Abstract The development of thermal convection is studied theoretically for a viscoplastic fluid. If the fluid has finite viscosity at zero shear rate, the critical Rayleigh number for convective instability takes the same value as for a Newtonian fluid with that viscosity. The subsequent weakly nonlinear behaviour depends on the degree of shear thinning: with a moderately shear-thinning nonlinear viscosity, the amplitude of convective overturning for a given temperature difference is increased relative to the Newtonian case. If the reduction in viscosity is sufficiently sharp the transition can even become subcritical (a detail particularly relevant to regularized constitutive laws). For a yield-stress fluid, the critical Rayleigh number for linear instability is infinite as the motionless layer is held rigid by the yield stress. Nonlinear convective overturning, however, still occurs and we trace out how the finite-amplitude solution branches develop from their Newtonian counterparts as the yield stress is increased from zero for the Bingham fluid. Preliminary laboratory experiments with a layer of Carbopol fluid heated from below, confirm that yield strength inhibits convection but convection will initiate with a sufficient kick to the system
