Journal of Fluid Mechanics

Experiments were performed studying the formation of a dip on the surface of an initially stationary liquid draining from a cylindrical tank through an axisym-metrically placed circular orifice. Based upon the information obtained from the experiments, a simple analytical expression was derived predicting the height of the liquid surface in the tank at which this dip forms. A comparison was made between the experimental data and the results of the analysis and good agreement was found between theory and data.

It is demonstrated that the basic stratification in a fluid region subject to thermal forcing may be predicted rather simply for a fairly wide class of boundary conditions. Explicit solutions are derived in certain cases. A useful experimental method for maintaining a stratified system with arbitrarily specified vertical variation of density emerges from the analysis. A preliminary laboratory experiment has demonstrated the efficiency of this method. The restrictions on the validity of the theory involve a limitation on the thermal forcing of the fluid, which may be expressed as an upper limit on the thermal conductance of the boundary of the region. Furthermore, the buoyancy frequency characterizing the solution must be sufficiently large to give rise to a boundary-layer-type flow pattern.

The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. The primary motivation for this work was to check numerical solution schemes by comparing the numerical results with laboratory measurements obtained with a rotating laser-Doppler velocimeter. The laser-Doppler technique is capable of high accuracy with small space and time resolution, and disturbances of the flow are virtually negligible. A series of measurements was made of the zonal flow over a range of Ekman numbers (1·06 × 10⁻³≤E≤ 3·30 × 10⁻³) and Rossby numbers (0·10 [les ]|ε| [les ] 0·33) at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.

A comparison is made between the theoretically predicted and the observed stratification in a container which is traversed by a prescribed flux of fluid. Two different geometries were used illustrating respectively a useful procedure for the control of a stratified laboratory system and a mechanism which is believed to be geophysically significant, e.g. for the control of the stratification in certain estuaries. The behaviour of the fluid system was in all cases characterized by an almost stagnant interior with a boundary layer at the non-horizontal wall of buoyancy-layer type. Agreement between theory and experiment was satisfactory within experimental errors, say 10% of the overall temperature difference.

The flow around a circular cylinder placed at various heights above a plane boundary has been investigated experimentally. The cylinder spanned the test section of a wind tunnel and was aligned with its axis parallel to a long plate and normal to the free stream. It was placed 36 diameters downstream of the leading edge of the plate and its height above the plate was varied from zero, the cylinder lying on the surface, to 3·5 cylinder diameters. The thickness of the turbulent boundary layer on the plate at the cylinder position, but with it removed from the tunnel, was equal to 0·8 of the cylinder diameter. Distributions of mean pressure around the cylinder and along the plate were measured at a Reynolds number, based on cylinder diameter, of 4·5 × 10⁴. Spectral analysis of hot-wire signals demonstrated that regular vortex shedding was suppressed for all gaps less than about 0·3 cylinder diameters. For gaps greater than 0·3 the Strouhal number was found to be remarkably constant and the only influence of the plate on vortex shedding was to make it a more highly tuned process as the gap was reduced. Flow-visualization experiments in a smoke tunnel revealed the wake structure at various gap-to-diameter ratios.

The distortion by a linear flow of the electric double layer around a small particle is studied for the case of a charge cloud which is thick in comparison with the particle radius and for arbitrary flow strengths, including those which are strong enough to produce a significant distortion of the cloud. For weak flows a second-order-fluid approximation is obtained for the stress contribution for a dilute suspension of such particles. For arbitrarily strong flows integral representations of the charge density and numerical calculations of the stress contribution are given for three representative flows: simple shear, axisymmetric strain and two-dimensional straining motion.

Previous studies of the distortion of the electric double layer around a charged sphere have assumed that the electric stresses are small compared with the viscous stresses. The flow around the particle is therefore changed only slightly by the presence of the charge cloud. This change is measured by the Hartmann number, and in § 6 we remove the restriction that it should be small. It is found that the previous linearized theory is sufficiently accurate for typical experimental values of the Hartmann number. Previous studies have also assumed that the potential at the surface of the particle is small. This assumption is removed in § 7 of this paper. For values of the non-dimensional surface potential smaller than 2 the predictions are altered by less than 10 %. For higher values the differences between linear and nonlinear theory are not negligible, especially when the charge cloud is thin compared with the radius of the charged sphere.

A charged particle suspended in an electrolyte solution attracts ions of opposite charge and repels those of like charge. The surface charge and the resulting distributed charge in the fluid comprise an electrical double layer. When a shear flow deforms the diffuse part of the double layer from equilibrium, stresses are generated which make the effective viscosity of the suspension greater than it would be if there were no charges present. In this paper these stresses are calculated for a dilute dispersion of spheres which have small surface charges and which are surrounded by thin double layers. The viscosity is predicted to be Newtonian in extensional flow but shear-thinning with non-zero normal-stress differences in shear flow. For more complex flows a constitutive equation couples the bulk stress directly to the micro-structural deformation responsible for non-Newtonian effects.

We study the primary electroviscous effect in a suspension of spheres when the double layer thickness κ⁻¹ is small compared with the particle radius a. The case of a 1–1 symmetric electrolyte is examined using the methods of Dukhin & coworkers (1974), whilst the asymmetric electrolyte is studied along lines similar to those of O'Brien (1983). Sherwood's (1980) asymptotic results for high surface potentials and high Hartmann numbers are extended and complemented.