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Dimensionless numbers in fluid mechanics

Topic: Physics

 From HandWiki - Reading time: 11 min

Short description: Dimension

Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena.[1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed. To compare a real situation (e.g. an aircraft) with a small-scale model it is necessary to keep the important characteristic numbers the same. Names and formulation of these numbers were standardized in ISO 31-12 and in ISO 80000-11.

Diffusive numbers in transport phenomena

Dimensionless numbers in transport phenomena
vs. Inertial Viscous Thermal Mass
Inertial vd Re Pe PeAB
Viscous Re−1 μ/ρ, ν Pr Sc
Thermal Pe−1 Pr−1 α Le
Mass PeAB−1 Sc−1 Le−1 D

As a general example of how dimensionless numbers arise in fluid mechanics, the classical numbers in transport phenomena of mass, momentum, and energy are principally analyzed by the ratio of effective diffusivities in each transport mechanism. The six dimensionless numbers give the relative strengths of the different phenomena of inertia, viscosity, conductive heat transport, and diffusive mass transport. (In the table, the diagonals give common symbols for the quantities, and the given dimensionless number is the ratio of the left column quantity over top row quantity; e.g. Re = inertial force/viscous force = vd/ν.) These same quantities may alternatively be expressed as ratios of characteristic time, length, or energy scales. Such forms are less commonly used in practice, but can provide insight into particular applications.

Droplet formation

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

Droplet formation mostly depends on momentum, viscosity and surface tension.[2] In inkjet printing for example, an ink with a too high Ohnesorge number would not jet properly, and an ink with a too low Ohnesorge number would be jetted with many satellite drops.[3] Not all of the quantity ratios are explicitly named, though each of the unnamed ratios could be expressed as a product of two other named dimensionless numbers.

List

All numbers are dimensionless quantities. See other article for extensive list of dimensionless quantities. Certain dimensionless quantities of some importance to fluid mechanics are given below:

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

References

  1. ↑ "ISO 80000-1:2009". International Organization for Standardization. https://www.iso.org/standard/30669.html. 
  2. ↑ Dijksman, J. Frits; Pierik, Anke (2012). "Dynamics of Piezoelectric Print-Heads". in Hutchings, Ian M.; Martin, Graham D.. Inkjet Technology for Digital Fabrication. John Wiley & Sons. pp. 45–86. doi:10.1002/9781118452943.ch3. ISBN 9780470681985. 
  3. ↑ 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. doi:10.1146/annurev-matsci-070909-104502. ISSN 1531-7331. Bibcode: 2010AnRMS..40..395D. https://pure.manchester.ac.uk/ws/files/174918681/DERBYwithfigures_2017_02_22_19_00_59_UTC_.pdf. 
  4. ↑ Bhattacharje, Subrata; Grosshandler, William L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". in Jacobs, Harold R.. National Heat Transfer Conference. 1. Houston, TX: American Society of Mechanical Engineers. pp. 711–716. Bibcode: 1988nht.....1..711B. 
  5. ↑ 5.0 5.1 "Table of Dimensionless Numbers". http://www.cchem.berkeley.edu/gsac/grad_info/prelims/binders/dimensionless_numbers.pdf. 
  6. ↑ Mahajan, Milind P.; Tsige, Mesfin; Zhang, Shiyong; Alexander, J. Iwan D.; Taylor, P. L.; Rosenblatt, Charles (10 January 2000). "Collapse Dynamics of Liquid Bridges Investigated by Time-Varying Magnetic Levitation". Physical Review Letters 84 (2): 338–341. doi:10.1103/PhysRevLett.84.338. PMID 11015905. Bibcode: 2000PhRvL..84..338M. http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf. 
  7. ↑ "Home". OnePetro. 2015-05-04. http://www.onepetro.org/mslib/servlet/onepetropreview?id=00020506. 
  8. ↑ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc.. pp. 132–134. ISBN 0-13-086885-X. https://archive.org/details/boundarylayerana00sche. 
  9. ↑ "Fanning friction factor". http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm. 
  10. ↑ Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science 56 (21–22): 6337. doi:10.1016/S0009-2509(01)00247-0. Bibcode: 2001ChEnS..56.6337T. 
  11. ↑ Stewart, David (February 2003). "The Evaluation of Wet Gas Metering Technologies for Offshore Applications, Part 1 – Differential Pressure Meters". Flow Measurement Guidance Note (Glasgow, UK: National Engineering Laboratory) 40. http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf. 
  12. ↑ Richardson number
  13. ↑ Schmidt number
  14. ↑ Ekerfors, Lars O. (1985). Boundary lubrication in screw-nut transmissions (PDF) (PhD). Luleå University of Technology. ISSN 0348-8373.
  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: 75–84. doi:10.1016/S0029-5493(99)00005-9. 
  16. ↑ Smith, Douglas E.; Babcock, Hazen P.; Chu, Steven (12 March 1999). "Single-Polymer Dynamics in Steady Shear Flow". Science (American Association for the Advancement of Science) 283 (5408): 1724–1727. doi:10.1126/science.283.5408.1724. PMID 10073935. Bibcode: 1999Sci...283.1724S. http://physics.ucsd.edu/~des/Shear1999.pdf. 
  17. ↑ "Comparison of Flow Measure Techniques during Continuous and Pulsatile Flow". Department of Bioengineering, University of Pennsylvania. May 2001. https://www.seas.upenn.edu/~belab/LabProjects/2001/be310s01m2.html. 
  • Tropea, C.; Yarin, A.L.; Foss, J.F. (2007). Springer Handbook of Experimental Fluid Mechanics. Springer-Verlag. 



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