Microfluidics for Biologists Fundamentals and Applications

Table 1.4 Dimensionless numbers in fluid mechanics

Dimensionless

number Details Formula

Reynolds Number Inertial force/Viscous force

convective momentum/viscous

momentum

Forced Convection

Re¼ρUL=η¼UL=ν

Prandtl Number

(heat)

Prandtl-Schmidt

Number (mass)

Momentum/Species diffusivity

Used to determine fluid or heat or

mass transfer boundary layer

thickness

Prheat¼ν=α¼ηCP=K

Prmass¼Sc¼ν=D¼η=ρD

Pe ́clet Number

(heat)

Pe ́clet Number

(mass)

Convection transport rate/Diffusion

transportation rate

Peheat¼RePr¼UL=α

α¼k=ρCP

PeMass¼RePr¼UL=D

Nusselt Number

(heat)

Nusselt-Sherwood

Number (mass)

Length scale/Diffusion boundary

layer thickness

Used to determine the heat (h) or

mass (hD) transfer coefficient

Nu¼ fεRe PrðÞ^1 =^3

hi

= 2

Nuheat¼hL=kfluid

NuMass¼hDL=Dfluid

L¼As=Pm

Grashof Number

(heat)

Grashof Number

(mass)

Natural convection buoyancy force/

Viscous force

Used to calculate Re for buoyant

flow

Controls the lengthscale to natural

convection boundary layer thick-

ness

Natural Convection

Grheat¼gβðÞTsTbL^3 =ν^2

GrMass¼gβCðÞCasCaaL^3 =ν^2

β¼ðÞ∂ρ=∂CαT,P

hi

=ρ

Rayleigh Number

(heat)

Rayleigh Number

(mass)

Natural convection/Diffusive heat

or mass transport

Used to determine the transition to

turbulence

Raheat¼GrPr¼gβΔðÞTL^3 =να

Ramass¼GrPr¼gβCðÞΔCL^3 =νD

Knudsen Number

(to analyze extent

of continuum)

Slip length/Macroscopic length Kn¼β=L

Richardson

Number

Buoyancy/Flow gradient Ri¼gðÞΔρ=ρU^2

E€otv€os (Eo) or

Bond Number

(Bo)

Body forces/Surface tension

Used together with Morton Number

to determine shape of drops or bub-

bles in surrounding fluid or contin-

uous phase

Eo¼Bo¼ ðÞΔρgL^3



=σ

Capillary Number Viscous forces/Interfacial forces Ca¼ηU=σ

Elasticity Number Elastic effects/Inertial effects El¼ θη=ρR^2 ¼Wi=Re

Weissenberg

Number

Viscous forces/Elastic forces Wi¼γ’:ts

Deborah Number Stress relaxation time/Time of

observation

ts/to

1 Fundamentals of Fluidics 5