C.J. Lawrence

Long-time behaviour of the drag on a body in impulsive motion

We consider the response of the hydrodynamic drag on a body in rectilinear motion to a change in the speed between two steady states, from U 1 to U 2 ≥0. We consider situations where the body generates no lift, such as occur for bodies with an axis of symmmetry aligned with the motion. At large times, the laminar wake consists of two quasi-steady regions - the new wake and the old wake - connected by a transition zone that is convected downstream with the mean speed U 2 . A global mass balance indicates the existence of a sink flow centred on the transition zone, and this is responsible for the leading-order behaviour of the unsteady force at long times. For the case of U 1 ≥0, the force is shown to decay algebraically with the inverse square of time for any finite Reynolds number (Re), and this result is also shown to hold for non-rectilinear motions. The cases of reversed flow (U 1 <0) and stopped flow (U 2 =0) are treated separately, and it is shown that the transient force is dominated by the effects of the old wake, leading to a slower decay as the simple inverse of time. The force is determined by the far regions of the flow field and so the results are valid for any (symmetric) particle, bubble or drop and (in an average sense) for any Re, provided τ»max {Re, Re -1 }, where the time τ is made dimensionless with the convection timescale. These are believed to be the first calculations which adequately resolve the transient far wake behind a bluff body at long times. The asymptotic result for the force is applied to determine that the approach to terminal velocity of a body in free fall is also as the inverse square of time

Axisymmetric wave regimes in viscous liquid film flow over a spinning disk

Finite-amplitude capillary waves, which can accompany the axisymmetric flow of a thin viscous film over a rotating disk, are considered. A system of approximate evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived, which contains two similarity parameters. In order to inspire confidence in this model, its steady solutions and their linear stability characteristics are compared to those of the full Navier–Stokes equations. Localized equations, which account for the presence of inertial, capillary, centrifugal and Coriolis forces, are obtained via truncation of the approximate system. Periodic solutions of these equations are then determined and found to be similar to those observed experimentally. Our results suggest that the steady quasi-periodic waves with largest amplitude compare well with experimentally observed wave profiles.

Pressure gradient and holdup in horizontal two-phase gas- liquid flows with low liquid loading

Abstract Pressure gradient and holdup data are presented for air–water and air–oil flows in a horizontal, 0.079 m diameter pipe. Addition of a very small liquid flow was found to result in a considerable increase in the pressure gradient compared with single phase gas flow. The pressure gradient and the holdup data were compared with predictions of the ‘apparent rough surface’ (ARS) and the ‘double-circle’ models. The ARS model generally gave better predictions for the holdup over the experimental range. Both models predicted the pressure gradient for air–water flows at high gas flowrates reasonably well. However, the predictions of both methods were unsatisfactory for air–water experiments at low gas flowrates and for air–oil experiments. Overall the ARS model was judged to be the more robust.

Physical and numerical modelling of ram extrusion of paste materials: conical die entry case

Abstract A ram extrusion process with a conical die entry is analysed using a physically based analysis and a numerical modelling procedure involving the finite element method. The aim of the study is to test the applicability of the elasto-viscoplastic constitutive model combined with an established boundary condition criterion for describing this forming operation. “Plasticine”, a commercial modelling clay which exhibits an elasto-viscoplastic flow response, is used to study the nature of the material deformation. For the numerical modelling, an elasto-viscoplastic finite element program has been implemented. Satisfactory agreement, between experiment and simulation, is obtained for the force–time data and the material displacement fields which indicates that the bulk and interfacial constitutive relationships adopted, along with the associated numerical parameters, are an appropriate description of the flow behaviour of the system. The evolution of the deformation, within the material during the extrusion process, is interpreted. The study indicates that the procedures described may ultimately provide a numerical rheometric tool from which the effects of various process boundary conditions, on the evolution of paste material deformation, can be examined and quantified for the ram extrusion process with conical die entries.

Near-wall particle depletion in a flowing colloidal suspension

In this paper a method based on attenuated total reflection–Fourier-transform infrared spectroscopy is described to measure the concentration of colloidal particles dispersed in a liquid near a solid boundary in pressure driven flows. The method has been used to obtain measurements of wall depletion in an aqueous suspension of ethylcellulose particles. The method is capable of measuring the particle concentration in a thin layer near the boundary, whose thickness is between 0.3 and 1.4 μm. A considerable decrease in the concentration of particles near the wall is found for a suspension with a particle volume fraction of 0.24 over a wide range of apparent shear rates. The results from the spectroscopic experiments are compared to rheological experiments on parallel disk torsional flow at different gap heights. With this method, the order of magnitude of the thickness of an equivalent fully depleted layer (a layer at the wall assumed to contain only pure solvent and no particles) can be obtained. The result...

