Decay estimates for steady magnetohydrodynamic pipe flow

Abstract

This paper establishes exponential decay bounds for steady magnetohydrodynamic pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of Ames and Payne (SIAM J. Math. Anal. 20 (1989) 789) and Horgan and Wheeler (SIAM J. Appl. Math. 35 (1978) 97), the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is fully developed at the exit section. Energy inequalities are derived that lead to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy.