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Mathematical simplification technique in physical sciences
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution
Nondimensionalization
Units defined only by physical constants
physics, however, this scruple may be set aside, by a process called nondimensionalization. The effective result is that many fundamental equations of physics
Planck_units
Dimensionless quantity related to transient heat conduction
if convection is present. The Fourier number arises naturally in nondimensionalization of the heat equation. The general definition of the Fourier number
Fourier_number
Model describing the departures from ideality in solutions of electrolytes and plasmas
a\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=Z_{0}} To obtain the nondimensionalized differential equation and initial conditions, use the π {\displaystyle
Debye–Hückel_theory
Units of measurement based on universal physical constants
for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light c may be set
Natural_units
Nondimensional heat transfer coefficient
The Nusselt number is a nondimensionalization of the convective heat transfer coefficient. Like the heat transfer coefficient, the Nusselt number may
Nusselt_number
Difference between a variable's observed value and a reference value
studentizing (e.g., Studentized residual). Another approach to nondimensionalization focuses on scaling by location rather than dispersion. The percent
Deviation_(statistics)
Dimensionless quantity in fluid dynamics
The Mach number arises naturally when the continuity equation is nondimensionalized for compressible flows. If density variations are related to pressure
Mach_number
Foundational principle in quantum physics
{\displaystyle \omega _{0}>0} describing the width of the distribution—cf. nondimensionalization. If the state is allowed to evolve in free space, then the time-dependent
Uncertainty_principle
Theorem in dimensional analysis
computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown. The Buckingham
Buckingham_pi_theorem
Quantum mechanical model
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
Quantum_harmonic_oscillator
Short "burst" or "envelope" of restricted wave action that travels as a unit
)[x-(B^{3}t^{2}/6m^{2})]},} where, for simplicity (and nondimensionalization), choosing ħ = 1, m = 1/2, and B an arbitrary constant results in
Wave_packet
Mechanism of spontaneous phase separation
spinodal time. The spinodal length and spinodal time can be used to nondimensionalize the equation of motion, resulting in universal scaling for spinodal
Spinodal_decomposition
Mathematical model
u ( r , t ) {\displaystyle \mathbf {q} _{u}(\mathbf {r} ,t)} are nondimensionalized noises. The choice of free energy function, f ( φ ) {\displaystyle
Phase-field_model
Dimensionless number in fluid mechanics
{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 ∇ ′
Weber_number
Movement of solid particles, typically by gravity and fluid entrainment
dimensionless critical shear stress τ c ∗ {\displaystyle \tau _{c}*} . The nondimensionalization is in order to compare the driving forces of particle motion (shear
Sediment_transport
Equations describing classical electromagnetism
system of quantities is chosen in the speed of light, c, is used for nondimensionalization, so that, for example, seconds and lightseconds are interchangeable
Maxwell's_equations
Distance an aircraft can fly between takeoff and landing
is often useful to separate these terms. Doing so completes the nondimensionalization of the range equation into fundamental design disciplines of aeronautics
Range_(aeronautics)
flow, including moving and deforming grids Dimensional units and nondimensionalization information Reference states Convergence history Association to
CGNS
Non-mainstream theory in physics
quantities are normalized against like-dimensioned quantities (called nondimensionalization), resulting in only dimensionless quantities remaining. Only variation
Variable_speed_of_light
velocity scale (used for nondimensionalization), R 0 {\displaystyle R_{0}} , the length scale (used for nondimensionalization), R = R ( t ) = R 0 2 + 4
Batchelor_vortex
German mechanical engineer and professor (1882–1957)
engineer whose work helped establish modern heat-transfer theory. His nondimensionalization of convective heat transfer led to a widely used framework for correlating
Wilhelm_Nusselt
Physical system that responds to a restoring force proportional to displacement
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)
Harmonic_oscillator
Physical property
types of turbines. To deal with this, nondimensionalization is applied to various qualities. Nondimensionalization allows one to make comparisons between
Wind-turbine_aerodynamics
Dimensionless number in fluid dynamics
squared. The number is named after Henry Darcy and is found from nondimensionalizing the differential form of Darcy's law. This number should not be confused
Darcy_number
Analysis of the dimensions of different physical quantities
for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation. Mathematica also
Dimensional_analysis
Streamlined body for generating lift
{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
Airfoil
Specific quantum state of a quantum harmonic oscillator
