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Difference between revisions of "Reynolds number"

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One of the non-dimensional numbers which provide criteria of similarity for flows of viscous liquids and gases. It characterizes the relation between the inertial forces and the forces of viscosity:
 
One of the non-dimensional numbers which provide criteria of similarity for flows of viscous liquids and gases. It characterizes the relation between the inertial forces and the forces of viscosity:
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$$
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\mathrm{Re} = \frac{\rho \, \nu \, l}{\mu}
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$$
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where $\rho$ is the density, $\mu$ is the dynamical coefficient of viscosity of the liquid or gas, $\nu$ is the typical rate of flow, and $l$ is the typical linear dimension.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817101.png" /></td> </tr></table>
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The Reynolds number also determines the mode of flow of a liquid in terms of a critical Reynolds number, $\mathrm{Re}_{\mathrm{cr}}$. When $\mathrm{Re} < \mathrm{Re}_{\mathrm{cr}}$, only laminar liquid flow is possible, whereas when $\mathrm{Re} > \mathrm{Re}_{\mathrm{cr}}$ the flow may become turbulent.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817102.png" /> is the density, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817103.png" /> is the dynamical coefficient of viscosity of the liquid or gas, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817104.png" /> is the typical rate of flow, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817105.png" /> is the typical linear dimension.
 
 
 
The Reynolds number also determines the mode of flow of a liquid in terms of a critical Reynolds number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817106.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817107.png" />, only laminar liquid flow is possible, whereas when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081710/r0817108.png" /> the flow may become turbulent.
 
  
 
The Reynolds number is named after O. Reynolds.
 
The Reynolds number is named after O. Reynolds.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. Sect. 4.7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.I. Vishik,  A.V. Fursikov,  "Mathematical problems of statistical hydromechanics" , Kluwer  (1988)  pp. Chapts. 3; 4; 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Fluid mechanics" , Pergamon  (1959)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. Sect. 4.7</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.I. Vishik,  A.V. Fursikov,  "Mathematical problems of statistical hydromechanics" , Kluwer  (1988)  pp. Chapts. 3; 4; 6  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Fluid mechanics" , Pergamon  (1959)  (Translated from Russian)</TD></TR>
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</table>

Revision as of 20:07, 12 October 2014

One of the non-dimensional numbers which provide criteria of similarity for flows of viscous liquids and gases. It characterizes the relation between the inertial forces and the forces of viscosity: $$ \mathrm{Re} = \frac{\rho \, \nu \, l}{\mu} $$ where $\rho$ is the density, $\mu$ is the dynamical coefficient of viscosity of the liquid or gas, $\nu$ is the typical rate of flow, and $l$ is the typical linear dimension.

The Reynolds number also determines the mode of flow of a liquid in terms of a critical Reynolds number, $\mathrm{Re}_{\mathrm{cr}}$. When $\mathrm{Re} < \mathrm{Re}_{\mathrm{cr}}$, only laminar liquid flow is possible, whereas when $\mathrm{Re} > \mathrm{Re}_{\mathrm{cr}}$ the flow may become turbulent.

The Reynolds number is named after O. Reynolds.


Comments

References

[a1] G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1974) pp. Sect. 4.7
[a2] M.I. Vishik, A.V. Fursikov, "Mathematical problems of statistical hydromechanics" , Kluwer (1988) pp. Chapts. 3; 4; 6 (Translated from Russian)
[a3] L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) (Translated from Russian)
How to Cite This Entry:
Reynolds number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reynolds_number&oldid=15768
This article was adapted from an original article by Material from the article "Reynolds number" in BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article