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 $\hat{\mathbf{t}}$ 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	$\mathbf{𝐮}=\mathbf{𝐮}(\mathbf{𝐫},t)$	m s−1	[L][T]−1
Velocity pseudovector field	ω	$\mathit{\omega }=\mathrm{\nabla }\times \mathbf{𝐯}$	s−1	[T]−1
Volume velocity, volume flux	φV (no standard symbol)	${\varphi }_V=\underset{S}{\int }\mathbf{𝐮}\cdot \mathrm{d}\mathbf{𝐀}$	m3 s−1	[L]3 [T]−1
Mass current per unit volume	s (no standard symbol)	$s=\mathrm{d}\rho /\mathrm{d}t$	kg m−3 s−1	[M] [L]−3 [T]−1
Mass current, mass flow rate	Im	$I_{\mathrm{m}}=\mathrm{d}m/\mathrm{d}t$	kg s−1	[M][T]−1
Mass current density	jm	$I_{\mathrm{m}}=\iint {\mathbf{𝐣}}_{\mathrm{m}}\cdot \mathrm{d}\mathbf{𝐒}$	kg m−2 s−1	[M][L]−2[T]−1
Momentum current	Ip	$I_{\mathrm{p}}=\mathrm{d}\left|\mathbf{𝐩}\right|/\mathrm{d}t$	kg m s−2	[M][L][T]−2
Momentum current density	jp	$I_{\mathrm{p}}=\iint {\mathbf{𝐣}}_{\mathrm{p}}\cdot \mathrm{d}\mathbf{𝐒}$	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	$\mathrm{\nabla }P=\rho \mathbf{𝐠}$
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 ${\mathbf{𝐅}}_{\mathrm{b}}=-{\rho }_f{V}_{\mathrm{i}\mathrm{m}\mathrm{m}}\mathbf{𝐠}=-{\mathbf{𝐅}}_{\mathrm{g}}$ Apparent weight ${\mathbf{𝐖}}_{\mathrm{a}\mathrm{p}\mathrm{p}}=\mathbf{𝐖}-{\mathbf{𝐅}}_{\mathrm{b}}$
Bernoulli's equation	pconstant is the total pressure at a point on a streamline	$p+\rho u^2/2+\rho gy=p_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$
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 $\otimes$ denotes the tensor product	$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{𝐮})=0$ $\frac{\partial \rho \mathbf{𝐮}}{\partial t}+\mathrm{\nabla }\cdot (\mathbf{𝐮}\otimes \left(\rho \mathbf{𝐮}\right))+\mathrm{\nabla }p=0$ $\frac{\partial E}{\partial t}+\mathrm{\nabla }\cdot \left(\mathbf{𝐮}(E+p)\right)=0$ $E=\rho (U+\frac{1}{2}{\mathbf{𝐮}}^2)$
Convective acceleration		$\mathbf{𝐚}=(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}$
Navier–Stokes equations	TD = Deviatoric stress tensor $\mathbf{𝐟}$ = volume density of the body forces acting on the fluid $\mathrm{\nabla }$ here is the del operator.	$\rho (\frac{\partial \mathbf{𝐮}}{\partial t}+\mathbf{𝐮}\cdot \mathrm{\nabla }\mathbf{𝐮})=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot {\mathbf{𝐓}}_{\mathrm{D}}+\mathbf{𝐟}$

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

• 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.