Annular Couette–Poiseuille flow and heat transfer of a power-law fluid – analytical solutions

Abstract

Analytical solutions involving hypergeometric functions are presented for the fully developed laminar flow of a Non-Newtonian fluid in an annulus with a stationary or a moving inner boundary. The solutions are valid for arbitrary values of the power law index in the Couette–Poiseuille flow under consideration, and consider scenarios with and without extrema in the velocity profile, in addition to flow against adverse pressure gradients. Criteria are identified for the disappearance and reappearance of an extreme point, development of flow reversal, and occurrence of an adverse pressure gradient. Fully developed Heat Transfer is also analysed with three different types of boundary conditions and analytical solutions are presented for the case of both inner and outer annulus boundaries maintained at constant but possibly different heat fluxes. A non-homogeneous Cauchy-Euler ODE is solved numerically for the case in which both annulus boundaries are at the same temperature. Correlations for the friction factor and for the Nusselt numbers at the inner and outer walls are presented in terms of the annulus radius ratio, the power law index and the ratio of the wall-to-bulk velocities. It is shown how a multiplicative correction can be used to extend the applicability of the API-RP-13D prescription for the friction factor to flow scenarios involving a moving inner wall as is common in many oil well circulating scenarios. Results of a parametric sensitivity analysis are presented and indicate that the influence of the power law index on the friction factor is significant only in conjunction with the inner wall velocity, and that its influence on the Nusselt numbers is largely minimal. The methodology is validated with exact closed form solutions for limiting cases, and with results in the literature. From a practical perspective, the results will prove useful in the context of hydraulics and heat transfer in complex oilfield wellbores, as well as in the process industry.