द्रव यांत्रिकी में आयामहीन संख्याएँ

अभिलक्षणिक संख्याएँ आयामहीन मात्राओं का एक समूह हैं जो तरल पदार्थों के व्यवहार और उनके प्रवाह के साथ-साथ अन्य परिवहन घटनाओं के विश्लेषण में महत्वपूर्ण भूमिका निभाते हैं। उनमें रेनॉल्ड्स संख्या और मैक संख्याएं शामिल हैं, जो द्रव के सापेक्ष परिमाण और घनत्व, चिपचिपाहट, ध्वनि की गति और वेग गति जैसी भौतिक प्रणाली विशेषताओं के अनुपात का वर्णन करती हैं।

किसी वास्तविक स्थिति (उदाहरण के लिए एक विमान) की तुलना छोटे पैमाने के मॉडल से करने के लिए महत्वपूर्ण विशेषता संख्याओं को समान रखना आवश्यक है। इन नंबरों के नाम और सूत्रीकरण आईएसओ 31-12 और आईएसओ 80000-11 में मानकीकृत किए गए थे।

परिवहन परिघटना में विवर्तनिक संख्याएँ

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

द्रव यांत्रिकी में आयामहीन संख्याएँ कैसे उत्पन्न होती हैं, इसके एक सामान्य उदाहरण के रूप में, द्रव्यमान के संरक्षण, गति के संरक्षण और ऊर्जा के संरक्षण की परिवहन घटनाओं में शास्त्रीय संख्याओं का मुख्य रूप से प्रत्येक परिवहन तंत्र में प्रभावी प्रसार के अनुपात द्वारा विश्लेषण किया जाता है। छह आयामहीन मात्राएँ जड़ता, श्यानता, ऊष्मा चालन और विसरणीय जन परिवहन की विभिन्न घटनाओं की सापेक्ष शक्तियाँ देती हैं। (तालिका में, विकर्ण मात्राओं के लिए सामान्य प्रतीक देते हैं, और दी गई आयाम रहित संख्या शीर्ष पंक्ति की मात्रा पर बाएं स्तंभ की मात्रा का अनुपात है; उदाहरण के लिए रेनॉल्ड्स संख्या = जड़त्व बल/श्यान बल = vd/श्यानता|ν।) ये समान मात्राओं को वैकल्पिक रूप से विशिष्ट समय, लंबाई या ऊर्जा पैमाने के अनुपात के रूप में व्यक्त किया जा सकता है। ऐसे फॉर्म आमतौर पर व्यवहार में कम उपयोग किए जाते हैं, लेकिन विशेष अनुप्रयोगों में अंतर्दृष्टि प्रदान कर सकते हैं।

बूंद निर्माण

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

बूंदों का निर्माण अधिकतर गति, चिपचिपाहट और सतह तनाव पर निर्भर करता है। उदाहरण के लिए, इंकजेट मुद्रण में, बहुत अधिक ओहनेसॉर्ज संख्या वाली स्याही ठीक से जेट नहीं होगी, और बहुत कम ओहनेसॉर्ज संख्या वाली स्याही कई उपग्रह बूंदों के साथ जेट होगी। सभी मात्रा अनुपातों को स्पष्ट रूप से नामित नहीं किया गया है, हालांकि प्रत्येक अनाम अनुपात को दो अन्य नामित आयामहीन संख्याओं के उत्पाद के रूप में व्यक्त किया जा सकता है।

सूची

सभी संख्याएँ [[आयामहीन मात्राएँ]] हैं। आयामहीन मात्राओं की विस्तृत सूची के लिए अन्य लेख देखें। द्रव यांत्रिकी के लिए कुछ महत्व की कुछ आयामहीन मात्राएँ नीचे दी गई हैं:

