Solid harmonics - Revision history

132.70.34.184: the hbar does not belong there; l here is defined as -ir cross del, and it is unitless. in quantum mechanics L is r cross p, and p is (-ihbar*del), which gives rise to hbar.

2014-12-16T11:27:26Z

the hbar does not belong there; l here is defined as -ir cross del, and it is unitless. in quantum mechanics L is r cross p, and p is (-ihbar*del), which gives rise to hbar.

← Older revision		Revision as of 13:27, 16 December 2014
Line 1:		Line 1:
	~~The~~ '''~~Dean number''' (~~'~~'D'') is~~ a ~~[[dimensionless quantity\|dimensionless group]] in [[fluid mechanics]]~~, ~~which occurs in the study of flow in curved pipes and channels~~. ~~It is named after the [[Great Britain\|British]] scientist [[W~~. R. ~~Dean]], who studied such flows in the 1920s~~ (~~Dean, 1927~~, ~~1928)~~.		Vous devez savoir de qui des ventes aux enchères d'automobiles peuvent obtenir d'innombrables acheteurs potentiels étant qu'ils sont vendus à-dire pas cher. méthode de Voyage, à compter principalement parce qu'il a fait de fournir un wagonmaking garder pendant au bercail, Coach Outlet Dans, sur 1852. 24/7, les collègues sont là pour aider toute usa de toutes les bonnes et les bad.Bottega Veneta Chaussures consommateurs, qui seront songent à acheter planche à laver avec un emplacement idéal de Mohali, peut faire face à la propriété ou à la maison realestate entreprises. Nous avons un ensemble de scripts (qui est définitivement autorisé!) Qui devrait vous avatars cliquables antérieures Sixtyfive en planifiant seulement facilement un voyage à n'importe quelle page Web, et ainsi de suite pratiquement toutes les relations qui va être remplies en une demi-minute.<br><br>

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	~~The Dean number is typically denoted by the symbol~~ ''D''. ~~For flow in a pipe or tube it is defined as:~~

	~~: <math>\mathit{D} = \frac{\rho V\! d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}</math>~~

	~~where~~
	* <math>\rho</math> is the density of the fluid
	* <math>\mu</math> is the dynamic viscosity
	* <math>V</math> is the axial velocity scale
	* <math>d</math> is the diameter (other shapes are represented by an equivalent diameter, ~~see [[Reynolds number]])~~
	* <math>R</math> is the radius of curvature of the path of the channel.

	~~The Dean number is therefore the product of the Reynolds number (based on axial flow~~ <~~math~~>V<~~/math~~> ~~through a pipe of diameter~~ <~~math~~>d~~</math>) and the square root of the curvature ratio~~.

	~~== The Dean Equations ==~~

	~~The Dean number appears in the so~~-~~called '''Dean Equations~~'''. ~~These are an approximation to the full [[Navier-Stokes equations\|Navier&ndash~~;~~Stokes equations]] for the steady axially uniform flow of a [[Newtonian fluid]] in a [[torus\|toroidal]] pipe, obtained by retaining just the [[leading-order\|leading order]] curvature effects (i~~.~~e. the~~ [~~[leading-order]] equations for <math>a~~/~~r \ll 1<~~/~~math>)~~.

	~~We use [[orthogonal coordinates]~~] <~~math~~>~~(x,y,z)~~<~~/math~~> ~~with corresponding unit vectors <math>(\hat{\boldsymbol{x}}~~,~~\hat{\boldsymbol{y}}~~,~~\hat{\boldsymbol{z}})</math> aligned with the centre-line of the pipe at each point~~. ~~The axial direction is <math>\hat{\boldsymbol{z}}</math>~~, ~~with <math>\hat{\boldsymbol{x}}</math> being the normal in the plane of the centre-line~~, ~~and <math>\hat{\boldsymbol{~~y~~}}</math> the [[binormal]]~~. ~~For an axial flow driven by a pressure gradient~~ <~~math~~>G<~~/math~~>, ~~the axial velocity <math>u_z</math> is scaled with <math>U=Ga^2/\mu</math>. The cross-stream velocities <math>u_x~~, ~~u_y</math> are scaled with <math>(a/R)^{1/2} U</math>~~, ~~and cross-stream pressures with <math>\rho a U^2/L</math>~~. ~~Lengths are scaled with the tube radius <math>a</math>~~.

