The hole pressure problem: On the Higashitani-Pritchard theory for transverse and axial slots

Abstract

Analysis of the Higashitani-Pritchard (HP) theory for a transverse slot provides insight into the nature of the errors involved. The effects of geometry, elasticity and asymmetry on the errors in the HP theory are also discussed. Inertial effects and the effect of asymmetry are included in the HP theory for a transverse slot, through modification of the pressure term in the momentum equation; the extended HP theory proposed in this work provides evidence as to the reasons for the failure of the Newtonian hole pressure as a correction term at higher Reynolds numbers. Finally, the HP theory is used to analyse a control volume of fluid in the axial slot geometry and deduce a rigorous sufficient condition for secondary flows to exist. This analysis illustrates the nature of the role played by the second normal-stress difference (N2) in causing secondary flows, and suggests the possibility of the direction of the secondary flows being related to the sign ofN2. A computational study of all three aspects of the HP theory investigated in this work seems worth-while.