Adimensional

Adimensional (or dimensionless) refers to a quantity that has no physical dimensions and is expressed as a pure number. Such quantities arise from ratios of like quantities or from nondimensionalization, where units are removed by scaling variables with characteristic reference values. Dimensionless quantities are invariant under changes of units and are central to comparing different systems on a common basis.

In practice, adimensionalization involves choosing characteristic scales for variables (for example, a characteristic length, time, or velocity) and forming dimensionless variables by dividing by these scales. Substituting these into governing equations yields a dimensionless form in which physical effects appear as dimensionless parameters. This process clarifies which effects dominate under given conditions and can reduce the number of independent parameters.

A formal approach to finding dimensionless groups is Buckingham Pi theorem, which states that any physically meaningful relation among a set of variables can be rewritten as a relationship among a set of dimensionless products (Pi terms). This method explains why many complex systems can be described by a small number of governing dimensionless parameters.

Common examples of adimensional numbers include the Reynolds number (ratio of inertial to viscous forces in a fluid), the Mach number (ratio of flow speed to sound speed), the Prandtl number (ratio of momentum to thermal diffusivity), and the Nusselt number (ratio of convective to conductive heat transfer). Simpler dimensionless ratios, such as length or time ratios, also fall into this category.

Applications of adimensional analysis span fluid dynamics, heat and mass transfer, chemical kinetics, and numerical simulation, where nondimensional forms simplify scaling, enable data collapse, and improve computational stability.