	~~In terms of these non-dimensional variables and coordinates~~, ~~the Dean equations are then~~
	~~:<math>D \left( \frac{\mathrm{D} u_x}{\mathrm{D} t} + u_z^2 \right) =~~ -~~D \frac{\partial p}{\partial x} + \nabla^2 u_x </math>~~
	:~~<math>D \frac{\mathrm{D} u_y}{\mathrm{D} t} = -D\frac{\partial p}{\partial y} + \nabla^2 u_y</math>~~
	:<~~math~~>~~D \frac{\mathrm{D} u_z}{\mathrm{D} t} = 1 + \nabla^2 u_z~~ <~~/math~~>
	~~:<math>\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0</math>~~
	~~where~~
	~~:<math>\frac{\mathrm{D}}{\mathrm{D} t} = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y}</math>~~
	~~is the [[convective derivative]].~~

	~~The Dean number ''D'' is the only parameter left in the system~~, ~~and encapsulates the [[leading-order\|leading order]] curvature effects. Higher-order approximations will involve additional parameters.~~

	~~For weak curvature effects (small ''D'')~~, ~~the Dean equations can be solved as a series expansion in~~ ''D''. The first correction to the leading-order axial [[Poiseuille flow]] is a pair of vortices in the cross-section carrying flow form the inside to the outside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number <~~math~~>~~D_c \approx 956~~<~~/math~~> ~~(Dennis & Ng 1982). For larger~~ ''~~D'', there are multiple solutions, many of which are unstable.~~

	~~== References ==~~
	* {{cite journal \| last1 = Berger \| first1 = S. A. \| last2 = Talbot \| first2 = L. \| last3 = Yao \| first3 = L. S. \| year = 1983 \| title = Flow in Curved Pipes \| url = \| journal = Ann. Rev. Fluid Mech. \| volume = 15 \| issue = \| pages = 461–512 \| doi = 10.1146/annurev.fl.15.010183.002333 \|bibcode = 1983AnRFM..15..461B }}
	* {{cite journal \| last1 = Dean \| first1 = W. R. \| year = 1927 \| title = Note on the motion of fluid in a curved pipe \| url = http:~~//www.informaworld.com/openurl?genre=~~article~~&issn=1941-5982&volume=4&issue=20&spage=208 \| journal = Phil. Mag. \| volume = 20 \| issue = \| pages = 208–223 }}~~
	* {{cite journal \| last1 = Dean \| first1 = W. R. \| year = 1928 \| title = The streamline motion of fluid in a curved pipe \| url = http://~~www.informaworld~~.com/~~openurl?genre=article&issn=1941-5982&volume=5&issue=30&spage=673 \| journal = Phil. Mag. (7) \| volume = 5 \| issue = \| pages = 673–695 }}~~
	* {{cite journal \| last1 = Dennis \| first1 = C. R. \| last2 = Ng \| first2 = M. \| year = 1982 \| title = Dual solutions for steady laminar-flow through a curved tube \| url = \| journal = Q. J. Mech. Appl. Math. \| volume = 35 \| issue = \| page = 305 }}
	* Mestel, J. [http://www.ma.ic.ac.uk/~ajm8/M4A33/Dean.pdf Flow in curved pipes: The Dean equations]~~, ''Lecture Handout for Course M4A33'', Imperial College~~.

	~~{{NonDimFluMech}}~~

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

en>Susilehtola: Moved example on atomic orbitals to Spherical harmonics.

2014-01-16T18:18:04Z

Moved example on atomic orbitals to Spherical harmonics.

← Older revision		Revision as of 20:18, 16 January 2014
Line 1:		Line 1:
	~~47 yrs old Quality Assurance Manager Arlie from Ayr~~, ~~usually spends time~~ with ~~interests for instance climbing~~, ~~injustice gods among us~~ and ~~collecting music albums~~. In the ~~last year has completed~~ a ~~journey~~ to ~~Colonial City~~ of ~~Santo Domingo~~.<br><br>~~Review my web~~-~~site~~ ... [http://~~Kacharya~~.~~Com~~.np/~~blog~~/~~JeramykfManciniry injustice gods among us ultimate edition xbox 360~~]		The '''Dean number''' (''D'') is a [[dimensionless quantity\|dimensionless group]] in [[fluid mechanics]], which occurs in the study of flow in curved pipes and channels. It is named after the [[Great Britain\|British]] scientist [[W. R. Dean]], who studied such flows in the 1920s (Dean, 1927, 1928).

			== Definition ==
			The Dean number is typically denoted by the symbol ''D''. For flow in a pipe or tube it is defined as:

			: <math>\mathit{D} = \frac{\rho V\! d}{\mu} \left( \frac{d}{2 R} \right)^{1/2}</math>

			where
			* <math>\rho</math> is the density of the fluid
			* <math>\mu</math> is the dynamic viscosity
			* <math>V</math> is the axial velocity scale
			* <math>d</math> is the diameter (other shapes are represented by an equivalent diameter, see [[Reynolds number]])
			* <math>R</math> is the radius of curvature of the path of the channel.

			The Dean number is therefore the product of the Reynolds number (based on axial flow <math>V</math> through a pipe of diameter <math>d</math>) and the square root of the curvature ratio.

