ALBERT

Notes: Summary When investigating the influence of an acceleration upon the velocity distribution and upon the resistance in a tube or canal, a distinction can be made between slowly varying motions where the resistance dominates, and quickly varying motions where the inertia dominates. When the motion varies quickly, practically all the liquid moves bodily, and the resistance only affects a small region near the walls. When the motion varies slowly, the velocity distribution differs from that of steady flow in that there is a phase lag of the central layers with respect to the peripheral layers. The following special types of motion were studied for laminar flow: forced oscillations in a round tube; forced oscillations in a crevice; the starting motion by a constant drop of potential in a round tube; free oscillations in a round tube. For free oscillations in a U-tube, the theory is checked by comparison with experiments (table I). General slow motion was studied in the following cases: laminar flow in tubes of elliptical (circular) and rectangular (square) cross-section, or in open canals of rectangular cross-section; fully developed turbulent flow in round tubes and wide open canals. The relation between drop of potential and total flow can be represented by the impedance of the tube or canal. For quick laminar motions the impedance is given by (52) and (53) and characterized by the high frequency inertanceH. For slow laminar motions the impedance is given by (49) and (50) and characterized by the resistanceR and the low frequency inertanceL. For slow turbulent motions the relation between drop of potential and total flow is given by (73). The low frequency inertance is always greater than the high frequency inertance. The difference represents the change in the resistance caused by the influence of the acceleration upon the distribution of the velocity. This difference betweenL andH amounts to 33% for laminar flow in a round tube, 38% in a square tube, and 20% in a crevice or a wide open canal. For turbulent flow it ranges from 1 to 8% in a rough round tube, from 1/2 to 3% in a smooth round tube, from 1/2 to 3 1/2% in a rough open canal or crevice, and from 1/4 to 1 1/2% in a smooth open canal or crevice (table III). For laminar flow the ratioL: H is equal to the ratio of the mean square and the square mean velocity in a steady flow.