Physics:List of equations in fluid mechanics

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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	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.