List of equations in fluid mechanics

Short description: None

Continuum mechanics

Laws

|  Conservations

* Mass
* Momentum
* Energy

Inequalities

* Clausius–Duhem (entropy)

Solid mechanics

* Deformation
* Elasticity
* linear
* Plasticity
* Hooke's law
* Stress
* Finite strain
* Infinitesimal strain
* Compatibility
* Bending
* Contact mechanics
* frictional
* Material failure theory
* Fracture mechanics

Fluid mechanics

|  Fluids

* Statics · Dynamics
* Archimedes' principle · Bernoulli's principle
* Navier–Stokes equations
* Poiseuille equation · Pascal's law
* Viscosity
* (Newtonian · non-Newtonian)
* Buoyancy · Mixing · Pressure

Liquids

* Surface tension
* Capillary action

Gases

* Atmosphere
* Boyle's law
* Charles's law
* Gay-Lussac's law
* Combined gas law

Plasma

Rheology

* Viscoelasticity
* Rheometry
* Rheometer

Smart fluids

* Electrorheological
* Magnetorheological
* Ferrofluids

Scientists

* Bernoulli
* Boyle
* Cauchy
* Charles
* Euler
* Gay-Lussac
* Hooke
* Pascal
* Newton
* Navier
* Stokes

* v
* t
* e

This article summarizes equations in the theory of fluid mechanics.

## Contents

* 1 Definitions
* 2 Equations
* 3 See also
* 4 Sources
* 5 Further reading

## Definitions

Flux F through a surface, dS is the 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 dS equivalently (resolved into components, θ is angle to normal n). F•dS 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]\displaystyle{ \mathbf{\hat{t}} \,\\! }[/math] is a unit vector in the direction of the flow/current/flux.

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Flow velocity vector field  | u | [math]\displaystyle{ \mathbf{u}=\mathbf{u}\left ( \mathbf{r},t \right ) \,\\! }[/math] | m s−1 | [L][T]−1
Velocity pseudovector field  | ω | [math]\displaystyle{ \boldsymbol{\omega} = \nabla\times\mathbf{v} }[/math] | s−1 | [T]−1
Volume velocity, volume flux  | φV (no standard symbol)  | [math]\displaystyle{ \phi_V = \int_S \mathbf{u} \cdot \mathrm{d}\mathbf{A}\,\\! }[/math] | m3 s−1 | [L]3 [T]−1
Mass current per unit volume | s (no standard symbol)  | [math]\displaystyle{ s = \mathrm{d}\rho / \mathrm{d}t \,\\! }[/math] | kg m−3 s−1 | [M] [L]−3 [T]−1
Mass current, mass flow rate | Im | [math]\displaystyle{ I_\mathrm{m} = \mathrm{d} m/\mathrm{d} t \,\\! }[/math] | kg s−1 | [M][T]−1
Mass current density  | jm | [math]\displaystyle{ I_\mathrm{m} = \iint \mathbf{j}_\mathrm{m} \cdot \mathrm{d}\mathbf{S} \,\\! }[/math] | kg m−2 s−1 | [M][L]−2[T]−1
Momentum current  | Ip | [math]\displaystyle{ I_\mathrm{p} = \mathrm{d} \left | \mathbf{p} \right |/\mathrm{d} t \,\\! }[/math] | kg m s−2 | [M][L][T]−2
Momentum current density  | jp | [math]\displaystyle{ I_\mathrm{p} =\iint \mathbf{j}_\mathrm{p} \cdot \mathrm{d}\mathbf{S} }[/math] | kg m s−2 | [M][L][T]−2

## Equations

Physical situation  | Nomenclature  | Equations

|
|

Fluid statics,
pressure gradient |

* r = Position
* ρ = ρ(r) = Fluid density at gravitational equipotential containing r
* g = g(r) = Gravitational field strength at point r
* ∇P = Pressure gradient

| [math]\displaystyle{ \nabla P = \rho \mathbf{g}\,\\! }[/math]
Buoyancy equations  |

