Bridging Power-Law and Newtonian Regimes in Sisko Fluid Flow and Heat Transfer

Pressure-driven flow and heat transfer of a Sisko fluid between parallel plates are investigated under steady, laminar, fully developed conditions with uniform wall heat flux and viscous dissipation. A dimensionless Sisko parameter, the ratio of the Newtonian to the power-law stress contribution, distinguishes power-law-dominated, transition, and Newtonian-dominated regimes and depends on the operating conditions as well as the fluid. Second-order perturbation solutions give closed forms for the velocity, temperature, mean velocity, bulk mean temperature, Nusselt number, and critical Brinkman number in the limiting regimes, and Padé approximations extend them to the transition regime. Comparisons with boundary-value solutions show close agreement within the tabulated validity ranges; for a power-law index of 0.6 and unit Brinkman number, the transition approximations keep the mean-velocity error below 5% and the Nusselt-number error below 0.8% for Sisko parameters up to 3. Stronger shear-thinning flattens the velocity profile and enhances heat transfer in the power-law-dominated regime, whereas a larger Newtonian contribution weakens the effects of the flow index and viscous dissipation. Under wall cooling, increasing the Newtonian contribution or shear-thinning shifts the critical Brinkman number toward more negative values. The closed-form results provide estimates of flow resistance and heat transfer for Sisko fluids in narrow channels.