and P, normally called field quadratures in quantum optics. (See Nondimensionalization.) These operators are related to the position and momentum operators
Coherent_state
Asymmetry between the two acting surfaces of an airfoil
(note that the lines over the variables indicates that they have been nondimensionalized by dividing through by the chord): Z ¯ ( x ) = a [ ( b − 1 ) x ¯ 3
Camber_(aerodynamics)
Operators useful in quantum mechanics
^{2}x^{2}\right)\psi (x)=E\psi (x).} Make a coordinate substitution to nondimensionalize the differential equation x = ℏ m ω q . {\displaystyle x\ =\ {\sqrt
Creation and annihilation operators
Creation_and_annihilation_operators
Type of aerodynamic or hydrodynamic drag
friction coefficient is a dimensionless skin shear stress which is nondimensionalized by the dynamic pressure of the free stream. The skin friction coefficient
Skin_friction_drag
Solution of Einstein field equations
tilted in the direction of rotation. For brevity, we further use nondimensionalized quantities normalized against G {\displaystyle G} , M {\displaystyle
Kerr–Newman_metric
Formulation of quantum mechanics
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
Matrix_mechanics
Dimensionless number; ratio of a fluid's flow inertia to the external field
densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the Richardson number which is more commonly
Froude_number
Topics referred to by the same term
produced by lifting the fingers one at a time from bottom to top Nondimensionalization, the removal of units from a mathematical equation This disambiguation
Natural_scale
Equation
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
Cauchy_momentum_equation
Mathematics and basic principles of three-phase electric power
_{i}P_{Li}\end{aligned}}} To simplify the mathematics, we define a nondimensionalized power for intermediate calculations, p = 1 V P 2 P T O T R {\displaystyle
Mathematics of three-phase electric power
Mathematics_of_three-phase_electric_power
Physics theorem about a swimmer's displacement
on the left-hand side tend to zero. This is made more apparent by nondimensionalizing the Navier–Stokes equations. By defining a characteristic velocity
Scallop_theorem
Ocean and atmospheric oscillations
contributing to the body's motion. These considerations are captured in the nondimensionalized Rossby number. In stability calculations, the rate of change of f
Coriolis_frequency
Quantum optical theoretical system
and Cummings found approximate eigenenergies and eigenstates for a nondimensionalized Hamiltonian in three different regimes of approximation: first, for
Tavis–Cummings_model
Dimensionless quantity in plasma physics
_{0}}}\left(\nabla \times \mathbf {B} \right)\times \mathbf {B} } We may nondimensionalize the momentum equation by introducing the nondimensional variables:[page needed]
Alfvén_Mach_number
Differential equation describing pressure distribution of thin viscous fluids
respectively. The equation can either be used with consistent units or nondimensionalized. The Reynolds Equation assumes: The fluid is Newtonian. Fluid viscous
Reynolds_equation
Hydrodynamic distortions of tides
{x}}}}=-{\frac {\partial {\tilde {\eta }}}{\partial {\tilde {x}}}}} The nondimensionalization shows that the nonlinear terms are very small if the average water
Nonlinear_tides
4th Edition, McGraw-Hill Education, 2018 (see p521, section 10.2. Nondimensionalized Equations of Motion). Doering, C.R.; Gibbon, J.D. (1995). Applied
Non-dimensionalization and scaling of the Navier–Stokes equations
Non-dimensionalization_and_scaling_of_the_Navier–Stokes_equations
dark matter Nonclassical light Noncommutative quantum field theory Nondimensionalization Nonequilibrium Gas and Plasma Dynamics Laboratory Nonequilibrium
Index_of_physics_articles_(N)
Surface representing the interface between two different fluids
{\displaystyle z} coordinate so that Δ p = 0 {\displaystyle \Delta p=0} . Nondimensionalized, the Young-Laplace equation is usually studied in the form κ z + λ
Capillary_surface
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Surname or Lastname
English and French
English and French : nickname from Middle English, Old French jay(e), gai ‘jay’ (the bird), probably referring to an idle chatterer or a showy person, although the jay was also noted for its thieving habits.The name is associated with a Huguenot family from La Rochelle, France, who settled in New Amsterdam. Peter Jay was the scion of the NY Jays; his son John (1745–1829) was a U.S. diplomat and first chief justice of the U.S. Supreme Court.
Girl/Female
Hebrew American German English
From the tower.
Boy/Male
American, Australian, British, Chinese, Christian, English
The Priest's Village; Priest's Town; Priest's Settlement
Girl/Female
Indian
Good Person
Girl/Female
Christian, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil
Godly; Divine
Male
Hebrew
(עַדְלַי) Hebrew name ADLAY means "justice of God." In the bible, this is the name of an ancestor of Shaphat, the overseer of David's herds.Â
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Oriya, Tamil, Telugu
Lord Hanuman
Female
Irish
Variant spelling of Irish Séarlait, SÉRLAIT means "man."
Boy/Male
American, Australian, Chinese, Jamaican, Swedish
Precious Green Stone
Male
Slovene
Slovene form of Latin Johannes, JANEZ means "God is gracious."
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