* ρf = Mass density of the fluid
* Vimm = Immersed volume of body in fluid
* Fb = Buoyant force
* Fg = Gravitational force
* Wapp = Apparent weight of immersed body
* W = Actual weight of immersed body

| Buoyant force

[math]\displaystyle{ \mathbf{F}_\mathrm{b} = - \rho_f V_\mathrm{imm} \mathbf{g} = - \mathbf{F}_\mathrm{g}\,\\! }[/math]

Apparent weight
[math]\displaystyle{ \mathbf{W}_\mathrm{app} = \mathbf{W} - \mathbf{F}_\mathrm{b}\,\\! }[/math]

Bernoulli's equation  | pconstant is the total pressure at a point on a streamline  | [math]\displaystyle{ p + \rho u^2/2 + \rho gy = p_\mathrm{constant}\,\\! }[/math]
Euler equations |

* ρ = fluid mass density
* u is the flow velocity vector
* E = total volume energy density
* U = internal energy per unit mass of fluid
* p = pressure
* [math]\displaystyle{ \otimes }[/math] denotes the tensor product

| [math]\displaystyle{ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0\,\\! }[/math]

[math]\displaystyle{ \frac{\partial\rho{\mathbf{u}}}{\partial t} + \nabla \cdot \left ( \mathbf{u}\otimes \left ( \rho \mathbf{u} \right ) \right )+\nabla p=0\,\\! }[/math]
[math]\displaystyle{ \frac{\partial E}{\partial t}+\nabla\cdot\left ( \mathbf u \left ( E+p \right ) \right ) = 0 \,\\! }[/math]
[math]\displaystyle{ E = \rho \left ( U + \frac{1}{2} \mathbf{u}^2 \right ) \,\\! }[/math]

Convective acceleration  |  | [math]\displaystyle{ \mathbf{a} = \left ( \mathbf{u} \cdot \nabla \right ) \mathbf{u} }[/math]
Navier–Stokes equations |

* TD = Deviatoric stress tensor
* [math]\displaystyle{ \mathbf{f} }[/math] = volume density of the body forces acting on the fluid
* [math]\displaystyle{ \nabla }[/math] here is the del operator.

| [math]\displaystyle{ \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

## Sources

* P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
* G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2. https://archive.org/details/cambridgehandboo0000woan.
* A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4.
* R.G. Lerner, G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4.
* C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3. https://archive.org/details/mcgrawhillencycl1993park.
* P.A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 978-1-4292-0265-7.
* L.N. Hand, J.D. Finch (2008). Analytical Mechanics. Cambridge University Press. ISBN 978-0-521-57572-0.
* T.B. Arkill, C.J. Millar (1974). Mechanics, Vibrations and Waves. John Murray. ISBN 0-7195-2882-8.
* H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons. ISBN 0-471-90182-2.

## Further reading

* L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0. https://archive.org/details/physicswithmoder0000gree.
* J.B. Marion, W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
* A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
* H.D. Young, R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 978-0-321-50130-1.

* v
* t
* e

SI units

Base units|

* ampere
* candela
* kelvin
* kilogram
* {{{sp}}}}}|us|er|re}}
* mole
* second

Derived units
with special names|

* becquerel
* coulomb
* degree Celsius
* farad
* gray
* henry
* hertz
* joule
* katal
* lumen
* lux
* newton
* ohm
* pascal
* radian
* siemens
* sievert
* steradian
* tesla
* volt
* watt
* weber

Other accepted units|

* astronomical unit
* dalton
* day
* decibel
* degree of arc
* electronvolt
* hectare
* hour
* {{{sp}}}}}|us|er|re}}
* minute
* minute and second of arc
* neper
* tonne

See also|

* Conversion of units
* Metric prefixes
* 2005–2019 definition
* 2019 redefinition
* Systems of measurement

* Book
* Category

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