John Harper

Victoria University of Wellington · School of Mathematics and Statistics

Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \inft...

A gas bubble rising steadily in a pure liquid otherwise at rest at a moderat e Weber number is, to a good approximation, of oblate spheroidal shape. Previous analytical calculations of that shape at high Reynolds numbers have ignored viscosity. This paper shows that if one includes viscosity by incorporating Rayleigh’s dissipation integral in Lagra...

In 1996 Kelsall et al. [5] reported electrophoretic experiments with oxygen bubbles in dilute solutions of several salts that were remarkably free of surfactants, but the experimental results agree with neither their own theory nor anyone else's known to the present author. This paper assumes a double-layer thickness much smaller than the bubble ra...

Diatoms were investigated as part of the world's first marine Bioblitz held on the south coast of Wellington, New Zealand, in October 2007. Two unusual diatoms were associated with the red alga Herposiphonia ceratoclada. They were examined by light and scanning electron microscopy. The first, previously described as »Cocconeis coelata« Arnott ex Gr...

When a gas bubble rises in a surfactant solution, the velocity field and the distribution of surfactant affect each other. This paper gives the theory for small Reynolds and internal Peclet numbers if the surfactant is gaseous or volatile, if its mass flux across the bubble and around its surface dominates its mass flux through the bulk liquid, and...

A dynamical theory of flow in the asthenosphere is developed by treating it as a thin layer of Newtonian viscous fluid, pushed horizontally by the dense sinking slabs attached to some plates and by the differences in heating under continental and oceanic crust. Two-dimensional lubrication theory is adapted to give the world-wide effects of inflow t...

In 1986 the author's previous paper describing a simple mathematical model for the forces driving the plates had an rms error of 1.62 per cent in its best case. The model included slab pull and its global and local reactions proportional to speed times (age)3/2, viscous drag due to shear under moving plates and in the flow towards mantle sinks at m...

We suppose that the plates are pulled along on top of an effectively viscous asthenosphere by their cold dense sinking leading edges, and that they also tend to slide down the flanks of ocean ridge systems. Using reasonable literature values of density, viscosity and thickness, we find that a typical strong subduction zone pulls about seven times a...

When a gas bubble rises in an impure liquid, its surface often has an upper spherical cap with negligible shear stress, a lower spherical cap with negligible tangential velocity, and a very small transition region between the two caps. This paper gives the diffusion boundary-layer theory for the distribution of surfactant around a stagnant-cap bu...

Over many years the author and others have given theories for bubbles rising in line in a liquid. Theory has usually suggested that the bubbles will tend towards a stable distance apart, but experiments have often showed them pairing off and sometimes coalescing. However, existing theory seems not to deal adequately with the case of bubbles growing...

Analytical support is given to Fornberg's numerical evidence that the steady axially symmetric flow of a uniform stream past a bluff body has a wake eddy which tends towards a large Hill's spherical vortex as the Reynolds number tends to infinity. The viscous boundary layer around the eddy resembles that around a liquid drop rising in a liquid, esp...

An analytical theory is given for the viscous wake behind a spherical bubble rising steadily in a pure liquid at high Reynolds number, and for that wake's effect on the motion of a second bubble rising underneath the first. Previous theoretical work on this subject consists of just two papers: a first approximation ignoring wake vorticity diffusion...

This paper gives an integral which can be evaluated faster for points close to the line

A simple method of reducing a parabolic partial differential equation to canonical form if it has only one term involving second derivatives is the following: find the general solution of the first-order equation obtained by ignoring that term and then seek a solution of the original equation which is a function of one more independent variable. Sp...

Trace impurities often collect on the upstream side of an obstacle in the surface of flowing liquid. The transition from practically free surface to surface sufficiently clogged to be treated as stationary can be quite sharp. The viscous flow underneath is nonlinearly coupled to the convective mass transfer of surface-active material. For two-dimen...

When gas bubbles rise in a surface-contaminated liquid, the upper parts of their surfaces may be almost free of contaminant and shear stress, while on the remainder of their surfaces there is enough contaminant to prevent tangential motion. Sadhal & Johnson solved the problem of Stokes flow of a uniform stream past a single spherical bubble. We ext...

