DBAR problem

Template:Continuum 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 though 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
Vorticity 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	$p+\rho v^2/2+\rho gy=p_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$
Euler equations	ρ = fluid mass density u is the fluid velocity vector E = total volume energy density U = internal energy per unit mass of fluid p = pressure $\otimes$ denotes the tensor product	$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{u})=0$ $\frac{\partial \rho \mathbf{u}}{\partial t}+\mathrm{\nabla }\cdot (\mathbf{u}\otimes \left(\rho \mathbf{u}\right))+\mathrm{\nabla }p=0$ $\frac{\partial E}{\partial t}+\mathrm{\nabla }\cdot \left(\mathbf{u}(E+p)\right)=0$ $E=\rho (U+\frac{1}{2}{\mathbf{v}}^2)$
Convective acceleration		$\mathbf{a}=(\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}$
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.	$\rho (\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla }\mathbf{v})=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot {\mathbf{T}}_{\mathrm{D}}+\mathbf{f}$

See also

• Defining equation (physical chemistry)
• List of equations in classical mechanics
• Table of thermodynamic equations
• List of relativistic equations
• List of equations in gravitation
• List of electromagnetism equations
• List of photonics equations
• List of equations in quantum mechanics
• List of equations in nuclear and particle physics

Footnotes

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Sources

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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