Laminar Bingham fluid flow between vertical parallel plates

Packer fluids for deep-water oil and gas wells are being developed currently to minimize the rate of heat transfer from the flowing production fluid to the outer casing annuli. In this work we study a gel that has a yield point capable of preventing or drastically reducing natural convective fluid flow and therefore the heat transfer that otherwise would occur from the production tubing to the production casing. The gel is modeled as a Bingham material. The tubing-to-casing annulus is geometrically modeled as vertical and large parallel plates. This modeling is appropriate as the radial extent of the annulus containing the fluid is usually small compared to the mean radius of the annulus. The flow is assumed to be laminar, and in order to provide a reference case the solution to the linear viscous flow is first presented. The natural convection problem of the Bingham fluid is described in five distinct regions within the gap between parallel plates, progressing from hotter to the cooler plate. The velocity and shear stress distributions with some examples from the oil industry are given. Equivalent dimensionless numbers are developed for the Bingham fluid in order to be able to use the available linear viscous correlation equations. The correlation results characterize the heat transfer performance of the gel.