View source for List of equations in fluid mechanics

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{{continuum mechanics}}
This article summarizes [[equation]]s in the theory of [[fluid mechanics]].

==Definitions==

[[File:General flux diagram.svg|thumb|300px|Flux '''F''' through a [[surface integral|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 through the surface, multiplied by the area of the surface (see [[dot product]]). For this reason flux represents physically a flow ''per unit area''.]]

Here <math> \mathbf{\hat{t}} \,\!</math> is a [[unit vector]] in the direction of the flow/current/flux.

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
| [[Flow velocity]] vector field
| '''u'''
| <math> \mathbf{u}=\mathbf{u}\left ( \mathbf{r},t \right ) \,\!</math> 
| m s<sup>−1</sup> 
| [L][T]<sup>−1</sup>
|-
| [[Velocity]] [[pseudovector]] field
| '''ω'''
| <math> \boldsymbol{\omega} = \nabla\times\mathbf{v} </math> 
| s<sup>−1</sup> 
| [T]<sup>−1</sup>
|-
|Volume velocity, volume flux
| ''φ<sub>V</sub>'' (no standard symbol)
|<math>\phi_V = \int_S \mathbf{u} \cdot \mathrm{d}\mathbf{A}\,\!</math>
| m<sup>3</sup> s<sup>−1</sup>
| [L]<sup>3</sup> [T]<sup>−1</sup>
|-
|[[Continuity equation#General equation|Mass current per unit volume]]
| ''s'' (no standard symbol)
|<math>s = \mathrm{d}\rho / \mathrm{d}t \,\!</math>
| kg m<sup>&minus;3</sup> s<sup>−1</sup>
| [M] [L]<sup>&minus;3</sup> [T]<sup>−1</sup>
|-
| Mass current, [[mass flow rate]]
| ''I<sub>m</sub>''
| <math> I_\mathrm{m} = \mathrm{d} m/\mathrm{d} t \,\!</math> 
| kg s<sup>−1</sup> 
| [M][T]<sup>−1</sup>
|-
| Mass current density 
| '''j'''<sub>m</sub> 
| <math> I_\mathrm{m} = \iint \mathbf{j}_\mathrm{m} \cdot \mathrm{d}\mathbf{S} \,\!</math> 
| kg m<sup>−2</sup> s<sup>−1</sup> 
| [M][L]<sup>−2</sup>[T]<sup>−1</sup>
|-
| Momentum current 
| ''I<sub>p</sub>'' 
| <math> I_\mathrm{p} = \mathrm{d} \left | \mathbf{p} \right |/\mathrm{d} t \,\!</math> 
| kg m s<sup>−2</sup> 
| [M][L][T]<sup>−2</sup>
|-
| Momentum current density 
| '''j'''<sub>p</sub> 
| <math> I_\mathrm{p} =\iint \mathbf{j}_\mathrm{p} \cdot \mathrm{d}\mathbf{S} </math> 
| kg m s<sup>−2</sup> 
| [M][L][T]<sup>−2</sup>
|-
|}

==Equations==

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! scope="row" | [[Fluid statics]],<br />[[pressure gradient]]
| {{plainlist}}
* '''r''' = Position
* ''ρ'' = ''ρ''('''r''') = Fluid density at gravitational equipotential containing '''r'''
* '''g''' = '''g'''('''r''') = Gravitational field strength at point '''r'''
* ∇''P'' = Pressure gradient
{{endplainlist}}
| <math> \nabla P = \rho \mathbf{g}\,\!</math>
|-
! scope="row" | Buoyancy equations
| {{plainlist}}
* ''ρ''<sub>''f''</sub> = Mass density of the fluid
* ''V''<sub>imm</sub> = Immersed volume of body in fluid
* '''F'''<sub>b</sub> = Buoyant force
* '''F'''<sub>g</sub> = Gravitational force
* '''W'''<sub>app</sub> = Apparent weight of immersed body
* '''W''' = Actual weight of immersed body
{{endplainlist}}
| [[Buoyant force]]<br />
<math>\mathbf{F}_\mathrm{b} = - \rho_f V_\mathrm{imm} \mathbf{g} = - \mathbf{F}_\mathrm{g}\,\!</math>

[[Apparent weight]]<br />
<math>\mathbf{W}_\mathrm{app} = \mathbf{W} - \mathbf{F}_\mathrm{b}\,\!</math>
|-
! [[Bernoulli's equation]]
| ''p''<sub>constant</sub> is the total pressure at a point on a streamline
| <math>p + \rho u^2/2 + \rho  gy = p_\mathrm{constant}\,\!</math>
|-
! [[Euler equations (fluid dynamics)|Euler equations]]
| {{plainlist}}
* ''ρ'' = fluid [[mass density]]
* '''u''' is the [[flow velocity]] [[Vector (geometric)|vector]]
* ''E'' =  total volume [[energy]] density
* ''U'' = [[internal energy]] per unit mass of fluid
* ''p'' = [[pressure]]
* ''<math>\otimes</math>'' denotes the [[tensor product]]
{{endplainlist}}
| <math>\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0\,\!</math><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 />
<math>\frac{\partial E}{\partial t}+\nabla\cdot\left ( \mathbf u \left ( E+p \right ) \right ) = 0 \,\!</math><br />
<math> E = \rho \left ( U + \frac{1}{2} \mathbf{u}^2 \right ) \,\!</math>
|-
! [[Convective acceleration]]
|
| <math>\mathbf{a} = \left ( \mathbf{u} \cdot \nabla \right ) \mathbf{u}</math>
|-
! [[Navier–Stokes equations]]
| {{plainlist}}
* '''T'''<sub>D</sub> = [[Deviatoric stress tensor]]
* <math>\mathbf{f} </math> = volume density of the [[body force]]s acting on the fluid
* <math>\nabla</math> here is the [[del]] operator.
{{endplainlist}}
| <math> \rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \nabla \cdot\mathbf{T}_\mathrm{D} + \mathbf{f} </math>
|}

==See also==

*[[Defining equation (physical chemistry)]]
*[[List of electromagnetism equations]]
*[[List of equations in classical mechanics]]
*[[List of equations in gravitation]]
*[[List of equations in nuclear and particle physics]]
*[[List of equations in quantum mechanics]]
*[[List of photonics equations]]
*[[List of relativistic equations]]
*[[Table of thermodynamic equations]]

{{Reflist}}

==Sources==

* {{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}}
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| url=https://archive.org/details/cambridgehandboo0000woan| url-access=registration| publisher=Cambridge University Press| year=2010| isbn=978-0-521-57507-2}}
* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill| year=1988| isbn=978-0-07-025734-4}}
* {{cite book|pages=12–13| author=[[Rita G. Lerner|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}}
* {{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book| 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=978-1-4292-0265-7}}
* {{cite book|title=Analytical Mechanics|author=L.N. Hand, J.D. Finch|publisher=Cambridge University Press |year=2008|isbn=978-0-521-57572-0}}
* {{cite book|title=Mechanics, Vibrations and Waves|author=T.B. Arkill, C.J. Millar|publisher=John Murray |year=1974|isbn=0-7195-2882-8}}
* {{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}}

==Further reading==

* {{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|url-access=registration|url=https://archive.org/details/physicswithmoder0000gree}}
* {{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}}
* {{cite book|title=Concepts of Modern Physics|edition=4th|author=A. Beiser|publisher=McGraw-Hill (International)|year=1987|isbn=0-07-100144-1}}
* {{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=978-0-321-50130-1}}

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