Dimensionless numbers in fluid mechanics

Definition

Fix base dimensions, such as mass $\mathsf{M}$ , length $\mathsf{L}$ , time $\mathsf{T}$ and temperature $\Theta$ . Each positive physical quantity $q$ of a flow has a dimension $[q] = \mathsf{M}^{a} \mathsf{L}^{b} \mathsf{T}^{c} \Theta^{d}$ with rational exponents $a, b, c, d$ . Let $q_1, \dots, q_n$ be quantities of the flow and let $k_1, \dots, k_n$ be rational numbers. The product $\Pi = q_1^{k_1} \cdots q_n^{k_n}$ is a dimensionless number if all the exponents of $[\Pi] = [q_1]^{k_1} \cdots [q_n]^{k_n}$ are zero.

Remarks

The value of a dimensionless number does not change when the units of the base dimensions are changed. Example: with density $\rho$ , speed $U$ , length $L$ and dynamic viscosity $\mu$ , $[\rho U L / \mu] = (\mathsf{M} \mathsf{L}^{-3})(\mathsf{L} \mathsf{T}^{-1})(\mathsf{L}) / (\mathsf{M} \mathsf{L}^{-1} \mathsf{T}^{-1}) = 1$ , so $\rho U L / \mu$ is dimensionless; it is the Reynolds number. Non-example: $\rho U L$ has dimension $\mathsf{M} \mathsf{L}^{-1} \mathsf{T}^{-1}$ , so it is not dimensionless. Most dimensionless numbers of fluid mechanics are ratios of two forces, two time scales or two transport rates.

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Fluid mechanics concept
Dimensionless numbers in fluid mechanics definition

Tag	`MB88KZ691Q`
URI	`m2://MB88KZ691Q`
Kind	definition
Writing	W0 stub
MSC	76M55, 00A71
Defines	Dimensionless numbers in fluid mechanics
Uses	Fluid mechanics
Changed	3 November 2026

Dimensionless numbers in fluid mechanics

Definition

Remarks

Reading path

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