Hydrodynamic stability and natural convection in Ostwald‐de Waele and Ellis fluids: The development of a numerical solution

TLDR

An algorithm was developed using finite‑difference methods to compute hydrodynamic stability and natural convection in Ostwald‑de Waele and Ellis non‑Newtonian fluids, with test calculations performed for roll‑cells under rigid and dragless vertical boundaries and thorough sensitivity analyses of time‑step and grid‑size. The simulations showed independence from initial conditions, reproduced experimental Nusselt and critical Rayleigh numbers for Newtonian fluids, and revealed that the Tien‑Tsuei‑Sun approximation underestimates the critical Rayleigh number for Ostwald‑de Waele fluids.

Abstract

Abstract An algorithm was developed for the finite‐difference computation of hydrodynamic stability and natural convection in non‐Newtonian fluids heated from below. Test calculations were carried out for fluids whose viscosity characteristics are described by the Ostwald‐de Waele (power‐law) and Ellis models and for roll‐cells with both rigid and dragless vertical boundaries. The effects of time‐step and grid‐size were tested thoroughly. The results were found to be independent of the assumed initial state. The computed values of the Nusselt number and the critical Rayleigh number for Newtonian fluids agree well with prior experimental results. The computations for the Ostwald‐de Waele model indicate that the approximate solution of Tien, Tsuei, and Sun may underestimate the critical Rayleigh Number.