科研之友
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摘要
The drift volume D refers to the volume of fluid enclosed between the initial and final profiles of an initially flat marked sheet of fluid that deforms due to the passage of a body. Classic investigations of the drift volume in an inviscid fluid show that D is comparable in magnitude to the volume of the translating body. The situation is fundamentally different for a viscous fluid. For instance, if the Reynolds number Re is zero, then D diverges with the distance traveled by the body and is typically orders of magnitude larger than the body volume. The goal of the present paper is to quantify the drift volume when Re is finite. We interpret the drift volume as the flux through a stationary plane bounded by a stream tube, thus allowing D to be computed without explicit reference to the trajectories of the marked fluid elements. We focus our analysis on a rigid sphere of radius a that is steadily towed through an unbounded viscous fluid by an external force. The initial profile of the marked fluid is taken to be a disk of finite height h, and h is furthermore assumed to be much larger than a. A two-term asymptotic expansion of D for h/a -> infinity is computed for Re = 0 and compared to the analogous result for inviscid flow in order to illustrate the fundamental differences between these two cases. Next, a leading-order approximation to D for a sphere translating at small Re is computed using Oseen's approximation to the flow. These results are extended to Re >= O(1) by taking advantage of the fact that, at distances greatly exceeding its radius, the flow around a sphere is described by the point force solution to Oseen's equations. Therefore, it is found universally that D diverges as the distance traveled by the sphere becomes large at finite Re. However, the exact nature of this divergence critically depends upon the ratio of the distance traveled by the sphere to the radius of the initially marked disk of fluid (h).
- 出版日期2017-6-9
