Abstract
A force balance is derived for a hemispherical particle in the viscous boundary layer at the wall of a horizontal pipe conveying Newtonian fluid; the hemisphere, of radius much less than that of the pipe, rests on the bottom with its flat face against the wall. The drag on the hemisphere is calculated from the creeping flow field of Price (Q. J. Mech. Appl. Math. Pt. 1 (1993)). This yields a prediction of the maximum velocity gradient at the wall for equilibrium, with limiting friction between the hemisphere and the wall. It is shown that the flow field of Price predicts a zero lift force but the validity of this, for actual flows, is questioned. Use of a hemisphere formulates a relevant well-posed problem, capable of mathematical solution. However, the flow field around real particles, e.g. sand, is complex, because of their irregular shapes, but the hemisphere work gives a qualitative indication of the behaviour of irregular particles. For turbulent flow in a pipe it is pertinent to consider a particle wholly within the viscous sub-layer, because it is isolated from significant turbulence and therefore hard to move; for such flow, the theory gives
Eq. (21) to predict the critical pipe velocity,
v
C
, for incipient motion of the hemisphere. For laminar flow, the wall shear rate is readily obtained from the parabolic velocity profile leading to Eq. (26) for
v
C
. The flow field of Price (and therefore the force acting on the hemisphere) is valid only for creeping flow (i.e. very low particle Reynolds number). Modifications to the force balance are tentatively suggested to account for inertial components to the drag force. The predictions of critical velocity are tested against our data for the incipient motion of small hemispheres at pipe walls in hydraulic conveying as well as new and previously published data for both hydraulic and pneumatic conveying. The new method of predicting incipient motion works well for both the pneumatic and hydraulic conveying of hemispheres and sand shaped particles but it overpredicts the critical velocity for more
rounded particles. The dependency of critical velocity on particle shape is under-researched.