Name	Standard symbol	Definition	Field of application
Archimedes number	Ar	$\mathrm {Ar} ={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}$	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	$\mathrm {Be} ={\frac {\Delta PL^{2}}{\mu \alpha }}$	fluid mechanics (dimensionless pressure drop along a channel)
Bingham number	Bm	$\mathrm {Bm} ={\frac {\tau _{y}L}{\mu V}}$	fluid mechanics, rheology (ratio of yield stress to viscous stress)
Biot number	Bi	$\mathrm {Bi} ={\frac {hL_{C}}{k_{b}}}$	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	$\mathrm {Bo} ={\frac {\rho aL^{2}}{\gamma }}$	geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number)
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)
Capillary number	Ca	$\mathrm {Ca} ={\frac {\mu V}{\gamma }}$	porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number	C	$\mathrm {C} ={\frac {B^{2}L^{2}}{\mu _{o}\mu D_{M}}}$	hydromagnetics (Lorentz force versus viscosity)
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	$\mathrm {D} ={\frac {\rho Vd}{\mu }}\left({\frac {d}{2R}}\right)^{1/2}$	turbulent flow (vortices in curved ducts)
Deborah number	De	$\mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}}$	rheology (viscoelastic fluids)
Drag coefficient	cd	$c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\,,$	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	$\mathrm {Er} ={\frac {\mu vL}{K}}$	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)
Fanning friction factor	f		fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)
Froude number	Fr	$\mathrm {Fr} ={\frac {U}{\sqrt {g\ell }}}$	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)
Hartmann number	Ha	$\mathrm {Ha} =BL\left({\frac {\sigma }{\rho \nu }}\right)^{\frac {1}{2}}$	magnetohydrodynamics (ratio of Lorentz to viscous forces)
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)
Jakob number	Ja	$\mathrm {Ja} ={\frac {c_{p,f}(T_{w}-T_{sat})}{h_{fg}}}$	heat transfer (ratio of sensible heat to latent heat during phase changes)
Karlovitz number	Ka	$\mathrm {Ka} =kt_{c}$	turbulent combustion (characteristic flow time times flame stretch rate)
Kapitza number	Ka	$\mathrm {Ka} ={\frac {\sigma }{\rho (g\sin \beta )^{1/3}\nu ^{4/3}}}$	fluid mechanics (thin film of liquid flows down inclined surfaces)
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)
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)
Mach number	M or Ma	$\mathrm {M} ={\frac {v}{v_{\mathrm {sound} }}}$	gas dynamics (compressible flow; dimensionless velocity)
Manning roughness coefficient	n		open channel flow (flow driven by gravity)
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)
Markstein number	Ma	$\mathrm {Ma} ={\frac {L_{b}}{l_{f}}}$	turbulence, combustion (Markstein length to laminar flame thickness)
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)
Péclet number	Pe	$\mathrm {Pe} ={\frac {Lu}{D}}$ or $\mathrm {Pe} ={\frac {Lu}{\alpha }}$	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	$\mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {c_{p}\mu }{k}}$	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	$\mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}$	heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number	Re	$\mathrm {Re} ={\frac {UL\rho }{\mu }}={\frac {UL}{\nu }}$	fluid mechanics (ratio of fluid inertial and viscous forces)
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)
Roshko number	Ro	$\mathrm {Ro} ={fL^{2} \over \nu }=\mathrm {St} \,\mathrm {Re}$	fluid dynamics (oscillating flow, vortex shedding)
Schmidt number	Sc	$\mathrm {Sc} ={\frac {\nu }{D}}$	mass transfer (viscous over molecular diffusion rate)
Shape factor	H	$H={\frac {\delta ^{*}}{\theta }}$	boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number	Sh	$\mathrm {Sh} ={\frac {KL}{D}}$	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)
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)
Strouhal number	St	$\mathrm {St} ={\frac {fL}{U}}$	Vortex shedding (ratio of characteristic oscillatory velocity to ambient flow velocity)
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)
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)
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)
Zel'dovich number	$\beta$	$\beta ={\frac {E}{RT_{f}}}{\frac {T_{f}-T_{o}}{T_{f}}}$	fluid dynamics, Combustion (Measure of activation energy)

संदर्भ

• Tropea, C.; Yarin, A.L.; Foss, J.F. (2007). Springer Handbook of Experimental Fluid Mechanics. Springer-Verlag.