Natural convection of a viscoplastic fluid in an enclosure filled with solid obstacles

In this work, the natural convection of a viscoplastic fluid inside an enclosure filled with many obstacles is solved numerically. The enclosure is characterized by differentially heated vertical walls, while the obstacles are heat-conducting square blocks. Due to the nature of the viscoplastic material, it is seen that if the buoyancy force is not strong enough, the whole fluid can be unyielded, and the heat transfer takes place only by conduction. The Bingham constitutive equation is chosen to represent the yield-stress material. Steady-state velocity and temperature fields are obtained for a wide range of Rayleigh and Bingham numbers and several numbers of blocks. We show that increasing the yield-stress has a quantitative effect on reducing the fluid circulation, and consequently, the heat transfer rate. At a certain yield-stress level, the fluid motion is abruptly interrupted. The critical Bingham number for complete unyielding does not depend on the Rayleigh or Prandtl numbers. However, the obstacles' configuration has a significant effect on this transition. Flow vs. no-flow diagrams and average Nusselt number correlations are proposed as aid for thermal engineering of similar arrangements.