Some applications of extremum principles to magnetohydrodynamic pipe flow

Abstract

This paper is concerned with the application of extremum principles to the laminar flow of a conducting fluid along a pipe with conducting walls. The extremum principles provide upper and lower bounds to the mass-flow rate Q . While these may supply numerical bounds for Q their main application lies in the construction of asymptotic series at large Hartmann numbers. The most important result is a formula for the leading coefficient in the asymptotic series for Q for a wide class of pipe sections with thick conducting walls. A number of examples are given. A particular example is the square channel with thin conducting walls and it is shown how the ‘thin wall’ approximation can be derived from the extremum principles.