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<p><a href="/index.php?title=WP:CHECKWIKI&action=edit&redlink=1" class="new" title="WP:CHECKWIKI (page does not exist)">WP:CHECKWIKI</a> error #52 + general fixes using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a> (8888)</p>
<p><b>New page</b></p><div>{{continuum mechanics}}<br />
This article summarizes [[equation]]s in the theory of [[fluid mechanics]].<br />
<br />
==Definitions==<br />
<br />
[[File:General flux diagram.svg|thumb|300px|Flux '''F''' through a [[surface]], d'''S''' is the [[Differential (infinitesimal)|differential]] [[vector area]] element, '''n''' is the [[unit normal]] to the surface. '''Left:''' No flux passes in the surface, the maximum amount flows normal to the surface. '''Right:''' The reduction in flux passing through a surface can be visualized by reduction in '''F''' or d'''S''' equivalently (resolved into [[Euclidean vector#Decomposition|components]], θ is angle to normal '''n'''). '''F'''•d'''S''' is the component of flux passing though the surface, multiplied by the area of the surface (see [[dot product]]). For this reason flux represents physically a flow ''per unit area''.]]<br />
<br />
Here <math> \mathbf{\hat{t}} \,\!</math> is a unit vector in the direction of the flow/current/flux.<br />
<br />
:{| class="wikitable"<br />
|-<br />
! scope="col" width="100" | Quantity (common name/s) <br />
! scope="col" width="100" | (Common) symbol/s <br />
! scope="col" width="300" | Defining equation <br />
! scope="col" width="125" | SI units <br />
! scope="col" width="100" | Dimension<br />
|-<br />
| [[Flow velocity]] vector field<br />
| '''u'''<br />
| <math> \mathbf{u}=\mathbf{u}\left ( \mathbf{r},t \right ) \,\!</math> <br />
| m s<sup>−1</sup> <br />
| [L][T]<sup>−1</sup><br />
|-<br />
| [[Vorticity]] [[pseudovector]] field<br />
| '''ω'''<br />
| <math> \boldsymbol{\omega} = \nabla\times\mathbf{v} </math> <br />
| s<sup>−1</sup> <br />
| [T]<sup>−1</sup><br />
|-<br />
|Volume velocity, volume flux<br />
| ''φ<sub>V</sub>'' (no standard symbol)<br />
|<math>\phi_V = \int_S \mathbf{u} \cdot \mathrm{d}\mathbf{A}\,\!</math><br />
| m<sup>3</sup> s<sup>−1</sup><br />
| [L]<sup>3</sup> [T]<sup>−1</sup><br />
|-<br />
|[[Continuity equation#General equation|Mass current per unit volume]]<br />
| ''s'' (no standard symbol)<br />
|<math>s = \mathrm{d}\rho / \mathrm{d}t \,\!</math><br />
| kg m<sup>&minus;3</sup> s<sup>−1</sup><br />
| [M] [L]<sup>&minus;3</sup> [T]<sup>−1</sup><br />
|-<br />
| Mass current, [[mass flow rate]]<br />
| ''I<sub>m</sub>''<br />
| <math> I_\mathrm{m} = \mathrm{d} m/\mathrm{d} t \,\!</math> <br />
| kg s<sup>−1</sup> <br />
| [M][T]<sup>−1</sup><br />
|-<br />
| Mass current density <br />
| '''j'''<sub>m</sub> <br />
| <math> I_\mathrm{m} = \iint \mathbf{j}_\mathrm{m} \cdot \mathrm{d}\mathbf{S} \,\!</math> <br />
| kg m<sup>−2</sup> s<sup>−1</sup> <br />
| [M][L]<sup>−2</sup>[T]<sup>−1</sup><br />
|-<br />
| Momentum current <br />
| ''I<sub>p</sub>'' <br />
| <math> I_\mathrm{p} = \mathrm{d} \left | \mathbf{p} \right |/\mathrm{d} t \,\!</math> <br />
| kg m s<sup>−2</sup> <br />
| [M][L][T]<sup>−2</sup><br />
|-<br />
| Momentum current density <br />
| '''j'''<sub>p</sub> <br />
| <math> I_\mathrm{p} =\iint \mathbf{j}_\mathrm{p} \cdot \mathrm{d}\mathbf{S} </math> <br />
| kg m s<sup>−2</sup> <br />
| [M][L][T]<sup>−2</sup><br />
|-<br />
|}<br />
<br />
==Equations==<br />
<br />
:{| class="wikitable"<br />
|-<br />
! scope="col" width="100" | Physical situation<br />
! scope="col" width="250" | Nomenclature<br />
! scope="col" width="10" | Equations<br />
|-<br />
! scope="row" | [[Fluid statics]],<br />[[pressure gradient]]<br />
| <div class="plainlist"><br />
* '''r''' = Position<br />
* ''ρ'' = ''ρ''('''r''') = Fluid density at gravitational equipotential containing '''r'''<br />
* '''g''' = '''g'''('''r''') = Gravitational field strength at point '''r'''<br />
* ∇''P'' = Pressure gradient<br />
</div><br />
| <math> \nabla P = \rho \mathbf{g}\,\!</math><br />
|-<br />
! scope="row" | Buoyancy equations<br />
| <div class="plainlist"><br />
* ''ρ''<sub>''f''</sub> = Mass density of the fluid<br />
* ''V''<sub>imm</sub> = Immersed volume of body in fluid<br />
* '''F'''<sub>b</sub> = Buoyant force<br />
* '''F'''<sub>g</sub> = Gravitational force<br />
* '''W'''<sub>app</sub> = Apparent weight of immersed body<br />
* '''W''' = Actual weight of immersed body<br />
</div><br />
| [[Buoyant force]]<br /><br />
<math>\mathbf{F}_\mathrm{b} = - \rho_f V_\mathrm{imm} \mathbf{g} = - \mathbf{F}_\mathrm{g}\,\!</math><br />
<br />
[[Apparent weight]]<br /><br />
