List of equations in fluid mechanics

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This article summarizes equations in the theory of fluid mechanics.


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


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
  • 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 u^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{u}^2 \right ) \,\!

Convective acceleration \mathbf{a} = \left ( \mathbf{u} \cdot \nabla \right ) \mathbf{u}
Navier–Stokes equations
 \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}

See also[edit]


  • P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1. 
  • G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2. 
  • A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4. 
  • R.G. Lerner, G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4. 
  • C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3. 
  • P.A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 9-781429-202657. 
  • L.N. Hand, J.D. Finch (2008). Analytical Mechanics. Cambridge University Press,. ISBN 978-0-521-57572-0. 
  • T.B. Arkill, C.J. Millar (1974). Mechanics, Vibrations and Waves. John Murray,. ISBN 0-7195-2882-8. 
  • H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons,. ISBN 0-471-90182-2. 

Further reading[edit]

  • L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0. 
  • J.B. Marion, W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2. 
  • A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1. 
  • H.D. Young, R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321-50130-6.