Natural convection flows of a Bingham fluid in a long vertical channel

Abstract

We analyze the 1D flow of a Bingham fluid between two differentially heated vertical plates, in the presence of a stabilizing vertical temperature gradient, imposed at the walls. The solution is parameterized by the Bingham number, B, and the stratification parameter Γ, and is surprisingly complex. When B⩾Bcr=1/16 the fluid is unyielded everywhere and heat transfer is via pure conduction. We refer to this as a 1-plug solution. For B≲Bcr, a perturbation solution shows that yielding starts at the walls and the centerline of the channel, breaking into two asymmetric counter-current streams and with a single plug in each stream (a 2-plug solution). We characterize the solution regimes in the Γ–B plane in terms of the number of plugs that are found. We identify the main characteristics of these solutions and provide data suitable for numerical benchmarking. For increasing Γ and decreasing B, we show that in principle, an arbitrarily large number of plugs can be found in the finite width channel. Primarily we solve for the 1-plug (conductive), 2-plug and 3-plug solutions, which are found to dominate the Γ–B parameter space.