List of equations in fluid mechanics

This article summarizes equations in the theory of fluid mechanics.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Flow velocity vector field	u	$\mathbf {u} =\mathbf {u} \left(\mathbf {r} ,t\right)\,\!$	m s⁻¹	[L][T]⁻¹
Velocity pseudovector field	ω	${\boldsymbol {\omega }}=\nabla \times \mathbf {v}$	s⁻¹	[T]⁻¹
Volume velocity, volume flux	φ_V (no standard symbol)	$\phi _{V}=\int _{S}\mathbf {u} \cdot \mathrm {d} \mathbf {A} \,\!$	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	I_m	$I_{\mathrm {m} }=\mathrm {d} m/\mathrm {d} t\,\!$	kg s⁻¹	[M][T]⁻¹
Mass current density	j_m	$I_{\mathrm {m} }=\iint \mathbf {j} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {S} \,\!$	kg m⁻² s⁻¹	[M][L]⁻²[T]⁻¹
Momentum current	I_p	$I_{\mathrm {p} }=\mathrm {d} \left\|\mathbf {p} \right\|/\mathrm {d} t\,\!$	kg m s⁻²	[M][L][T]⁻²
Momentum current density	j_p	$I_{\mathrm {p} }=\iint \mathbf {j} _{\mathrm {p} }\cdot \mathrm {d} \mathbf {S}$	kg m s⁻²	[M][L][T]⁻²

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	$\nabla P=\rho \mathbf {g} \,\!$
Buoyancy equations	ρ_f = Mass density of the fluid V_imm = Immersed volume of body in fluid F_b = Buoyant force F_g = Gravitational force W_app = Apparent weight of immersed body W = Actual weight of immersed body	Buoyant force $\mathbf {F} _{\mathrm {b} }=-\rho _{f}V_{\mathrm {imm} }\mathbf {g} =-\mathbf {F} _{\mathrm {g} }\,\!$ Apparent weight $\mathbf {W} _{\mathrm {app} }=\mathbf {W} -\mathbf {F} _{\mathrm {b} }\,\!$
Bernoulli's equation	p_constant is the total pressure at a point on a streamline	$p+\rho u^{2}/2+\rho gy=p_{\mathrm {constant} }\,\!$
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 $\otimes$ denotes the tensor product	${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0\,\!$ ${\frac {\partial \rho {\mathbf {u} }}{\partial t}}+\nabla \cdot \left(\mathbf {u} \otimes \left(\rho \mathbf {u} \right)\right)+\nabla p=0\,\!$ ${\frac {\partial E}{\partial t}}+\nabla \cdot \left(\mathbf {u} \left(E+p\right)\right)=0\,\!$ $E=\rho \left(U+{\frac {1}{2}}\mathbf {u} ^{2}\right)\,\!$
Convective acceleration		$\mathbf {a} =\left(\mathbf {u} \cdot \nabla \right)\mathbf {u}$
Navier–Stokes equations	T_D = Deviatoric stress tensor $\mathbf {f}$ = volume density of the body forces acting on the fluid $\nabla$ here is the del operator.	$\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla p+\nabla \cdot \mathbf {T} _{\mathrm {D} }+\mathbf {f}$

v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
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List of equations in fluid mechanics

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