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

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.