List_of_equations_in_fluid_mechanics Knowpia

This article summarizes equations in the theory of fluid mechanics.

Definitions edit

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 through 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
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

Equations edit

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

Sources 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 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.

Summary

Definitions edit

Equations edit

See also edit

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Further reading edit