Mathematical simplification technique in physical sciences
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Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by differential equations. One important use is in the analysis of control systems.
One of the simplest characteristic units is the doubling time of a system experiencing exponential growth, or conversely the half-life of a system experiencing exponential decay; a more natural pair of characteristic units is mean age/mean lifetime, which correspond to base e rather than base 2.
Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following:
List of dynamical systems and differential equations topics
List of partial differential equation topics
Differential equations of mathematical physics
Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization in statistics.
Measuring devices are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.
and 22 Related for: Nondimensionalization information
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution...
if convection is present. The Fourier number arises naturally in nondimensionalization of the heat equation. The general definition of the Fourier number...
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calculate the initiation of motion of sediment in a fluid flow. It is a nondimensionalization of a shear stress, and is typically denoted τ ∗ {\displaystyle \tau...
physics, however, this scruple may be set aside, by a process called nondimensionalization. The effective result is that many fundamental equations of physics...
which can be used to simplify the problem. These can be found by nondimensionalization. The result is that, if energy is measured in units of ħω and distance...
c^{2}}}{Dp \over {Dt}}=\nabla \cdot {\bf {u}}} The next step is to nondimensionalize the variables as such: x ∗ = x / L , t ∗ = U t / L , u ∗ = u / U ...
be reduced to this form.[citation needed] This is done through nondimensionalization. If the forcing function is f(t) = cos(ωt) = cos(ωtcτ) = cos(ωτ)...
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quantities are normalized against like-dimensioned quantities (called nondimensionalization), resulting in only dimensionless quantities remaining. Physicists...
flow, including moving and deforming grids Dimensional units and nondimensionalization information Reference states Convergence history Association to...
{x}}} and, assuming high Reynolds number flow, it is possible to nondimensionalize the variables with the scalings: p d = ρ V 2 p d ′ , ∇ = L − 1 ∇ ′...
for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation. Mathematica also...
spinodal time. The spinodal length and spinodal time can be used to nondimensionalize the equation of motion, resulting in universal scaling for spinodal...
{1+\cos(\theta )}{2}},} and then expanding both dy⁄dx and γ(x) as a nondimensionalized Fourier series in θ with a modified lead term: d y d x = A 0 + A 1...
and P, normally called field quadratures in quantum optics. (See Nondimensionalization.) These operators are related to the position and momentum operators...
the mass and frequency of the oscillator are equal to one (see nondimensionalization), the energy of the oscillator is H = 1 2 ( P 2 + X 2 ) . {\displaystyle...
units can be entered into them or units can be removed at all by nondimensionalization. Note that only we use column vectors (in the Cartesian coordinate...
contributing to the body's motion. These considerations are captured in the nondimensionalized Rossby number. In stability calculations, the rate of change of f...
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