ALBERT

Description: Taylor-Couette flow is ideal for studying drag-reducing polymer additives because, unlike turbulent boundary layers, the instabilities are better understood. Video records of laser-induced fluorescence experiments with and without polymers will be presented. Polyethylene-oxide (PEO) ‘oceans’ were used in concentrations of 20 and 100 p.p.m. In the Taylor number range, 3 x 104 ≤Ta ≤ 108, Newtonian flow consisted of Taylor vortices which span the gap between cylinders and much smaller Görtler vortices at the inner cylinder wall. Measurements of core-to-core separation between counter-rotating vortices were made to estimate the Görtler instability wavenumber. These measurements show that PEO addition increases the Görtler wavenumber for a given Taylor number. At the lower Taylor numbers, Görtler vortex formation was suppressed by PEO. This implies that polymers directly affect the evolution of centrifugal instabilities. © 1995, Cambridge University Press. All rights reserved.

Description: Gas flow through fractured nanoporous shale formations is complicated by a hierarchy of structural features (ranging from nanopores to microseismic and hydraulic fractures) and by several transport mechanisms that differ from the standard viscous flow used in reservoir modelling. In small pores, self-diffusion becomes more important than advection; also, slippage effects and Knudsen diffusion might become relevant at low densities. We derive a nonlinear effective diffusion coefficient that describes the main transport mechanisms in shale-gas production. In dimensionless form, this coefficient depends only on a geometric factor (or dimensionless permeability) and on the kinetic model that describes the gas. To simplify the description of the complex structure of fractured shales, we observe that the production rate is controlled by the flow from the shale matrix (which has the smallest diffusivity) into the fracture network, which is assumed to produce instantaneously. Therefore, we propose to model the flow in the shale matrix and estimate the production rate with a simple bundle-of-dual-tubes model (BoDTM), in which each tube is characterized by two diameters (one for transport and the other for storage). The solution of a single tube is approximately self-similar at early time, but not at late time, when the gas flux decays exponentially owing to the finite length of the tube. To construct a BoDTM, a reliable estimate of the joint statistics of the matrix-porosity parameters is required. This can be either inferred from core measurements or postulated on the basis of somea prioriassumptions when information from laboratory and field measurements is scarce. By comparison with field production data from the Barnett shale-gas field, we demonstrate that BoDTM can be calibrated to estimate structural parameters of the shale formation and to predict the cumulative production of shale gas. Our framework has enough flexibility to construct models of increasing complexity that can be employed in the presence of a complex dataset or when more information is available.

Description: The motion of a sphere in the presence of a fluid-fluid interface is studied. First, a solution is derived for a point force near a plane interface. Then the solution is extended to include the higher-order terms which are required to describe the motion of a solid sphere. Singularities of higher orders at the centre of the sphere are obtained by using the method of reflexions. For a fluid–fluid interface with an arbitrary viscosity ratio, the drag force and the hydrodynamic torque are calculated for the special cases of motion of a sphere perpendicular and parallel to the interface. In addition, the rotational motion of a sphere is also investigated. © 1979, Cambridge University Press

Description: A general solution for Stokes' equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids. © 1980, Cambridge University Press. All rights reserved.

Description: Numerical results are presented for the approach of a rigid sphere normal to a deformable fluid-fluid interface in the velocity range for which inertial effects may be neglected. Both the case of a sphere moving with constant velocity, and that of a sphere moving under the action of a constant non-hydrodynamic body force are considered for several values of the viscosity ratio, density difference and interfacial tension between the two fluids. Two distinct modes of interface deformation are demonstrated: a film drainage mode in which fluid drains away in front of the sphere leaving an ever-thinning film, and a tailing mode where the sphere passes several radii beyond the plane of the initially undeformed interface, while remaining encapsulated by the original surrounding fluid which is connected with its main body by a thin thread-like tail behind the sphere. We consider the influence of the viscosity ratio, density difference, interfacial tension and starting position of the sphere in determining which of these two modes of deformation will occur. © 1986, Cambridge University Press. All rights reserved.

Description: A numerical implementation of the method of matched asymptotic expansions is proposed to analyse two-dimensional uniform streaming flow at low Reynolds number past a straight cylinder (or cylinders) of arbitrary cross-sectional shape. General solutions for both the Stokes and Oseen equations in two dimensions are expressed in terms of a boundary distribution of fundamental single-and double-layer singularities. These general solutions are then converted to integral equations for the unknown distributions of singularity strengths by application of boundary conditions at the cylinder surface, and matching conditions between the Stokes and Oseen solutions. By solving these integral equations, using collocation methods familiar from three-dimensional application of ‘boundary integral’ methods for solutions of Stokes equation, we generate a uniformly valid approximation to the solution for the whole domain. We demonstrate the method by considering, as numerical examples, uniform flow past an elliptic cylinder, uniform flow past a cylinder of rectangular cross-section, and uniform flow past two parallel cylinders which may be either equal in radius, or of different sizes. © 1986, Cambridge University Press. All rights reserved.

Description: The evolution of small counter-rotating circumferential vortices in Taylor-Couette flow was examined using the laser induced fluorescence and alumina particle flow visualization techniques. The objective of the study was to critically evaluate the hypothesis of Barcilon et al. (1979) and Barcilon & Brindley (1984) that Gortler vortices form close to the cylinder walls at moderately high Taylor numbers. Three radius ratios spanning an order of magnitude, 0.084 〈 i?in/i?out〈 0.877, were examined over a Taylor number range of 3 x 104^ Ta ^ 3 x 108. Still-photograph sequences taken from video records of the LIF experiments are presented showing vortex pairs close to the inner cylinder wall at Taylor numbers an order of magnitude smaller than those reported by Barcilon and co-workers. Measurements of the core-to-core separation between counter-rotating vortices were made in order to estimate the wavenumber of the instability. These measurements agree remarkably well with the theoretical analysis of Barcilon and co-workers particularly for the small and medium-gap experiments. The present measurements indicate that there is a — § power law relationship between the Gortler wavelength and Taylor number. This is consistent with the work of Barcilon & Brindley (1984). However, the present study indicates that the Gortler vortices first form at the inner cylinder wall, and that a full theoretical treatment must include inner-wall effects. © 1992, Cambridge University Press. All rights reserved.