Quantities on Encyclopedia of dimensionless numbers

angular velocity

Mon, 01 Jan 0001 00:00:00 +0000

angular velocity#

$$\Omega \sim \text T^{-1}$$

Rate of rotation of a frame, body, or flow. It sets the Coriolis and rotational time scale in geophysical, turbomachinery, and rotating-container flows.

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bulk modulus

Mon, 01 Jan 0001 00:00:00 +0000

bulk modulus#

$$K \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Elastic resistance to uniform compression, defined as the pressure increase required to produce a relative decrease in volume. It sets the stress scale for compressibility and elastic wave response.

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Cauchy number

characteristic length

Mon, 01 Jan 0001 00:00:00 +0000

characteristic length#

$$L \sim \text L$$

A geometric scale representative of the problem: pipe diameter, boundary-layer thickness, chord length, or droplet radius. Its choice determines which physical effects are captured by a given dimensionless group.

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convective heat transfer coefficient

Mon, 01 Jan 0001 00:00:00 +0000

convective heat transfer coefficient#

$$h \sim \text M\,\Theta^{-1}\,\text T^{-3}$$

Rate of convective heat transfer per unit surface area per unit temperature difference between a surface and the adjacent fluid. Encapsulates geometry, flow conditions, and fluid properties into a single engineering quantity.

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diffusivity

Mon, 01 Jan 0001 00:00:00 +0000

diffusivity#

$$D \sim \text L^{2}\,\text T^{-1}$$

Proportionality constant in Fick’s first law, relating the diffusive mass flux of a species to its concentration gradient. Depends strongly on temperature, pressure, and the solute–solvent pair.

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dynamic viscosity

Mon, 01 Jan 0001 00:00:00 +0000

dynamic viscosity#

$$\mu \sim \text L^{-1}\,\text M\,\text T^{-1}$$

Resistance of a fluid to shear deformation. In a Newtonian fluid it relates shear stress linearly to the velocity gradient. It decreases with temperature for liquids and increases for gases.

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force

Mon, 01 Jan 0001 00:00:00 +0000

force#

$$F \sim \text L\,\text M\,\text T^{-2}$$

Mechanical load exerted on a body or control surface. In aerodynamic and hydrodynamic coefficients it is normalized by dynamic pressure and reference area.

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frequency

Mon, 01 Jan 0001 00:00:00 +0000

frequency#

$$f \sim \text T^{-1}$$

Number of complete oscillation cycles per unit time. In fluid mechanics it characterises periodic phenomena such as vortex shedding, acoustic resonance, or externally imposed oscillations.

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gravitational acceleration

Mon, 01 Jan 0001 00:00:00 +0000

gravitational acceleration#

$$g \sim \text L\,\text T^{-2}$$

Acceleration imparted by gravity at the Earth’s surface (~9.81 m/s²). Sets the restoring force for surface waves and drives buoyancy-induced flows wherever density gradients are present.

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kinematic viscosity

Mon, 01 Jan 0001 00:00:00 +0000

kinematic viscosity#

$$\nu \sim \text L^{2}\,\text T^{-1}$$

Momentum diffusivity of a fluid, equal to dynamic viscosity divided by density. It measures how quickly viscous momentum disturbances spread through the fluid.

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local-reference pressure difference

Mon, 01 Jan 0001 00:00:00 +0000

local-reference pressure difference#

$$p - p_\infty \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Difference between local static pressure and a reference or free-stream static pressure. It is the pressure contrast used to express local pressure loading on a body or surface.

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Pressure coefficient

mass density

Mon, 01 Jan 0001 00:00:00 +0000

mass density#

$$\rho \sim \text L^{-3}\,\text M$$

Mass per unit volume of a substance. A fundamental thermodynamic property that depends on temperature and pressure; its spatial variation underpins buoyancy-driven flows and stratification.

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mass transfer coefficient

Mon, 01 Jan 0001 00:00:00 +0000

mass transfer coefficient#

$$k_m \sim \text L\,\text T^{-1}$$

Convective mass-transfer analogue of the heat-transfer coefficient. Relates the convective species flux to the concentration difference between an interface and the bulk fluid; depends on geometry, flow conditions, and diffusivity.

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Sherwood number

mean free path

Mon, 01 Jan 0001 00:00:00 +0000

mean free path#

$$\lambda \sim \text L$$

Average distance a molecule travels between collisions. It provides the microscopic length scale used to decide whether a gas can be treated as a continuum.

