Manchester Applied Mathematics and Numerical Analysis Seminars Winter 1999/2000

November 10, 1999, 2.30 pm

Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Displacing Visco-plastic Fluids in Long Channels

Dr. Ian Frigaard, Etudes et Productions, Schlumberger-Dowell, Clamart, France

Using a lubrication approximation, it is shown that sufficient conditions for the non-existence of a static wall layer can be computed simply in terms of two dimensionless parameters: the Bingham number for the displacing fluid (B_1) and the ratio of the yield stresses of the two fluids (j_Y). When these conditions are not met, it is possible to compute the maximum possible static wall layer thickness h_max, which depends on B_1, j_Y and on a third dimensionless parameter j_B, a buoyancy to yield stress ratio.

On computing displacements using the lubrication approximation, the interface is observed to asymptotically approach the maximum static layer thickness . Results from fully two-dimensional displacement computations are also presented. These indicate that the displacement front propagates at a steady speed along the channel, leaving behind a static layer which is significantly thinner than h_max . Surprisingly, the computed static layer thickness is observed to decrease with a parametric increase in the dimensionless yield stress of the displaced fluid.

To explain these results we analyse the streamline configuration close to a steadily advancing displacement front. We demonstrate heuristically that the local visco-plastic dissipation functional will be approximately minimised by a critical layer thickness at which the displaced fluid begins to recirculate ahead of the displacement front. Comparison of the critical recirculation limit with the static layer thickness computed from the fully transient model gives a very close agreement, suggesting that a form of energy minimisation is responsible in this case for selecting the static layer thickness.

Considering now the problem of a steady state propagation, for a given fixed interface, chosen from a wide class of physically sensible interface shapes, we show that there exists a unique solution. As well as flexibility in the exact shape of the interface, the residual static layer thickness is also non-unique. Typically layer thicknesses h_l(h_min, h_max) admit a physically sensible static layer solution, where h_min and h_max are easily computable functions of the dimensionless problem parameters. The dependency of h_min and h_max on the dimensionless problem parameters is explained and example solutions are computed for different static residual thicknesses.

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).