Poiseuille flow

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