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Dimensionless numbers in fluid mechanics

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Dimensionless numbers in fluid mechanics

Dimensionless numbers in fluid mechanics are a set of dimensionless quantities that have an important role in the behaviour of fluids.

Contents

  • Droplet formation 1
  • Airfoil design 2
  • Microfluidics 3
  • Combustion 4
  • Turbulent flow 5
  • List of dimensionless numbers in fluid mechanics and related fields 6
  • References 7

Droplet formation

Dimensionless numbers in droplet formation
vs. Momentum Viscosity Kinetic energy Surface tension Gravity
Momentum ρvd Re
Viscosity Re−1 η, μ Oh, Ca Ga−1
Kinetic energy ρv2d We
Surface tension Oh−1, Ca−1 We−1 σ Bo−1
Gravity Ga Bo g

Droplet formation mostly depends on momentum, viscosity and surface tension.[1] In

  1. ^ Dijksman, J. Frits; Pierik, Anke (2014). "Dynamics of Piezoelectric Print-Heads". pp. 45–86.  
  2. ^ Derby, Brian (2010). "Inkjet Printing of Functional and Structural Materials: Fluid Property Requirements, Feature Stability, and Resolution". Annual Review of Materials Research 40 (1): 395–414.  
  3. ^ Bhattacharjee S., Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD 96: 711–6. 
  4. ^ a b "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05. 
  5. ^ Bond number
  6. ^ "Home". OnePetro. 2015-05-04. Retrieved 2015-05-08. 
  7. ^ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134.  
  8. ^ Fanning friction factor
  9. ^ Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science 56 (21–22): 6337.  
  10. ^ Lockhart–Martinelli parameter
  11. ^ Manning coefficient PDF (109 KB)
  12. ^ Richardson number
  13. ^ Schmidt number
  14. ^ Sommerfeld number
  15. ^ Petritsch, G.; Mewes, D. (1999). "Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor". Nuclear Engineering and Design 188: 75.  
  16. ^ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390.  
  17. ^ Weissenberg number
  18. ^ Womersley number

