Numerical simulation of thermal plumes in a Herschel–Bulkley fluid

We present a three dimensional numerical study of thermal plumes, developing from a localized heat source in a yield stress and shear thinning fluid. We assume that the fluid viscosity follows a Herschel–Bulkley law with a low shear rate viscosity plateau. Comparison of the plume onset time and morphology observed in the numerical study and in laboratory experiments with Carbopol shows good agreement. An extensive parameter study allows us to identify two local non-dimensional parameters that determine whether a plume rises through the fluid. The first parameter is the Bingham number, Bi, which compares the yield stress to the viscous stress. The second parameter, the yield number Ψ, compares the stress induced by the buoyancy of an equivalent hot sphere to the yield stress. We find that a plume develops only if Ψ > Ψc = 5 ± 1.2 and Bi < Bic = 1. As the plume rises it loses its buoyancy due to heat diffusion. So the upward progression of the plume halts as soon as Ψ < Ψc or Bi > 1. Hot fluid continues to rise from the bottom of the tank but spreads under an unyielded, high viscosity region at the top of the box.

► Onset time strongly depends on yield parameter, comparing thermally induced stresses to yield stress.
► Finger-like thermal instability.
► Critical local parameter that allow for the plume to rise are determined.
► Unyielded regions coexist with thermal instability.