Poiseuille flow

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

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

where 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 . 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