# Poiseuille Flow

Poiseuille flow is

*pressure-induced*flow ( Channel Flow) in a long duct, usually a pipe. It is distinguished from drag-induced flow such as Couette Flow. Specifically, it is assumed that there is Laminar Flow of an incompressible Newtonian Fluid of viscosity η) induced by a constant positive pressure difference or pressure drop Δp in a pipe of length L and radius R << L. By a pipe is meant a right circular cylindrical duct that is a duct with a circular crosssection normal to its axis or generator.
Because of the geometry, Poiseuille flow is analyzed using cylindrical polar coordinates (r, θ, z) with origin on the centerline of the pipe entrance and z-direction aligned with the centerline (see Figure 1). Symmetry means that Poiseuille flow is swirl-free and axisymmetric. Thus, the only nonzero components of the velocity u are the radial component u

_{r}and the axial component u_{z}: the angular component uθ = 0. Moreover, u_{r}and u_{z}are independent of θ, as is the pressure p. Because the pipe is long, Poiseuille flow is fully developed, that is the velocity u is independent of axial position z everywhere except near the entrance (z = 0) and exit (z = L) of the pipe, from which it follows that u_{r}= 0. Solution of the mass and linear momentum*conservation equations,*specifically the*Navier-Stokes equations*, with boundary conditions of no-slip at the pipe wall (r = R) and symmetry at the centerline (r = 0) yields [see Richardson (1989)]
(1)

Thus the axial velocity profile is parabolic (see Figure 2). The maximum axial velocity u

_{rmax}occurs at the center-line (r = 0) and is given by
(2)

whereas the mean axial velocity ū

_{z}is given by
(3)

from which it follows that ū

_{z}= u_{rmax}/2 . The volumetric flow rate through the pipe is given by
(4)

This is the

*Hagen-Poiseuille Equation*, also known as*Poiseuille's Law.*Experimentally, Eq. (4) is found to be corroborated provided the Reynolds Number (Re) given by
(5)

is less than some critical value Re

_{c}= 2000, so that there is laminar flow and not Turbulent Flow, though it should be noted that Re_{c}appears to be very much larger than 2000 if particular care is taken to minimize disturbances which might cause flow instabilities; and provided L/R >> C Re where C ~ 0.1, so that the pipe is long enough for entrance and exit effects to be negligible and hence for u to be fully-developed.
Poiseuille flow is a shear flow with shear-rate γ given by

(6)

The

*viscous dissipation rate*ε is given by
(7)

Dissipation of mechanical energy, that is conversion of mechanical energy (specifically pressure energy) into thermal energy by viscous action, increases the temperature T of the fluid. Assuming, that the temperature T is fully developed, the solution of the energy conservation equation with boundary conditions of specified temperature T

_{w}at the pipe wall (r = R) and symmetry at the center-line (r = 0) yields [see Richardson (1989)]
(8)

where λ denotes thermal conductivity. Experimentally, (8) is found to be corroborated provided L/R >> C Pe where C ~ 0.1 and the Peclet Number (Pe) is given by

(9)

where c denotes specific heat. The pipe is then long enough for entrance and exit effects to be negligible and hence for T to be fully-developed.

For Poiseuille flow in channels or ducts of noncircular crosssection, analogous expressions can be obtained for velocity and temperature [see Happel and Brenner (1973) and Shah and London (1978)].