Suppose that a gas bubble rises steadily in a very dilute surface-active solution, with very small Weber number, very large Péclet number, and either very small or very large Reynolds number. Define k, the dimensionless surface activity, as the ratio of the amount of contaminant absorbed on the surface to the amount dissolved in the diffusion bound...

New data on mantle viscosity variation and plate motions suggest that the author's previous theory for the forces driving and resisting plate tectonics needs revision. The flow in the mantle associated with plate movements probably pervades the whole mantle instead of being almost entirely in a thin asthenosphere at the top of it. The first task un...

Morgan's and Blake and Chwang's image system gives the flow due to a point force in a viscous fluid in the neighbourhood of a plane rigid wall. I show that it can provide a good simple approximation for the flow in the Earth's mantle and the forces on the surface plates in two cases of geophysical interest: the uplift associated with cessation of s...

A theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symme...

Consider the two-dimensional Stokes flow in a semi-infinite strip of viscous fluid due to a transverse velocity of its end wall. This is an idealization of the motion in the asthenosphere due to a subducting slab. We give an iterative scheme for calculating the flow, prove existence and uniqueness, and obtain numerical approximations, using develop...

This paper reviews recent progress in the theories of the surface boundary conditions of adsorbed solutes in liquids, and of the effects of those solutes on the steady motion of a bubble or drop in the liquid. Both singular perturbation theory and numerical solutions have useful roles in this problem, and their relationship is explored. In addition...

A world-wide set of plate boundaries and relative velocities was calculated for anomaly 23 time (about 55 m.y. ago, at the Paleocene/Eocene boundary, just before the breakup of Australia—Antarctica). The set was used as input data for Harper's recently published model of plate driving and resisting forces for the present day. The model enabled esti...

Sinking slabs of lithosphere must push the asthenosphere if they pull the plates. The resulting flow in the asthenosphere then goes rapidly round the ends of a slab, pulling the lithosphere with it. The motion can be analyzed with the help of Hele?Shaw's mathematical analogy between viscous flow in a thin layer of fluid and classical two-dimensiona...

The effects of small quantities of a soluble surface-active agent on flow around a spherical bubble rising in a liquid are investigated theoretically, with the assumptions of large Peclet number, small effect on the surfactant on the motion, and either large or small Reynolds number. Previously available theories fail near the rear stagnation point...

Surface-active impurities may collect as a stationary film on the lowest part of a bubble rising in liquid while the remainder of the surface moves freely. Numerical approximations for the motion are available if the Reynolds number is low, but they fail for small films. We give the steady-state asymptotic solution for that case, and obtain the per...

Chemical engineers, metallurgists, geologists, brewers and cooks all try to understand processes in which bubbles or drops move through liquids. Because interfacial phenomena affect the motion in a number of different ways, there are many cases to consider. The simplest case is a bubble rising in a Newtonian liquid, far enough from boundaries to be...

Conditions for two gas bubbles in a liquid to rise steadily in a vertical line are derived theoretically with these assumptions: large Reynolds number, no surface contamination, spherical shape, negligible gas density and viscosity. Drag coefficients are found, and are lower than for single bubbles. The bubbles have equilibrium distances apart, whi...

Diatoms were investigated as part of the world’s first marine Bioblitz held on the south coast of Wellington, New Zealand, in October 2007. Two unusual diatoms were associated with the red alga Herposiphonia ceratoclada. They were examined by light and scanning electron microscopy. The first, previously described as »Cocconeis coelata« Arnott ex Gr...

The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent....

Adhesive and tractive forces of diatoms have been measured by the bending of a glass fibre. The results show that motility depends on attachment to the substrate. Movement on girdle bands occurs when the raphe acts against lumps of trail substance. Resistance to shear stress in laminar flow through a capillary tube is found to be particularly high...

The boundary conditions at the surface of a small bubble rising in a liquid are examined theoretically, and it is shown by order-of-magnitude arguments, which are confirmed by detailed calculation in a special case, that although surfacetension gradients must always exist around the bubble, they are too small to affect the motion appreciably unless...

The starting-point for this paper is the suggestion (Batchelor 1956 b ) that the wake behind a bluff body in a uniform stream may consist principally of two eddies rotating in opposite directions. The fluid is assumed to be incompressible and in two-dimensional steady motion at a very high Reynolds number. Along the boundary between the eddies, a v...