List of equations in fluid mechanics

Short description: None

This article summarizes equations in the theory of fluid mechanics.

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	φ_V (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	I_m	[math]\displaystyle{ I_\mathrm{m} = \mathrm{d} m/\mathrm{d} t \,\! }[/math]	kg s⁻¹	[M][T]⁻¹
Mass current density	j_m	[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	I_p	[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	j_p	[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	ρ_f = Mass density of the fluid V_imm = Immersed volume of body in fluid F_b = Buoyant force F_g = Gravitational force W_app = 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	p_constant 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	T_D = 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]

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.

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

List of equations in fluid mechanics

Topic: Physics

Contents

Definitions

Equations

See also

Sources

Further reading