# 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 ${\displaystyle \mathbf {\hat {t}} \,\!}$ 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 ${\displaystyle \mathbf {u} =\mathbf {u} \left(\mathbf {r} ,t\right)\,\!}$ m s−1 [L][T]−1
Velocity pseudovector field ω ${\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {v} }$ s−1 [T]−1
Volume velocity, volume flux φV (no standard symbol) ${\displaystyle \phi _{V}=\int _{S}\mathbf {u} \cdot \mathrm {d} \mathbf {A} \,\!}$ m3 s−1 [L]3 [T]−1
Mass current per unit volume s (no standard symbol) ${\displaystyle s=\mathrm {d} \rho /\mathrm {d} t\,\!}$ kg m−3 s−1 [M] [L]−3 [T]−1
Mass current, mass flow rate Im ${\displaystyle I_{\mathrm {m} }=\mathrm {d} m/\mathrm {d} t\,\!}$ kg s−1 [M][T]−1
Mass current density jm ${\displaystyle I_{\mathrm {m} }=\iint \mathbf {j} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {S} \,\!}$ kg m−2 s−1 [M][L]−2[T]−1
Momentum current Ip ${\displaystyle I_{\mathrm {p} }=\mathrm {d} \left|\mathbf {p} \right|/\mathrm {d} t\,\!}$ kg m s−2 [M][L][T]−2
Momentum current density jp ${\displaystyle I_{\mathrm {p} }=\iint \mathbf {j} _{\mathrm {p} }\cdot \mathrm {d} \mathbf {S} }$ 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
${\displaystyle \nabla P=\rho \mathbf {g} \,\!}$
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

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

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

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

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

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

## Sources

• 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 978-1-4292-0265-7.
• 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.