Spatial Decay Estimates in Time-Dependent Stokes Flow

Abstract

This paper considers the time-dependent slow flow of an incompressible viscous fluid in a semi-infinite cylindrical pipe of smooth cross section. An exponential decay estimate in terms of the distance from the finite end of the pipe is obtained from a second-order differential inequality for a weighted energy integral defined on the solutions of the system. The decay constant depends only on the geometry and the first positive eigenvalues for the fixed and free membrane problems for the cross sectional geometry. The paper also indicates how to bound the total weighted energy.