E. O. Tuck

A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows

Some draining or coating fluid-flow problems, in which surface tension forces are important, can be described by third-order ordinary differential equations. Accurate computations are provided here...

Shallow-water flows past slender bodies

The problem solved concerns the disturbance to a stream of shallow water due to an immersed slender body, with special application to the steady motion of ships in shallow water. Formulae valid to first order in slenderness are given for the wave resistance and vertical forces at both sub- and supercritical speeds. The vertical forces are used to predict sinkage and trim of ships and satisfactory comparisons with model experiments are made.

Unsteady spreading of thin liquid films with small surface tension

The method of matched asymptotic expansions is used to solve for the free surface of a thin liquid drop draining down a vertical wall under gravity. The analysis is based on the smallness of the surface tension term in the lubrication equation. In a region local to the front of the drop, where the surface curvature is large, surface tension forces are significant. Everywhere else, the surface curvature is small, and surface tension plays a negligible role. A numerical time‐marching scheme, which makes no small surface tension assumptions, is developed to provide a datum from which to gauge the accuracy of the small surface tension theory. Agreement between the numerical scheme and the small surface tension theory is good for small values of surface tension. Extension to the propagation of drops by spinning and by blowing with a jet of air is also discussed. It is shown that there are inherent similarities between all three spreading mechanisms.

Calculation of unsteady flows due to small motions of cylinders in a viscous fluid

SummaryThe problem of small oscillations of a cylinder of general cross-section in a viscous fluid is formulated in terms of integral equations. Numerical solutions of the integral equation are presented for the special case of a ribbon of zero thickness.

A cusp-like free-surface flow due to a submerged source or sink

Abstract : We consider the steady flow induced by a line source placed at a given depth beneath the undisturbed level of a free-surface. We assume that the flow bifurcates at a definite point somewhere between the source and the undisturbed free-surface level. The free-surface at this point is cusp-like, the tip of the cusp pointing toward the source. The model is motivated by hydraulic problems of water entry or extraction from a reservoir. The problem is solved numerically by collocation. A unique solution is obtained whose cusp lies at 74.938% of the depth of the source. (Author)

On the oscillations of harbours of arbitrary shape

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.

Some methods for flows past blunt slender bodies

It is suggested that the use of prolate spheroidal co-ordinates in certain problems involving slender bodies may lead to results which not only are more likely to be uniformly valid for blunt bodies, but in many cases require less complicated analysis than results obtained by standard methods which use cylindrical co-ordinates. The method is developed for a simple problem in potential theory and is then applied also to a problem in Stokes flow, yielding a procedure for obtaining the Stokes drag on a slender body of arbitrary shape. For comparison purposes, consideration is also given to the use of both cylindrical and di-polar co-ordinates, and as a by-product of the comparison of results on cylindrical and spheroidal systems some new simple formulae involving Legendre polynomials are obtained heuristically, and then rigorously proved.

Matching Problems Involving Flow through Small Holes

Publisher Summary This chapter explains the matching process in intuitive terms. A collection of examples of solutions to problems involving small holes, slits, gaps, clearances, passages, and gratings are given. Extensive use of the concept of the “effective size” of holes is used. The effective size is a single parameter, of the dimensions of a length, that characterizes the hole in its far field. The chapter presents illustration of matching, using the effective size to compute the amount of leakage through a small central slit in a flat plate normal to a uniform stream. The chapter includes the exact solution of that problem and a complementary exact derivation of the effective size of a symmetric double slit, where the spacing of the two slits relative to their size is not necessarily large. It also mentions the case when there are two or more small slits or holes in a wall, whose sizes are all small compared with the spacing among them. The task is to find the overall effective size of such a combination of holes, when the individual sizes and spacing are given. The chapter considers some real-fluid effects. In the first place, unsymmetric flows through holes are considered in which, although the flow is sinklike on one side, separation takes place at the edges of the hole and a free jet, rather than a source, appears on the other side. Then effects of viscosity are considered, but only for the very restricted case of small-amplitude unsteady flow, in which the Navier-Stokes equation can be legitimately linearized.

A note on a swimming problem

The rate of self-propulsion of a doubly-infinite flexible sheet due to transverse waving oscillations in a viscous fluid is shown to decrease with increasing frequency, at a fixed (small) wave amplitude. This result differs from that of Reynolds (1965) who included local inertia, and thereby predicted that the swimming speed increases above the limiting value given by Taylor (1951) at zero frequency. The error in Reynolds’ work is due to his neglect of the simultaneous effect of convection, which induces a non-uniform mean second-order flow, with direction such as to oppose propulsion. Some other results concerning swimming sheets are presented.

Sliding sheets: lubrication with comparable viscous and inertia forces

A rigid plane thin sheet is sliding steadily at speed U close to a plane wall, in a fluid of kinematic viscosity v. The sheet is infinitely wide and is of length L in the direction of motion, and its leading edge is a distance h 0 [Lt ] L from the wall. A solution is sought for arbitrary finite values of R = Uh 2 0 /ν L . In the limit as e = h 0 / L →0, the problem reduces to that of solving the boundary-layer equation in the gap region between sheet and wall, and this is done here both by an empirical linearization, and by direct numerical methods. The solutions have the property that they reduce to those predicted by lubrication theory when R is small, and to those predicted by an inviscid small-gap theory when R is large. Special attention is paid to the correct entrance and exit conditions, and to the location of the centre of pressure.