List of equations in fluid mechanics

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 $\widehat{𝐭}$ 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	$𝐮=𝐮(𝐫,t)$	m s−1	[L][T]−1
Velocity pseudovector field	ω	$𝝎=\mathrm{\nabla }\times 𝐯$	s−1	[T]−1
Volume velocity, volume flux	φV (no standard symbol)	${\varphi }_V={\int }_S𝐮\cdot \mathrm{d}𝐀$	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	$I_{\mathrm{m}}=\iint {𝐣}_{\mathrm{m}}\cdot \mathrm{d}𝐒$	kg m−2 s−1	[M][L]−2[T]−1
Momentum current	Ip	$I_{\mathrm{p}}=\mathrm{d}\left|𝐩\right|/\mathrm{d}t$	kg m s−2	[M][L][T]−2
Momentum current density	jp	$I_{\mathrm{p}}=\iint {𝐣}_{\mathrm{p}}\cdot \mathrm{d}𝐒$	kg m s−2	[M][L][T]−2

Equations

Physical situation	Nomenclature	Equations
Fluid statics, pressure gradient	Template:Plainlist r = Position ρ = ρ(r) = Fluid density at gravitational equipotential containing r g = g(r) = Gravitational field strength at point r ∇P = Pressure gradient	$\mathrm{\nabla }P=\rho 𝐠$
Buoyancy equations	Template:Plainlist 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 ${𝐅}_{\mathrm{b}}=-{\rho }_f{V}_{\mathrm{i}\mathrm{m}\mathrm{m}}𝐠=-{𝐅}_{\mathrm{g}}$ Apparent weight ${𝐖}_{\mathrm{a}\mathrm{p}\mathrm{p}}=𝐖-{𝐅}_{\mathrm{b}}$
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	Template:Plainlist 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	$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho 𝐮)=0$ $\frac{\partial \rho 𝐮}{\partial t}+\mathrm{\nabla }\cdot (𝐮\otimes \left(\rho 𝐮\right))+\mathrm{\nabla }p=0$ $\frac{\partial E}{\partial t}+\mathrm{\nabla }\cdot \left(𝐮(E+p)\right)=0$ $E=\rho (U+\frac{1}{2}{𝐮}^2)$
Convective acceleration		$𝐚=(𝐮\cdot \mathrm{\nabla })𝐮$
Navier–Stokes equations	Template:Plainlist TD = Deviatoric stress tensor $𝐟$ = volume density of the body forces acting on the fluid $\mathrm{\nabla }$ here is the del operator.	$\rho (\frac{\partial 𝐮}{\partial t}+𝐮\cdot \mathrm{\nabla }𝐮)=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot {𝐓}_{\mathrm{D}}+𝐟$

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

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Further reading

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