a1 Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, Quebec, Canada H3C 3J7 email: firstname.lastname@example.org
a2 Department of Mathematics and Department of Mechanical Engineering, University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, British Columbia, Canada V6T 1Z4 email: email@example.com
a3 Department of Computer Science, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria email: Otmar.Scherzer@uibk.ac.at
a4 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2 email: firstname.lastname@example.org
We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.
(Received July 18 2006)
(Revised March 21 2007)