Abstract The efficiency of the heat transfer mechanism for the fully developed steady Poiseiulle flow of a viscus incompressible fluid through a pipe of uniform but arbitrary cross section, when the wall of the pipe is maintained at a constant temperature, is measured by the average Nusselt number Nu . It is shown that Nu is proportional to the well known Rayleigh quotient; and thus a lower bound on Nu , which is independent of the physical or geometrical parameters of the flow, is derived using the results in isoperimetric inequalities. Lower and upper bounds on Nu , dependent only on geometrical parameters of the pipe are also obtained with a view to get some bounds on the range of efficiency of the heat transfer.