			== The Dean Equations ==

			The Dean number appears in the so-called '''Dean Equations'''. These are an approximation to the full [[Navier-Stokes equations\|Navier–Stokes equations]] for the steady axially uniform flow of a [[Newtonian fluid]] in a [[torus\|toroidal]] pipe, obtained by retaining just the [[leading-order\|leading order]] curvature effects (i.e. the [[leading-order]] equations for <math>a/r \ll 1</math>).

			We use [[orthogonal coordinates]] <math>(x,y,z)</math> with corresponding unit vectors <math>(\hat{\boldsymbol{x}},\hat{\boldsymbol{y}},\hat{\boldsymbol{z}})</math> aligned with the centre-line of the pipe at each point. The axial direction is <math>\hat{\boldsymbol{z}}</math>, with <math>\hat{\boldsymbol{x}}</math> being the normal in the plane of the centre-line, and <math>\hat{\boldsymbol{y}}</math> the [[binormal]]. For an axial flow driven by a pressure gradient <math>G</math>, the axial velocity <math>u_z</math> is scaled with <math>U=Ga^2/\mu</math>. The cross-stream velocities <math>u_x, u_y</math> are scaled with <math>(a/R)^{1/2} U</math>, and cross-stream pressures with <math>\rho a U^2/L</math>. Lengths are scaled with the tube radius <math>a</math>.

			In terms of these non-dimensional variables and coordinates, the Dean equations are then
			:<math>D \left( \frac{\mathrm{D} u_x}{\mathrm{D} t} + u_z^2 \right) = -D \frac{\partial p}{\partial x} + \nabla^2 u_x </math>
			:<math>D \frac{\mathrm{D} u_y}{\mathrm{D} t} = -D\frac{\partial p}{\partial y} + \nabla^2 u_y</math>
			:<math>D \frac{\mathrm{D} u_z}{\mathrm{D} t} = 1 + \nabla^2 u_z </math>
			:<math>\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0</math>
			where
			:<math>\frac{\mathrm{D}}{\mathrm{D} t} = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y}</math>
			is the [[convective derivative]].

			The Dean number ''D'' is the only parameter left in the system, and encapsulates the [[leading-order\|leading order]] curvature effects. Higher-order approximations will involve additional parameters.

			For weak curvature effects (small ''D''), the Dean equations can be solved as a series expansion in ''D''. The first correction to the leading-order axial [[Poiseuille flow]] is a pair of vortices in the cross-section carrying flow form the inside to the outside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number <math>D_c \approx 956</math> (Dennis & Ng 1982). For larger ''D'', there are multiple solutions, many of which are unstable.

			== References ==
			* {{cite journal \| last1 = Berger \| first1 = S. A. \| last2 = Talbot \| first2 = L. \| last3 = Yao \| first3 = L. S. \| year = 1983 \| title = Flow in Curved Pipes \| url = \| journal = Ann. Rev. Fluid Mech. \| volume = 15 \| issue = \| pages = 461–512 \| doi = 10.1146/annurev.fl.15.010183.002333 \|bibcode = 1983AnRFM..15..461B }}
			* {{cite journal \| last1 = Dean \| first1 = W. R. \| year = 1927 \| title = Note on the motion of fluid in a curved pipe \| url = http://www.informaworld.com/openurl?genre=article&issn=1941-5982&volume=4&issue=20&spage=208 \| journal = Phil. Mag. \| volume = 20 \| issue = \| pages = 208–223 }}
			* {{cite journal \| last1 = Dean \| first1 = W. R. \| year = 1928 \| title = The streamline motion of fluid in a curved pipe \| url = http://www.informaworld.com/openurl?genre=article&issn=1941-5982&volume=5&issue=30&spage=673 \| journal = Phil. Mag. (7) \| volume = 5 \| issue = \| pages = 673–695 }}
			* {{cite journal \| last1 = Dennis \| first1 = C. R. \| last2 = Ng \| first2 = M. \| year = 1982 \| title = Dual solutions for steady laminar-flow through a curved tube \| url = \| journal = Q. J. Mech. Appl. Math. \| volume = 35 \| issue = \| page = 305 }}
			* Mestel, J. [http://www.ma.ic.ac.uk/~ajm8/M4A33/Dean.pdf Flow in curved pipes: The Dean equations], ''Lecture Handout for Course M4A33'', Imperial College.

			{{NonDimFluMech}}

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

en>RockMagnetist: /* Derivation, relation to spherical harmonics */ added subtitle

2012-03-27T18:46:50Z

Derivation, relation to spherical harmonics: added subtitle

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47 yrs old Quality Assurance Manager Arlie from Ayr, usually spends time with interests for instance climbing, injustice gods among us and collecting music albums. In the last year has completed a journey to Colonial City of Santo Domingo.<br><br>Review my web-site ... [http://Kacharya.Com.np/blog/JeramykfManciniry injustice gods among us ultimate edition xbox 360]