Drift, partial drift and Darwin's proposition
A body moves at uniform speed in an unbounded inviscid fluid. Initially, the body is infinitely far upstream of an infinite plane of marked fluid; later, the body moves through and distorts the plane and, finally, the body is infinitely far downstream of the marked plane. Darwin (1953) suggested that the volume between the initial and final positions of the surface of marked fluid (the drift volume) is equal to the volume of fluid associated with the ‘added-mass’ of the body.
We re-examine Darwin's (1953) concept of drift and, as an illustration, we study flow around a sphere. Two lengthscales are introduced: ρmax, the radius of a circular plane of marked particles; and x0, the initial separation of the sphere and plane. Numerical solutions and asymptotic expansions are derived for the horizontal Lagrangian displacement of fluid elements. These calculations show that depending on its initial position, the Lagrangian displacement of a fluid element can be either positive – a Lagrangian drift – or negative – a Lagrangian reflux. By contrast, previous investigators have found only a positive horizontal Lagrangian displacement, because they only considered the case of infinite x0. For finite x0, the volume between the initial and final positions of the plane of marked fluid is defined to be the ‘partial drift volume’, which is calculated using a combination of the numerical solutions and the asymptotic expansions. Our analysis shows that in the limit corresponding to Darwin's study, namely that both x0 and ρmax become infinite, the partial drift volume is not well-defined: the ordering of the limit processes is important. This explains the difficulties Darwin and others noted in trying to prove his proposition as a mathematical theorem and indicates practical, as well as theoretical, criteria that must be satisfied for Darwin's result to hold.
We generalize our results for a sphere by re-considering the general expressions for Lagrangian displacement and partial drift volume. It is shown that there are two contributions to the partial drift volume. The first contribution arises from a reflux of fluid and is related to the momentum of the flow; this part is spread over a large area. It is well-known that evaluating the momentum of an unbounded fluid is problematic since the integrals do not converge; it is this first term which prevented Darwin from proving his proposition as a theorem. The second contribution to the partial drift volume is related to the kinetic energy of the flow caused by the body: this part is Darwin's concept of drift and is localized near the centreline. Expressions for partial drift volume are generalized for flow around arbitrary-shaped two- and three-dimensional bodies. The partial drift volume is shown to depend on the solid angles the body subtends with the initial and final positions of the plane of marked fluid. This result explains why the proof of Darwin's proposition depends on the ratio ρmax/x0.
An example of drift due to a sphere travelling at the centre of a square channel is used to illustrate the differences between drift in bounded and unbounded flows.(Published Online April 26 2006)
(Received September 27 1993)
(Revised April 8 1994)
p1 Present address: Department of Meteorology, University of Reading, Reading, RG6 2AU, UK