Application of the Hagen—Poiseuille Equation to Fluid Feeding through Short Tubes

Abstract

The Hagen-Poiseuille equation has been widely applied to the study of fluid feeding by insects that have sucking (haustellate) mouthparts. However, the equation is valid only when the length of the cylinder is much longer than the entrance length (the length of the entrance region within which the flow is not fully developed). To estimate under what circumstances the flow through short tubular mouthparts would be slower than predicted from the Hagen-Poiseuille equation, we determined the empirical relationship between the relevant parameters using physical models made of glass micropipets. Fluid flow was measured while applying known pressure differences to cylinders of different geometries for a range of entrance lengths. We found a simple relationship between the flow and the entrance length in dimensionless terms for the range of experimental conditions (length:diameter range, 5:1-150:1; Re range, 18-1100). Thus, it is now possible to estimate easily correction factors resulting from the effects of the entrance region. The resistance to flow within the entrance region is sufficiently high that if the entrance length is one-half of the total length, the expected flow is only ≈80% of that predicted from the Hagen-Poiseuille equation. Reported flow rates through channels in insect mouthparts are sufficiently slow that the predicted entrance lengths are extremely small relative to their total lengths. Therefore, with the possible exception of some gigantic extinct insects, the Hagen-Poiseuille equation may be applied safely to fluid flow through even short feeding tubular structures (when the other assumptions are satisfied).