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Knudsen number

pressure drop

Mon, 01 Jan 0001 00:00:00 +0000

pressure drop#

$$p_1 - p_2 \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Static pressure decrease between an upstream station and a downstream station. It acts as the driving force for conduit flow and measures losses due to friction, fittings, or geometry changes.

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pressure margin to vapor pressure

Mon, 01 Jan 0001 00:00:00 +0000

pressure margin to vapor pressure#

$$p - p_v \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Difference between the local or reference static pressure and the liquid vapor pressure. It measures how much pressure head remains before vapor cavities can form.

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Cavitation number

reference area

Mon, 01 Jan 0001 00:00:00 +0000

reference area#

$$A \sim \text L^{2}$$

Characteristic surface or projected area used to normalize forces, fluxes, or loads. The appropriate reference area is set by the geometry and convention for the coefficient being formed.

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shear stress

Mon, 01 Jan 0001 00:00:00 +0000

shear stress#

$$\tau \sim \text L^{-1}\,\text M\,\text T^{-2}$$

Tangential stress transmitted across a surface by viscous or turbulent momentum transport. It is commonly normalized by dynamic pressure in wall-friction and sediment-transport parameters.

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solid-fluid density difference

Mon, 01 Jan 0001 00:00:00 +0000

solid-fluid density difference#

$$\rho_s - \rho_f \sim \text L^{-3}\,\text M$$

Density of a solid particle or sediment grain minus the surrounding fluid density. It sets the submerged-weight contrast for settling, suspension, and sediment transport.

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specific heat capacity

Mon, 01 Jan 0001 00:00:00 +0000

specific heat capacity#

$$c_p \sim \text L^{2}\,\Theta^{-1}\,\text T^{-2}$$

Energy required to raise the temperature of unit mass by one kelvin at constant pressure. It sets the sensible heat storage scale in convective heat transfer and compressible-flow heating estimates.

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speed of sound

Mon, 01 Jan 0001 00:00:00 +0000

speed of sound#

$$c \sim \text L\,\text T^{-1}$$

Phase speed of small-amplitude compressional waves in a medium. Determined by the ratio of the isentropic bulk modulus to density; sets the compressibility scale and serves as the reference speed in the Mach number.

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surface tension

Mon, 01 Jan 0001 00:00:00 +0000

surface tension#

$$\sigma \sim \text M\,\text T^{-2}$$

Energy per unit interfacial area (equivalently, force per unit contact-line length) arising from the imbalance of intermolecular cohesive forces at a phase boundary. Drives capillary rise, droplet formation, and interface stabilisation.

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surface-ambient temperature difference

Mon, 01 Jan 0001 00:00:00 +0000

surface-ambient temperature difference#

$$T_s - T_\infty \sim \Theta$$

Temperature difference between a surface and the surrounding or free-stream fluid. It is the driving temperature contrast for many convection, buoyancy, and wall-heating estimates.

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thermal conductivity

Mon, 01 Jan 0001 00:00:00 +0000

thermal conductivity#

$$k \sim \text L\,\text M\,\Theta^{-1}\,\text T^{-3}$$

Intrinsic ability of a material to transfer heat by conduction. Relates the conductive heat flux to the local temperature gradient via Fourier’s law; generally weakly temperature-dependent for solids, but strongly so for gases and liquids.

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thermal diffusivity

Mon, 01 Jan 0001 00:00:00 +0000

thermal diffusivity#

$$\alpha \sim \text L^{2}\,\text T^{-1}$$

Ratio of thermal conductivity to volumetric heat capacity; governs the rate at which temperature disturbances propagate through a material. A high value indicates rapid thermal equilibration relative to heat storage.

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thermal expansion coefficient

Mon, 01 Jan 0001 00:00:00 +0000

thermal expansion coefficient#

$$\beta \sim \Theta^{-1}$$

Fractional density or volume change per unit temperature change. In buoyancy-driven flow it converts a temperature difference into a density contrast under the Boussinesq approximation.

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time

Mon, 01 Jan 0001 00:00:00 +0000

time#

$$t \sim \text T$$

Elapsed time, used as the independent variable in transient analyses. Sets the temporal scale against which diffusion, convection, or oscillation rates are compared in unsteady dimensionless groups.

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velocity

Mon, 01 Jan 0001 00:00:00 +0000

velocity#

$$U \sim \text L\,\text T^{-1}$$

Rate of displacement of a fluid element or body. In dimensionless analysis it typically represents a characteristic (bulk, free-stream, or mean) speed that sets the advective time scale.