math>,
			Dimensionless Pressure Term: <math>P' = {P / {{\rho}V^2}}\,\!</math>,
			Dimensionless Time Term: <math>t' = t{\omega}\,\!</math>.
			Dividing the Navier-Stokes equation by <math>\frac{V^2}{L}\,\!</math> (Convective Inertial Force term) gives:

			: <math>\frac{{N_W}^2}{N_R} \left (\frac{{\partial}u'}{{\partial}t'}\right)+\left (u'\frac{{\partial}u'}{{\partial}x'}+v'\frac{{\partial}u'}{{\partial}y'}+w'\frac{{\partial}u'}{{\partial}z'}\right )= -\frac{{\partial}P'}{{\partial}x'}+\frac{1}{N_R}\left( \frac{{\partial^2}u'}{{\partial}x'^2}+\frac{{\partial^2}v'}{{\partial}y'^2}+\frac{{\partial^2}w'}{{\partial}z'^2}\right)\,\!</math>,

			With the addition of the dimensionless continuity equation (seen below) in any incompressible fluid flow problem the Reynolds and Womersley Numbers are the only two physical parameters that are in the two equations
			:<math>\frac{{\partial}u'}{{\partial}x'}+\frac{{\partial}v'}{{\partial}y'}+\frac{{\partial}w'}{{\partial}z'}=0 \,\!</math>, <ref>van de Vosse, Frans M. "Pulse Wave Propagation in the Arterial Tree.," ''Annual Review of Fluid Mechanics'', 43(2011)467:499.</ref>

			==Boundary layer thickness==
			Reynolds and Womersley Numbers are also used to calculate the thicknesses of the [[boundary layers]] that can form from the fluid flow’s viscous effects. Reynolds number is used to calculate the convective inertial boundary layer thickness that can form, and Womersley number is used to calculate the transient inertial boundary thickness that can form. From the Womersley Number it can be shown that the transient inertia force is represented by <math>{\rho}{\omega}V\,\!</math>, and from the last term in the non-modified Navier-Stokes equation that viscous force is represented by <math>{\mu}{V}\over {{\delta}_1}^2\,\!</math> (subscript one indicates that the boundary layer thickness is that of the transient boundary layer). Setting the two forces equal to each other yields:
			<math>{\rho}{\omega}V={{\mu}{V}\over {{\delta}_1}^2}\,\!</math>
			Solving for <math>{\delta}_1\,\!</math> yields:
			<math>{\delta}_1=\sqrt{{\mu}\over {\rho}{\omega}}\,\!</math>
			Adding a characteristic length (L) to both sides gives the ratio:
			<math>{L\over{\delta}_1}=L\sqrt{{\rho}{\omega}\over {\mu}} =L\sqrt{{\omega}\over {\nu}}=N_W \,\!</math>
			Therefore it can be seen that when the flow has a high Womersley Number the transient boundary layer thickness is very small, when compared to the characteristic length, which for circular vessels is the radius. As shown earlier the convective inertial force is represented by the term <math>{\rho}{V^2}\over L\,\!</math>, equating that to the viscous force term yields:
			<math style="vertical-align:-100%;"> {{\rho}{V^2}\over L}={{\mu}{V}\over {{{\delta}_2}^2}}\,\!</math>
			Solving for the convective boundary layer thickness yields:
			<math>{\delta}_2=\sqrt{{\mu}L\over {\rho}V}\,\!</math>
			Factoring in a characteristic length gives the ratio:
			<math>{L\over{\delta}_2}=\sqrt{{\rho}VL\over {\mu}} =\sqrt{{VL\over\nu}}=\sqrt{N_R} \,\!</math>
			From the equation it is shown that for a flow with a large Reynolds Number there will be a correspondingly small convective boundary layer compared to the vessel’s characteristic length.<ref>Skalak, Richard. "Biofluid Mechanics," ''Annual Review Fluid Mechanics'', 21(1989)167:204.</ref> By knowing the Reynolds and Womersley Numbers for a given flow it is able to calculate both transient and convective boundary layer thicknesses, and relate them to a flow in another system. Boundary layer thickness is also useful in knowing when the fluid can be treated as an ideal fluid. This is at a distance that is larger than both boundary layer thicknesses.<ref>Taylor,M G. "Hemodynamics," ''Annual Review of Physiology'', 35(1973)87:116.</ref>

			== See also ==
			* [[Similitude (model)]]
			* [[Dimensionless Number]]
			* [[Reynolds Number]]
			* [[Womersley number]]
			* [[Boundary Layer]]
			* [[Navier-Stokes equation]]

			== References ==
			<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
			{{Reflist}}

			[[Category:Dimensionless numbers]]
			[[Category:Fluid dynamics]]
			[[Category:Biomechanics]]

75.150.161.113: Version info and feature info updated. "Competing software" and "See also" added.

2012-08-25T11:29:40Z

Version info and feature info updated. "Competing software" and "See also" added.

New page

Hi there! :) My name is Zulma, I'm a student studying Computer Science from Raversbeuren, Germany.<br><br>Here is my web page :: [http://tinyurl.com/mqsp9rh http://tinyurl.com/mqsp9rh]