Sedimentation motion of sand particles in moving water (I): The resistance on a small sphere moving in non-uniform flow

The Maxey-Riley equation, which describes the force and motion of a single small-Reynolds-number spherical particle in an unsteady and non-uniform flow field, forms the basis for the study of particulate two-phase flows. The equation was published by Martin R. Maxey and James J. Riley in Physics of Fluids (26(4), 883-889) in 1983 and has had an important impact on the following research (as of August 29, 2022, the paper has been cited by 2250 times, including 129 citations for year 2021). However, as early as 1956-1957, when Prof. Shu-tang Tsai worked on the sediment settlement in rivers under the guidance of Prof. Pei-Yuan Chou, he conducted in-depth research on the basic hydrodynamic problem related to the force of spherical particles in the flow field, and published a series of results in the Acta Physica Sinica in Chinese, including the non-uniform sedimentation motion of particles in hydrostatic water (1956, 12(5), 409-418), the force on particles in arbitrary flow field (1957, 13(5), 388-398, henceforth referred as 1957a), and sediment settlement in laminar flows (1957, 13(5), 399-408). Among them, the title of the 1957a paper is “Sedimentation motion of sand particles in moving water (I) The resistance on a small sphere moving in non-uniform flow”. In this paper, the equation governing the force and motion of a single small spherical particle under general flow conditions was derived, and is essentially identical to what is now known as the Maxey-Riley equation. Since Mr. Tsai’s paper was published in Chinese, it has long been unknown to the international academic community. In this issue, we publish the English version of the 1957a paper, which was translated by Prof. Haitao Xu from Tsinghua University, to memorize Prof. Shu-tang Tsai and let more people aware of his pioneering work in the field of particulate two-phase flows. In hydraulics, when we deal with the problem of sand particles moving relative to the surrounding water, Stokes’ formula of resistance has usually been used to render the velocity of sedimentation of the particles. But such an approach has not been proved rigorously, and its accuracy must be carefully considered. In this paper, we discuss the problem of a sphere moving in a non-uniform flow field, on the basis of the fundamental theory of hydrodynamics. We introduce two assumptions: i) the diameter of the sphere is much smaller than the linear dimension of the flow field, and ii) the velocity of the sphere relative to the sur