## 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 center-line of the pipe entrance and z-direction aligned with the center-line (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 center-line (r = 0) yields [see Richardson (1989)]

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

whereas the mean axial velocity ū_{z} is given by

from which it follows that ū_{z} = u_{rmax}/2 . The volumetric flow rate through the pipe is given by

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

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

The *viscous dissipation rate***ε** is given by

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)]

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

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)].

#### REFERENCES

Happel, J. and Brenner, H. (1973) *Low Reynolds Number Hydrodynamics*, Noordhoff, Leyden.

Richardson, S. M. (1989) *Fluid Mechanics*, Hemisphere, New York.

Shah, R. K. and London, A. L. (1978) Laminar flow forced convection in ducts. *Advances in Heat Transfer, Supplement 1,* Academic Press, New York.

#### References

- Happel, J. and Brenner, H. (1973)
*Low Reynolds Number Hydrodynamics*, Noordhoff, Leyden. - Richardson, S. M. (1989)
*Fluid Mechanics*, Hemisphere, New York. - Shah, R. K. and London, A. L. (1978) Laminar flow forced convection in ducts.
*Advances in Heat Transfer, Supplement 1,*Academic Press, New York.

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