<math>\mathbf{W}_\mathrm{app} = \mathbf{W} - \mathbf{F}_\mathrm{b}\,\!</math><br />
|-<br />
! [[Bernoulli's equation]]<br />
| ''p''<sub>constant</sub> is the total pressure at a point on a streamline<br />
| <math>p + \rho v^2/2 + \rho gy = p_\mathrm{constant}\,\!</math><br />
|-<br />
! [[Euler equations (fluid dynamics)|Euler equations]]<br />
| <div class="plainlist"><br />
* ''ρ'' = fluid [[mass density]]<br />
* '''u''' is the fluid [[velocity]] [[Vector (geometric)|vector]]<br />
* ''E'' = total volume [[energy]] density<br />
* ''U'' = [[internal energy]] per unit mass of fluid<br />
* ''p'' = [[pressure]]<br />
* ''<math>\otimes</math>'' denotes the [[tensor product]]<br />
</div><br />
| <math>\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0\,\!</math><br/ ><br />
<math>\frac{\partial\rho{\mathbf{u}}}{\partial t} + \nabla \cdot \left ( \mathbf{u}\otimes \left ( \rho \mathbf{u} \right ) \right )+\nabla p=0\,\!</math><br /><br />
<math>\frac{\partial E}{\partial t}+\nabla\cdot\left ( \bold u \left ( E+p \right ) \right ) = 0 \,\!</math><br /><br />
<math> E = \rho \left ( U + \frac{1}{2} \mathbf{v}^2 \right ) \,\!</math><br />
|-<br />
! [[Navier-stokes equations#Convective acceleration|Convective acceleration]]<br />
|<br />
| <math>\mathbf{a} = \left ( \mathbf{v} \cdot \nabla \right ) \mathbf{v}</math><br />
|-<br />
! [[Navier-stokes equations]]<br />
| <div class="plainlist"><br />
* '''T'''<sub>D</sub> = [[Stress (physics)#Stress deviator tensor|Deviatoric]] stress [[tensor field|tensor]]<br />
* <math>\mathbf{f} </math> = volume density of the [[body force]]s acting on the fluid<br />
* <math>\nabla</math> here is the [[del]] operator.<br />
</div><br />
| <math> \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbf{T}_\mathrm{D} + \mathbf{f} </math><br />
|}<br />
<br />
==See also==<br />
<br />
*[[Defining equation (physical chemistry)]]<br />
*[[List of equations in classical mechanics]]<br />
*[[Table of thermodynamic equations]]<br />
*[[List of relativistic equations]]<br />
*[[List of equations in gravitation]]<br />
*[[List of electromagnetism equations]]<br />
*[[List of photonics equations]]<br />
*[[List of equations in quantum mechanics]]<br />
*[[List of equations in nuclear and particle physics]]<br />
<br />
==Footnotes==<br />
<br />
{{Reflist}}<br />
<br />
==Sources==<br />
<br />
* {{cite book| author=P.M. Whelan, M.J. Hodgeson| title=Essential Principles of Physics| publisher=John Murray|edition=2nd| year=1978 | isbn=0-7195-3382-1}}<br />
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| publisher=Cambridge University Press|edition=| year=2010| isbn=978-0-521-57507-2}}<br />
* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill|edition=| year=1988| isbn=978-0-07-025734-4}}<br />
* {{cite book|pages=12–13| author=R.G. Lerner, G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers, Hans Warlimont, Springer|edition=2nd| year=2005| isbn=978-0-07-025734-4}}<br />
* {{cite book|page=| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill|edition=2nd| year=1994| isbn=0-07-051400-3}}<br />
* {{cite book|page=| author=P.A. Tipler, G. Mosca| title=Physics for Scientists and Engineers: With Modern Physics| publisher=W.H. Freeman and Co|edition=6th| year=2008| isbn=9-781429-202657}}<br />
* {{cite book|title=Analytical Mechanics|author=L.N. Hand, J.D. Finch|publisher=Cambridge University Press, |year=2008|isbn=978-0-521-57572-0}}<br />
* {{cite book|title=Mechanics, Vibrations and Waves|author=T.B. Arkill, C.J. Millar|publisher=John Murray, |year=1974|isbn=0-7195-2882-8}}<br />
* {{cite book|title=The Physics of Vibrations and Waves|edition=3rd|author=H.J. Pain|publisher=John Wiley & Sons, |year=1983|isbn=0-471-90182-2}}<br />
<br />
==Further reading==<br />
<br />
* {{cite book|title=Physics with Modern Applications|author=L.H. Greenberg|publisher=Holt-Saunders International W.B. Saunders and Co|year=1978|isbn=0-7216-4247-0}}<br />
* {{cite book|title=Principles of Physics|author=J.B. Marion, W.F. Hornyak|publisher=Holt-Saunders International Saunders College|year=1984|isbn=4-8337-0195-2}}<br />
* {{cite book|title=Concepts of Modern Physics|edition=4th|author=A. Beiser|publisher=McGraw-Hill (International)|year=1987|isbn=0-07-100144-1}}<br />
* {{cite book|title=University Physics – With Modern Physics|edition=12th|author=H.D. Young, R.A. Freedman|publisher=Addison-Wesley (Pearson International)|year=2008|isbn=0-321-50130-6}}<br />
<br />
{{SI units navbox}}<br />
<br />
[[Category:Physical quantities]]<br />
[[Category:SI units]]<br />
[[Category:Physical chemistry]]<br />
[[Category:Equations of physics]]</div>
en>Yobot