Non-Newtonian viscoplastic (or yield stress) fluids are materials that behaves like a rigid solid body when a small shear stress is applied. Once the yield stress exceeds a certain critical value τ0 then the materiel deforms and flows as a viscous fluid. Practically, such fluid flow appears in many industrial processes, as in the food industry (Ketchup, mayonnaise, chocolate,...), the petroleum industry (drilling mud, cement, waxy crude oil,...), the biological industry (mucus, blood clot,...), the natural phenomena (flows of slurries, debris and lava,...), and in various complex printing and coating processes. The two-parameter Bingham fluid was the first studied model and in different geometries. Later, the researchers proposed three-parameter viscoplastic models such as the Herscel-Bulkley, Casson or Vocadlo (sometimes called Robertson-Stiff one) fluids, etc.

In this paper, we present a comparative study of the peristaltic pumping between the Herschel-Bulkley and the Vocadlo models in the presence of heat transfer through a deformable planar channel with a sinusoidal wave traveling down its walls. This work presents an interaction fluid-structure problem. The flow of the blood or of the food bolus in the human body, the transportation of high-viscosity fluids and solid-liquid mixtures, include cement plants, sewage treatment plants, sanitary fluid, food plants are some physiological and industrial applications of peristalsis.

Experimentally, Beirute and Flumerfelt (1977) presented a comparative data (for cement slurry along with fitting curves) between the two proposed fluids. The authors found that the Vocadlo (or RS) model is somewhat superior and they proposed it as an improved one for cement slurries even for different cement slurry of higher yield. Theoretically, we will compare (through graphs) the axial velocity profile, pressure gradient, pressure rise, frictional force, mechanical efficiency, temperature profile, entropy generation and the Bejan number between these two fluids or versus other fluids such as the Newtonian, power-law and Bingham models. In addition, we will investigate the effects of the rheological parameters (the yield stress or the threshold τ and the power index of fluid n) on the last physical quantities.