List of equations in fluid mechanics explained

This article summarizes equations in the theory of fluid mechanics.

Definitions

Here

\hat{t } \,\! 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	u = u \left ( r ,t \right )	m s^−1	[L][T]^−1
Velocity pseudovector field	ω	\boldsymbol{\omega} = \nabla x v	s^−1	[T]^−1
Volume velocity, volume flux	φV (no standard symbol)	\phi _{V} = \int _{S} u ⋅ d A	m^3 s^−1	[L]^3 [T]^−1
Mass current per unit volume	s (no standard symbol)	s = d \rho / d t	kg m^-3 s^−1	[M] [L]^-3 [T]^−1
Mass current, mass flow rate	Im	I _{m} = d m/ d t	kg s^−1	[M][T]^−1
Mass current density	jm	I _{m} = \iint j _{m} ⋅ d S	kg m^−2 s^−1	[M][L]^−2[T]^−1
Momentum current	Ip	I _{p} = d \left	\mathbf \right	/\mathrm t \,\!	kg m s^−2	[M][L][T]^−2
Momentum current density	jp	I _{p} = \iint j _{p} ⋅ d S	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	\nabla P = \rho g
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 F _{b} = - \rho _{f} V _{imm} g = - F _{g} Apparent weight W _{app} = W - F _{b\}
Bernoulli's equation	pconstant is the total pressure at a point on a streamline	p + \rho u ^{2/2} + \rho gy = p _{constant}
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 ⊗ denotes the tensor product	(\partial\rho)/(\partial t) + \nabla ⋅ (\rho u ) =0 (\partial\rho{ u)/() } + \nabla \cdot \left (\mathbf\otimes \left (\rho \mathbf \right) \right)+\nabla p=0\,\	(\partial E)/(\partial t) + \nabla ⋅ \left ( u \left ( E + p \right ) \right ) = 0 E = \rho \left ( U + (1)/(2) u ^{2} \right ) \
Convective acceleration		a = \left ( u ⋅ \nabla \right ) u
Navier–Stokes equations	TD = Deviatoric stress tensor f = volume density of the body forces acting on the fluid\nabla here is the del operator.	\rho \left( (\partial u)/(\partial t) + u ⋅ \nabla u \right) = - \nabla p + \nabla ⋅ T _{D} + f

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

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

Further reading

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