Dimensionless Numbers

Home , Archimedes number, Atwood number, Cauchy number, Deborah number, Eckert number, Graetz number

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A. Salih

Dept. of Aerospace Engineering IIST, Thiruvananthapuram

The nondimensionalization of the governing equations of fluid flow is important for both theoretical and computational reasons. Nondimensional scaling provides a method for developing dimensionless groups that can provide physical insight into the importance of various terms in the system of governing equations. Computationally, dimensionless forms have the added benefit of providing numerical scaling of the system discrete equations, thus providing a physically linked technique for improving the ill-conditioning of the system of equations. Moreover, dimensionless forms also allow us to present the solution in a compact way. Some of the important dimensionless numbers used in fluid mechanics and heat transfer are given below.

Nomenclature

Archimedes Number: 3 Re 2 gL ρ(ρs − ρ) Ar = = Fr µ2 Atwood Number:

(ρ1 − ρ2) A = (ρ1 + ρ2) Note: Used in the study of density stratified flows.

Biot Number: hL conductive resistance in solid Bi = = Ks convective resistance in thermal boundary layer

Bond Number:

We ρgL 2 Bo = = Fr σ

Brinkman Number:

µU2 Br = K(Tw − T o) Note: Brinkman number is related to heat conduction from a wall to a flowing viscous fluid. It is commonly used in polymer processing.

Capillary Number: We Uµ Ca = = Re σ

Cauchy Number:

ρU2 Ca = M2 = K

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Centrifuge Number:

We ρΩ 2L3 Ce = = Ro 2 σ

Dean Number: Re De = (R ⁄ h )1/2 Note: Dean number deals with the stability of two- dimensional flows in a curved channel with mean radius R and width 2h .

Deborah Number: τ relaxation time De = = tp characteristic time scale Note: Commonly used in rheology to characterize how "fluid" a material is. The smaller the De , the more the fluid the material appears.

Eckert Number:

U2 Ec = cp∆T Note: Eckert number represents the kinetic energy of the flow relative to the boundary layer enthalpy difference. Ec plays an important role in high speed flows for which viscous dissipation is significant.

Ekman Number: µ viscous force Ek = = ρσ L2 Coriolis force E�tv �s Number: We ∆ρ gL 2 Eo = = Fr σ

Euler Number: ∆p pressure force Eu = = ρU2 inertial force

Fourier Number: αt rate of heat conduction Fo = = L2 rate of thermal energy stored Note: Fourier number represents the dimensionless time. It may be interpreted as the ratio of current time to time to reach steady-state.

Froude Number: U2 inertial force Fr = = gL gravitational force

Galileo Number:

Re 2 ρ2gL 3 Ga = = Fr µ2

Graetz Number:

Gz = di Pe = Ud i

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L ν

Grashof Number:

gβ(T − T )L 3 buoyancy force Gr = hot ref = ν2 viscous force

Hagen Number:

dp ρL3 Hg = − dx µ2 Note: It is the forced flow equivalent of Grashof number.

Jakob Number:

cp(Tw − T sat ) Ja = hfg Note: Jakob number represents the ratio of sensible heat to latent heat absorbed (or released) during the phase change process.

Knudsen Number: λ length of mean free path Kn = = L characteristic dimension

Laplace Number:

Re 2 ρσ L La = = We µ2

Lewis Number: α thermal diffusivity Le = = DAB mass diffusivity

Mach Number: U inertial force M = = a elastic (compressibility) force

Marangoni Number: dσ L∆T Ma = − dT µα Note: Marangoni number is the ratio of thermal surface tension force to the viscous force.

Morton Number:

We 3 gµ4 Mo = = Fr Re 4 ∆ρσ 3

Nusselt Number: hL Nu = Kf Note: Nusselt number represents the dimensionless temperature gradient at the solid surface.

Ohnesorge Number:

We 1/2 µ Oh = = Re (ρσ L)1/2

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Peclet Number: UL inertia (convection) Pe = = α Diffusion

Prandtl Number: ν momentum diffusivity Pr = = α thermal diffusivity

Rayleigh Number:

gβ(T − T )L 3 buoyancy Ra = Gr Pr = hot ref = να viscous × rate of heat diffusion

Reynolds Number: ρUL inertial force Re = = µ viscous force

Richardson Number:

Gr gβ(Thot − T ref )L buoyancy force Ri = = = Re 2 U2 inertial force

Rossby Number: U inertial force Ro = = ΩL Coriolis force

Rotating Froude Number:

Fr Ω2L Fr R = = Ro 2 g

Schmidt Number: ν momentum diffusivity Sc = Le Pr = = DAB mass diffusivity

Sherwood Number:

hmL Sh = DAB Note: Sherwood number represents the dimensionless concentration gradient at the solid surface.

Stanton Number: Nu h St = = Re Pr ρUc p Note: Stanton number is the modified Nusselt number. It is used in analogy between heat transfer and viscous transport in boundary layers.

Stefan Number:

cpdT specific heat St = = Lm latent heat Note: Stefan number is useful in the study of heat transfer during phase change.

Stokes Number:

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τUo stopping distance of a particle Stk = = dc characteristic dimension of the obstacle Note: Commonly used in particles suspended in fluid. For Stk << 1, the particle negotiates the obstacle. For Stk >> 1, the particle travels in straightline and eventually collides with obstacle.

Strouhal Number (for oscillatory flow): L inertia (local) St = = Ut ref inertia (convection)

Note: If tref is taken as the reciprocal of the circular frequency ω of the system, then ωL St = U Taylor Number: ρ2Ω 2L4 Ta = i µ2 3 1/4 where L = [r i(ro − ri) ]

Weber Number: ρU2L inertial force We = = σ surface tension force

Womersley Number: (ρω )1/2 α = ( π Re St )1/2 = L µ1/2 Note: Womersley number is used in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to the viscous effects.

Nomenclature:

a → speed of sound

cp → specific heat at constant pressure

DAB → mass diffusivity coefficient dT → temperature difference between phases

dc → characteristic dimension of the obstacle

di → hydraulic diameter of the duct g → gravitational acceleration h → heat transfer coefficient h → width of the channel

hfg → latent heat of condensation

hm → mass transfer coefficient K → bulk modulus of elasticity K → thermal conductivity of fluid

Kf → thermal conductivity of fluid

Ks → thermal conductivity of solid L → characteristic length scale

Lm → latent heat of melting R → radius of the channel

ri → radius of the inner cylinder

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ro → radius of the outer cylinder

Thot → temperature of the hot wall

Tref → reference temperature

To → bulk fluid temperature

Tsat → saturation temperature

Tw → wall temperature

T∞ → quiescent temperature of the fluid t → time

tref → reference time

tp → characteristic time scale U → characteristic velocity scale

Uo → fluid velocity far away from the object dp/dx → pressure gradient dσ/dT → rate of change of surface tension with temperature α → thermal diffusivity of fluid β → volumetric thermal expansion coefficient ∆p → characteristic pressure difference of flow ∆T → characteristic temperature difference ∆ρ → difference in density of the two phases λ → length of mean free path µ → viscosity of fluid ν → kinematic viscosity of fluid ρ → density of fluid

ρ1 → density of heavier fluid

ρ2 → density of lighter fluid

ρl → density of solid σ → surface tension τ → relaxation time Ω → angular velocity ω → circular frequency

ωi → angular velocity of inner cylinder

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