View source for 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 $x$ 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|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====
<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>