Laminar stratified pipe flow

Abstract The Boundary Element Method (BEM) has been used to evaluate the integral and local flow properties of two-phase laminar–laminar stratified flow in a pipe for interface shapes determined by exact solution of the Young–Laplace equation. Results are presented for the volumetric flow rates, the interface and wall shear stresses and the velocity profiles on the interface and through the cross-section of the pipe. For the case of a flat planar interface, the results obtained from the BEM were compared against those from the analytical solution for a 50% holdup (i.e. with a flat interface passing through the horizontal axis) and with previous numerical solutions obtained using the bipolar coordinate system. Comparisons for the case of a circular interface were made with the Fourier integral method described by others. The solutions from the boundary integral method agree very well with these published results for special cases. Solutions are presented for a range of parameter values. The interface shape is determined by the Bond number, holdup or contact angle. The flow is controlled by these parameters, as well as the viscosity ratio, and the pipe inclination, which determines the relative driving forces on the two fluids. It is concluded that the BEM offers great flexibility and accuracy in addressing this class of flows.

Creeping flows of power-law fluids through periodic arrays of elliptical cylinders

Abstract Results from numerical simulations and lubrication theory are presented for creeping flows of power-law fluids through periodic arrays of elliptical cylinders. Flows are considered in the plane perpendicular to the axes of the cylinders, both along an axis of the array (on-axis flow) and at an angle to the axes of the array (off-axis flows). Results are presented for the apparent permeability tensor and for the dimensionless velocity variances (which can also be used to approximate the added mass coefficient for a cylinder in the array). The apparent permeability values obtained for on-axis flows of power-law fluids are shown to obey a simple scaling, which relates the apparent permeability tensor for power-law fluids to the corresponding permeability for Newtonian fluids; this scaling arises because of the choice of length scale used in the definition of the apparent permeability tensor for power-law fluids. The off-axis flow results are shown to be related to the on-axis results in a straightforward manner. The results are summarised in the form of closure relations for the apparent permeability tensor and velocity variances for off-axis flows of power-law fluids through arrays of elliptical cylinders for a range of aspect ratios using look-up graphs for only a few scalars.

Spatial Distribution of Oil and Water in Horizontal Pipe Flow

This paper reports a series of experiments to quantify the spatial distribution and identify the flow pattern of liquid-liquid flow in a horizontal 25.4mm (nominal one-inch) tube. Experimental results are presented for Kerosene (EXXOL D80) and tap water at room temperature. Two different measurement techniques (a high frequency impedance probe and a gamma densitometer system) were applied for measuring the volume fraction distribution across the tube and to obtain tomographic results for phase distribution. The use of the gamma densitometer system to obtain the tomographic results in liquid-liquid co-current flow is believed to be the first in this field. These methods are more precise than other techniques such as visualization and help to distinguish certain differences in flow patterns for different superficial velocities and liquid fractions. The two sets of measurements were compared and it was concluded that the gamma densitometer system was a more reliable method to measure the volume fraction. Two important phenomena in liquid-liquid flow were observed; a) Oil encapsulation by water at low mixture velocity, b) Droplets concentrated at the center of the pipe in the dispersed flow regime. Some possible explanations are given regarding these phenomena.

Chloride transport models for wick action in concrete at large Peclet number

Wick action is the transport of water and any species it may contain through a concrete element from a face in contact with water to a drying face. Analytical models are presented for chloride transport in concrete due to wick action. Two cases are considered for convection-controlled systems: (1) No precipitation of chloride salt at a wet-dry interface, and (2) precipitation of chloride salt at the interface. For both cases, a constant concentration of chloride salt is assumed for the face of the concrete in contact with salt solution, while at the interface continuity of fluxes is assumed for Case 1, and a constant chloride saturation level for Case 2. Perturbation techniques are used to solve the transient problems for the two cases investigated. The perturbation solution is compared with experimental and numerical results and good agreement is found when chloride binding is accounted for.

The flow of thin liquid films over spinning disks: Hydrodynamics and mass transfer

We study the hydrodynamics and mass transfer associated with gas absorption into a thin liquid film flowing over a spinning disk. We use the thin-layer approximation in conjunction with the Karman–Polhausen method to derive evolution equations for the film thickness and the volumetric flow rates in the radial and azimuthal directions. We also use the integral balance method to derive evolution equations for the thickness of the diffusion boundary layer as well as the concentration of solute at the disk surface. Numerical solutions of these partial differential equations, which govern the hydrodynamics and the associated mass transfer, reveal the formation of large finite-amplitude waves and elucidate their significant effect on the mass-transfer characteristics. We illustrate this dependence quantitatively by examining the effect of system parameters on the time-averaged and spatially averaged Sherwood numbers. The results are assessed by comparison with computations of the parabolized convective diffusio...