References

Name Standard symbol Definition Field of application
Archimedes number Ar \mathrm{Ar} = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2} fluid mechanics (motion of fluids due to density differences)
Atwood number A \mathrm{A} = \frac{\rho_1 - \rho_2} {\rho_1 + \rho_2} fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bejan number
(fluid mechanics)
Be \mathrm{Be} = \frac{\Delta P L^2} {\mu \alpha} fluid mechanics (dimensionless pressure drop along a channel)[3]
Bingham number Bm \mathrm{Bm} = \frac{ \tau_y L }{ \mu V } fluid mechanics, rheology (ratio of yield stress to viscous stress)[4]
Biot number Bi \mathrm{Bi} = \frac{h L_C}{k_b} heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B \mathrm{B} = \frac{u \rho}{\mu (1 - \epsilon) D} geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond number Bo \mathrm{Bo} = \frac{\rho a L^2}{\gamma} geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [5]
Brinkman number Br \mathrm{Br} = \frac {\mu U^2}{\kappa (T_w - T_0)} heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK \mathrm{N}_\mathrm{BK} = \frac{u \mu}{k_\mathrm{rw}\sigma} fluid mechanics (combination of capillary number and Bond number) [6]
Capillary number Ca \mathrm{Ca} = \frac{\mu V}{\gamma} porous media, fluid mechanics (viscous forces versus surface tension)
Colburn J factors JM, JH, JD turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Damkohler number Da \mathrm{Da} = k \tau chemistry (reaction time scales vs. residence time)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Dean number D \mathrm{D} = \frac{\rho V d}{\mu} \left( \frac{d}{2 R} \right)^{1/2} turbulent flow (vortices in curved ducts)
Deborah number De \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}} rheology (viscoelastic fluids)
Drag coefficient cd c_\mathrm{d} = \dfrac{2 F_\mathrm{d}}{\rho v^2 A}\, , aeronautics, fluid dynamics (resistance to fluid motion)
Eckert number Ec \mathrm{Ec} = \frac{V^2}{c_p\Delta T} convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Eötvös number Eo \mathrm{Eo}=\frac{\Delta\rho \,g \,L^2}{\sigma} fluid mechanics (shape of bubbles or drops)
Ericksen number Er \mathrm{Er}=\frac{\mu v L}{K} fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu \mathrm{Eu}=\frac{\Delta{}p}{\rho V^2} hydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient \Theta_r \Theta_r = \frac{c_p (T-T_e)}{U_e^2/2} heat transfer, fluid dynamics (change in internal energy versus kinetic energy)[7]
Fanning friction factor f fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[8]
Froude number Fr \mathrm{Fr} = \frac{v}{\sqrt{g\ell}} fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei number Ga \mathrm{Ga} = \frac{g\, L^3}{\nu^2} fluid mechanics (gravitational over viscous forces)
Görtler number G \mathrm{G} = \frac{U_e \theta}{\nu} \left( \frac{\theta}{R} \right)^{1/2} fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz \mathrm{Gz} = {D_H \over L} \mathrm{Re}\, \mathrm{Pr} heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number Gr \mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2} heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hagen number Hg \mathrm{Hg} = -\frac{1}{\rho}\frac{\mathrm{d} p}{\mathrm{d} x}\frac{L^3}{\nu^2} heat transfer (ratio of the buoyancy to viscous force in forced convection)
Iribarren number Ir \mathrm{Ir} = \frac{\tan \alpha}{\sqrt{H/L_0}} wave mechanics (breaking surface gravity waves on a slope)
Karlovitz number Ka \mathrm{Ka} = k t_c turbulent combustion (characteristic flow time times flame stretch rate)
Keulegan–Carpenter number KC \mathrm{K_C} = \frac{V\,T}{L} fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn \mathrm{Kn} = \frac {\lambda}{L} gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number Ku \mathrm{Ku} = \frac{U_h \rho_g^{1/2}}{\left({\sigma g (\rho_l - \rho_g)}\right)^{1/4}} fluid mechanics (counter-current two-phase flow)[9]
Laplace number La \mathrm{La} = \frac{\sigma \rho L}{\mu^2} fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number Le \mathrm{Le} = \frac{\alpha}{D} = \frac{\mathrm{Sc}}{\mathrm{Pr}} heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL C_\mathrm{L} = \frac{L}{q\,S} aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter \chi \chi = \frac{m_\ell}{m_g} \sqrt{\frac{\rho_g}{\rho_\ell}} two-phase flow (flow of wet gases; liquid fraction)[10]
Mach number M or Ma \mathrm{M} = \frac} gas dynamics (compressible flow; dimensionless velocity)
Manning roughness coefficient n open channel flow (flow driven by gravity)[11]
Marangoni number Mg \mathrm{Mg} = - {\frac{\mathrm{d}\sigma}{\mathrm{d}T}}\frac{L \Delta T}{\eta \alpha} fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Morton number Mo \mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3} fluid dynamics (determination of bubble/drop shape)
Nusselt number Nu \mathrm{Nu} =\frac{hd}{k} heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number Oh \mathrm{Oh} = \frac{ \mu}{ \sqrt{\rho \sigma L }} = \frac{\sqrt{\mathrm{We}}}{\mathrm{Re}} fluid dynamics (atomization of liquids, Marangoni flow)
Prandtl number Pr \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficient CP C_p = {p - p_\infty \over \frac{1}{2} \rho_\infty V_\infty^2} aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Rayleigh number Ra \mathrm{Ra}_{x} = \frac{g \beta} {\nu \alpha} (T_s - T_\infin) x^3 heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number Re \mathrm{Re} = \frac{vL\rho}{\mu} fluid mechanics (ratio of fluid inertial and viscous forces)[4]
Richardson number Ri \mathrm{Ri} = \frac{gh}{u^2} = \frac{1}{\mathrm{Fr}^2} fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[12]
Roshko number Ro \mathrm{Ro} = {f L^{2}\over \nu} =\mathrm{St}\,\mathrm{Re} fluid dynamics (oscillating flow, vortex shedding)
Schmidt number Sc \mathrm{Sc} = \frac{\nu}{D} mass transfer (viscous over molecular diffusion rate)[13]
Shape factor H H = \frac {\delta^*}{\theta} boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh \mathrm{Sh} = \frac{K L}{D} mass transfer (forced convection; ratio of convective to diffusive mass transport)
Sommerfeld number S \mathrm{S} = \left( \frac{r}{c} \right)^2 \frac {\mu N}{P} hydrodynamic lubrication (boundary lubrication)[14]
Stanton number St \mathrm{St} = \frac{h}{c_p \rho V} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}} heat transfer and fluid dynamics (forced convection)
Stokes number Stk or Sk \mathrm{Stk} = \frac{\tau U_o}{d_c} particles suspensions (ratio of characteristic time of particle to time of flow)
Stuart number N \mathrm{N} = \frac {B^2 L_{c} \sigma}{\rho U} = \frac{\mathrm{Ha}^2}{\mathrm{Re}} magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta \mathrm{Ta} = \frac{4\Omega^2 R^4}{\nu^2} fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Ursell number U \mathrm{U} = \frac{H\, \lambda^2}{h^3} wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Wallis parameter j* j^* = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} multiphase flows (nondimensional superficial velocity)[15]
Weaver flame speed number Wea \mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100 combustion (laminar burning velocity relative to hydrogen gas)[16]
Weber number We \mathrm{We} = \frac{\rho v^2 l}{\sigma} multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number Wi \mathrm{Wi} = \dot{\gamma} \lambda viscoelastic flows (shear rate times the relaxation time)[17]
Womersley number \alpha \alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2} biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[18]

All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:

List of dimensionless numbers in fluid mechanics and related fields

Turbulent flow

Combustion

Microfluidics

Airfoil design

[2]

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