Dimensionless Numbers Of Fluid Mechanics 502k41

DIMENSIONLESS NUMBERS OF FLUID MECHANICS12 Name(s)

Symbol

Definition

Alfv´en, K´ arm´an Bond

Al, Ka

VA /V

Bd

(ρ0 − ρ)L2 g/Σ

Boussinesq

B

V /(2gR)1/2

Brinkman Capillary Carnot

Br Ca

µV 2 /k∆T µV /Σ (T2 − T1 )/T2

Cauchy, Hooke Chandrasekhar Clausius

Cy, Hk

ρV 2 /Γ = M2

Ch

B 2 L2 /ρνη

Cl

LV 3 ρ/k∆T

Cowling Crispation

C Cr

(VA /V )2 = Al2 µκ/ΣL

Dean

D

D3/2 V /ν(2r)1/2

[Drag coefficient] Eckert

CD

(ρ0 − ρ)Lg/ ρ0 V 2 V 2 / ∆T

Ekman

Ek

Euler

Eu

Froude

Fr

Gay–Lussac

Ga

V /(gL)1/2 V /N L 1/β∆T

Grashof [Hall coefficient]

Gr CH

gL3 β∆T /ν 2 λ/rL

E

(ν/2ΩL2 )1/2 = (Ro/Re)1/2 ∆p/ρV 2

*(†) Also defined as the inverse (square) of the quantity shown.

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Significance *(Magnetic force/ inertial force)1/2 Gravitational force/ surface tension (Inertial force/ gravitational force)1/2 Viscous heat/conducted heat Viscous force/surface tension Theoretical Carnot cycle efficiency Inertial force/ compressibility force Magnetic force/dissipative forces Kinetic energy flow rate/heat conduction rate Magnetic force/inertial force Effect of diffusion/effect of surface tension Transverse flow due to curvature/longitudinal flow Drag force/inertial force Kinetic energy/change in thermal energy (Viscous force/Coriolis force)1/2 Pressure drop due to friction/ dynamic pressure †(Inertial force/gravitational or buoyancy force)1/2 Inverse of relative change in volume during heating Buoyancy force/viscous force Gyrofrequency/ collision frequency

Name(s)

Symbol

Definition 1/2

Hartmann

H

Knudsen

Kn

BL/(µη) = (Rm Re C)1/2 λ/L

Lewis

Le

κ/D

Lorentz Lundquist

Lo Lu

Mach

M

V /c µ0 LVA /η = Al Rm V /CS

Magnetic Mach Magnetic Reynolds Newton Nusselt

Mm

V /VA = Al−1

Rm

µ0 LV /η

Nt N

F/ρL2 V 2 αL/k

P´eclet Poisseuille Prandtl

Pe Po Pr

LV /κ D2 ∆p/µLV ν/κ

Rayleigh Reynolds Richardson

Ra Re Ri

gH 3 β∆T /νκ LV /ν (N H/∆V )2

Rossby Schmidt

Ro Sc

V /2ΩL sin Λ ν/D

Stanton

St

α/ρ V

Stefan Stokes

Sf S

σLT 3 /k ν/L2 f

Strouhal Taylor

Sr Ta

Thring, Boltzmann Weber

Th, Bo

f L/V (2ΩL2 /ν)2 R1/2 (∆R)3/2 ·(Ω/ν) ρ V /σT 3

W

ρLV 2 /Σ

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Significance (Magnetic force/ dissipative force)1/2 Hydrodynamic time/ collision time *Thermal conduction/molecular diffusion Magnitude of relativistic effects J × B force/resistive magnetic diffusion force Magnitude of compressibility effects (Inertial force/magnetic force)1/2 Flow velocity/magnetic diffusion velocity Imposed force/inertial force Total heat transfer/thermal conduction Heat convection/heat conduction Pressure force/viscous force Momentum diffusion/ heat diffusion Buoyancy force/diffusion force Inertial force/viscous force Buoyancy effects/ vertical shear effects Inertial force/Coriolis force Momentum diffusion/ molecular diffusion Thermal conduction loss/ heat capacity Radiated heat/conducted heat Viscous damping rate/ vibration frequency Vibration speed/flow velocity Centrifugal force/viscous force (Centrifugal force/ viscous force)1/2 Convective heat transport/ radiative heat transport Inertial force/surface tension

Nomenclature: B Cs , c D = 2R F f g H, L k = ρ κ N = (g/H)1/2 R r rL T V VA = B/(µ0 ρ)1/2 α β Γ ∆R, ∆V, ∆p, ∆T η κ, D Λ λ µ = ρν µ0 ν ρ ρ0 Σ σ Ω

Magnetic induction Speeds of sound, light Specific heat at constant pressure (units m2 s−2 K−1 ) Pipe diameter Imposed force Vibration frequency Gravitational acceleration Vertical, horizontal length scales Thermal conductivity (units kg m−1 s−2 ) Brunt–V¨ ais¨ al¨a frequency Radius of pipe or channel Radius of curvature of pipe or channel Larmor radius Temperature Characteristic flow velocity Alfv´en speed ∂T = α∆T Newton’s-law heat coefficient, k ∂x Volumetric expansion coefficient, dV /V = βdT Bulk modulus (units kg m−1 s−2 ) Imposed differences in two radii, velocities, pressures, or temperatures Surface emissivity Electrical resistivity Thermal, molecular diffusivities (units m2 s−1 ) Latitude of point on earth’s surface Collisional mean free path Viscosity Permeability of free space Kinematic viscosity (units m2 s−1 ) Mass density of fluid medium Mass density of bubble, droplet, or moving object Surface tension (units kg s−2 ) Stefan–Boltzmann constant Solid-body rotational angular velocity

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