# A complete model of shear dispersion in pipes

• G. N. Mercer
• A. J. Roberts
Article

## Abstract

High order models of the longitudinal dispersion of a passive contaminant in Poiseuille pipe flow are derive and their validity discussed. The derivation is done using centre manifold theory which provides a systematic and consistent approach to calculating each successive approximation. A stable, non-negative finite difference scheme is formulated which matches the evolution equation to a predetermined order. The limitations imposed by this matching is investigated. The appropriate initial conditions to use for the Taylor model of shear dispersion in pipes are derived. It is shown that the commonly used initial condition of simply taking the cross-sectional average is only a first approximation to the correct initial condition. In a similar manner the correct boundary conditions to be used at the inlet and outlet of a finite length pipe are derived. The generalisation to a pipe with varying cross-section and varying flow properties is studied and the resultant modifications to the advection velocity and the effective dispersion coefficient are calculated. An example of an exponentially varying pipe is considered and the differences between this approach and the classical Taylor theory are examined.

### Key words

shear dispersion centre manifold pipes

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