# List of equations in fluid mechanics

Part of a series on :
Continuum mechanics :
$J=-D{\frac {d\varphi }{dx}}$Fick's laws of diffusion :
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Mass
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Clausius–Duhem (entropy) :
Solid mechanics
Deformation
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linear
Plasticity
Hooke's law
Stress
Strain
Finite strain
Infinitesimal strain
Compatibility
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Contact mechanics
frictional
Material failure theory
Fracture mechanics :
Fluid mechanics
Fluids
Statics · Dynamics
Archimedes' principle · Bernoulli's principle
Navier–Stokes equations
Poiseuille equation · Pascal's law
Viscosity
(Newtonian · non-Newtonian)
Buoyancy · Mixing · Pressure
Liquids
Adhesion
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Chromatography
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Surface tension
Gases
Atmosphere
Boyle's law
Charles's law
Combined gas law
Fick's law
Gay-Lussac's law
Graham's law
Plasma :
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Viscoelasticity
Rheometry
Rheometer
Smart fluids
Electrorheological
Magnetorheological
Ferrofluids :
Scientists
Bernoulli
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Cauchy
Charles
Euler
Fick
Gay-Lussac
Graham
Hooke
Newton
Navier
Noll
Pascal
Stokes
Truesdell :
vte :
This article summarizes equations in the theory of fluid mechanics.

## Definitions

Here $\mathbf {\hat {t}} \,\!$ 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 | $\mathbf {u} =\mathbf {u} \left(\mathbf {r} ,t\right)\,\!$ | 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) | $\phi _{V}=\int _{S}\mathbf {u} \cdot \mathrm {d} \mathbf {A} \,\!$ | m$^3$ 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]$^{−2}$ |

## 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 | $\nabla P=\rho \mathbf {g} \,\!$ |
| 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. | $\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla p+\nabla \cdot \mathbf {T} _{\mathrm {D} }+\mathbf {f}$ |