The constant flow rate problem for fluids with increasing yield stress in a pipe

Experiments show that the degradation effect observed during both stirring and pipelining tests of some coal-water slurries is mainly to be ascribed to the increase of the yield stress. Regardless of the particular mathematical model adopted to investigate the dynamics of these fluids, engineering applications force us to consider the problem of how long a constant flow rate can be maintained during the pipelining process. We choose a Bingham model where the yield stress is assumed to increase with the dissipated energy as in [5]. It is first shown that the constant flow rate problem is equivalent to solving a nonlinear functional equation in the unknown pressure gradient that generalizes the classical algebraic Buckingham equation for the same problem with constant rheological parameters. By means of a fixed-point argument we also prove that the functional equation has one and only one solution which is local in time. We finally find an estimate from below of the interval of the interval of existence. Numerical results are rather good and agree with those expected from the engineering point of view.