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Difference between revisions of "Poiseuille flow"

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The flow of a homogeneous viscous incompressible fluid in a long tube of circular cross section. For a steady flow in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732601.png" /> direction the flow equation is
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The flow of a homogeneous viscous incompressible fluid in a long tube of circular cross section. For a steady flow in the $x$ direction the flow equation is
  
<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/p/p073/p073260/p0732602.png" /></td> </tr></table>
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$$\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=-\mu^{-1}G,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732603.png" /> is the pressure gradient and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732604.png" /> is the viscosity. For Poiseuille flow the flow is assumed to have the same axial symmetry as the boundary conditions, hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732605.png" /> is a function of the distance from the axis of the tube only. The solution with boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732606.png" /> at the boundary of the tube and no singularity at the axis is
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where $G$ is the pressure gradient and $\mu$ is the viscosity. For Poiseuille flow the flow is assumed to have the same axial symmetry as the boundary conditions, hence $u$ is a function of the distance from the axis of the tube only. The solution with boundary value $0$ at the boundary of the tube and no singularity at the axis is
  
<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/p/p073/p073260/p0732607.png" /></td> </tr></table>
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$$u(r)=\frac{G}{4\mu}(a^2-r^2),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732608.png" /> is the radius of the tube. This flow was studied by G. Hagen in 1839 and by J.L.M. Poiseuille in 1940.
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where $a$ is the radius of the tube. This flow was studied by G. Hagen in 1839 and by J.L.M. Poiseuille in 1940.
  
The Poiseuille flow is stable for a small [[Reynolds number|Reynolds number]], and becomes unstable at higher Reynolds numbers. This was established experimentally by O. Reynolds in 1883. For Poiseuille flow the critical Reynolds number is around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073260/p0732609.png" />. For a discussion of hydrodynamic instability and bifurcation of Poiseuille flow and other laminar flows, such as Couette flow (the steady circular flow of a liquid between two rotating co-axial cylinders) see [[#References|[a1]]], [[#References|[a2]]]. See also [[Orr–Sommerfeld equation|Orr–Sommerfeld equation]].
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The Poiseuille flow is stable for a small [[Reynolds number|Reynolds number]], and becomes unstable at higher Reynolds numbers. This was established experimentally by O. Reynolds in 1883. For Poiseuille flow the critical Reynolds number is around $2\cdot10^3$. For a discussion of hydrodynamic instability and bifurcation of Poiseuille flow and other laminar flows, such as Couette flow (the steady circular flow of a liquid between two rotating co-axial cylinders) see [[#References|[a1]]], [[#References|[a2]]]. See also [[Orr–Sommerfeld equation|Orr–Sommerfeld equation]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Chandrasekhar,  "Hydrodynamics and hydrodynamic stability" , Dover, reprint  (1981)  pp. Chapt. VII</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Th.J.R. Hughes,  J.E. Marsden,  "A short course on fluid mechanics" , Publish or Perish  (1976)  pp. §18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. 180ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Chandrasekhar,  "Hydrodynamics and hydrodynamic stability" , Dover, reprint  (1981)  pp. Chapt. VII</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Th.J.R. Hughes,  J.E. Marsden,  "A short course on fluid mechanics" , Publish or Perish  (1976)  pp. §18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1974)  pp. 180ff</TD></TR></table>

Latest revision as of 07:09, 12 August 2014

The flow of a homogeneous viscous incompressible fluid in a long tube of circular cross section. For a steady flow in the $x$ direction the flow equation is

$$\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=-\mu^{-1}G,$$

where $G$ is the pressure gradient and $\mu$ is the viscosity. For Poiseuille flow the flow is assumed to have the same axial symmetry as the boundary conditions, hence $u$ is a function of the distance from the axis of the tube only. The solution with boundary value $0$ at the boundary of the tube and no singularity at the axis is

$$u(r)=\frac{G}{4\mu}(a^2-r^2),$$

where $a$ is the radius of the tube. This flow was studied by G. Hagen in 1839 and by J.L.M. Poiseuille in 1940.

The Poiseuille flow is stable for a small Reynolds number, and becomes unstable at higher Reynolds numbers. This was established experimentally by O. Reynolds in 1883. For Poiseuille flow the critical Reynolds number is around $2\cdot10^3$. For a discussion of hydrodynamic instability and bifurcation of Poiseuille flow and other laminar flows, such as Couette flow (the steady circular flow of a liquid between two rotating co-axial cylinders) see [a1], [a2]. See also Orr–Sommerfeld equation.

References

[a1] S. Chandrasekhar, "Hydrodynamics and hydrodynamic stability" , Dover, reprint (1981) pp. Chapt. VII
[a2] Th.J.R. Hughes, J.E. Marsden, "A short course on fluid mechanics" , Publish or Perish (1976) pp. §18
[a3] G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1974) pp. 180ff
How to Cite This Entry:
Poiseuille flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poiseuille_flow&oldid=16158