Time-averaged Quantities Research Articles

A long wave approximation to peristaltic motion

The problem of motion of a viscous incompressible fluid induced by travelling wave motions of the confining walls has been studied analytically for the two-dimensional geometry. The analysis is aimed at the possible application to urine flow in human ureters. The wave length, λ, of the peristaltic waves is assumed to be large compared to the half channel width, d, whereas the amplitude of the wave, a, need not be small compared to d. A systematic approach based on an asymptotic expansion of the solution in terms of the small parameter d λ has been used and solutions up to O( d 2 λ 2 ) have been presented in closed forms. The limiting solution has been studied including the effects of externally applied pressure gradients, and the criteria for backward flow have been established and discussed in detail. Higher order solutions which include the effects of non-linear inertial terms of the Navier-Stokes equations, have been studied for the case of zero mean volume flow. The results indicate that the peristaltic waves on the wall do not contribute to the time-averaged flow quantities to O( d λ ) , but they do give rise to a mean axial velocity of O( d 2 λ 2 ) , and a mean pressure gradient of O( d 2 λ 2 ) is associated with the motion. Comparisons with some earlier results have been made and certain agreements are brought out.

Second‐order chemical reactions in a nonhomogeneous turbulent fluid

AbstractVery rapid, rapid, and slow second‐order reactions were studied in an isothermal turbulent flow reactor. The two aqueous reactant solutions were separately introduced through many alternate jets and the reaction took place in the resulting nonhomogeneous mixture. Very rapid reactions were diffusion controlled and were in agreement with earlier theory. All reactions followed second‐order rate laws based on time average quantities. The apparent reaction velocity constant was controlled by the mixing for very rapid reactions, by the chemical kinetics for slow reactions, and by both mechanisms for rapid reactions.