# List of equations in 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 $\mathbf{\hat{t}} \,\!$ 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
Vorticity pseudovector field ω $\boldsymbol{\omega} = \nabla\times\mathbf{v}$ s−1 [T]−1
Volume velocity, volume flux φV (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 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,
• r = Position
• ρ = ρ(r) = Fluid density at gravitational equipotential containing r
• g = g(r) = Gravitational field strength at point r
$\nabla P = \rho \mathbf{g}\,\!$
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

$\mathbf{F}_\mathrm{b} = - \rho_f V_\mathrm{imm} \mathbf{g} = - \mathbf{F}_\mathrm{g}\,\!$

Apparent weight
$\mathbf{W}_\mathrm{app} = \mathbf{W} - \mathbf{F}_\mathrm{b}\,\!$

Bernoulli's equation pconstant is the total pressure at a point on a streamline $p + \rho v^2/2 + \rho gy = p_\mathrm{constant}\,\!$
Euler equations
$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0\,\!$

$\frac{\partial\rho{\mathbf{u}}}{\partial t} + \nabla \cdot \left ( \mathbf{u}\otimes \left ( \rho \mathbf{u} \right ) \right )+\nabla p=0\,\!$
$\frac{\partial E}{\partial t}+\nabla\cdot\left ( \bold u \left ( E+p \right ) \right ) = 0 \,\!$
$E = \rho \left ( U + \frac{1}{2} \mathbf{v}^2 \right ) \,\!$

Convective acceleration $\mathbf{a} = \left ( \mathbf{v} \cdot \nabla \right ) \mathbf{v}$
Navier–Stokes equations
• TD = Deviatoric stress tensor
• $\mathbf{f}$ = volume density of the body forces acting on the fluid
• $\nabla$ here is the del operator.
$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbf{T}_\mathrm{D} + \mathbf{f}$

## Sources

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