On capillary gravity-wave motion in two-layer fluids

Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary–gravity waves in the presence of interfacial tension are derived.     

On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field

Heterogeneous incompressible fluid flows with jumps in the viscous properties are solved with the particle finite element method using continuous and discontinuous pressure fields. We show the importance of using discontinuous pressure fields to avoid errors in the incompressibility condition near the interface.     

Undular hydraulic jumps arising in non-developed turbulent flows

Turbulent plane flow over a bottom of constant slope is considered for very large Reynolds numbers, very small slopes of the bottom, and Froude numbers close to the critical value 1. In contrast to a previous work (Grillhofer and Schneider, Phys Fluids 15:730–735, 2003), it is not assumed that the flow far upstream is fully developed. The first-order perturbation equations contain unknown functions that are determined from a solvability condition of the second-order equations. Without making use of turbulence modeling or empirical parameters, a third-order ordinary differential equation is obtained for the shape of the free surface. Slow changes of amplitudes and wave lengths, respectively, associated with a small damping parameter are described by a multiple-scales solution, which also reveals the source of peculiarities of numerical solutions. A universal diagram of solutions and a universal map of the initial conditions that lead to undular jumps are given. Both numerical and multiple-scales solutions are compared with experimental data.     

On the concepts of admissibility and coherence

In the present paper it will be argued that if a parameter value assigns zero probability to an open set containing the actual response, then that parameter value should be excluded from the parameter space. When this is done, the model becomes a restricted model, and sampling theory inferences should be focused on the sampling distribution of an hypothetical independent response from the restricted model. The above situation may arise when probability densities vanish outside a compact set. This phenomenon arises frequently in the real world, but it is usually ignored for reasons of mathematical convenience. Although many statistical procedures remain substantially the same when we consider this restricted model, admissibility propertics may be drastically changed, and inferences which are known to be inadmissible may turn out to be really admissible. Thus, Stein's phenomenon concerning the inadmissibility of the sample mean as an estimator of the population mean of ap-variate normal distribution whenp≥3may be explained by the fact that the distribution has a compact support but this has been ignored by reasons of mathematical convenience. The introduction of a restricted model is also important in the study of coherence. Thus, it will be shown that Brunk's theory of countably additive coherence, which admits the use of countably additive improper priors, can be improved with the introduction of a restricted model. Thus, the theory will be unified because it will be proved that the posterior is really coherent if and only if it is a Bayes posterior, and it will be simplified because it will not be required that the prior be minimally compatible with the model.     

On transverse variation of velocity and bed shear stress in hydraulic jumps in a rectangular open channel

Summary This paper is concerned with an experimental investigation on the transverse variation of the velocity and bed shear stress in hydraulic jumps in a rectangular open channel. The velocity is measured by means of a Pitot tube and small propeller-type flowmeter, while the bed shear stress is determined by the Preston's method. The experimental parameter is the supercritical Froude number , whereU ₀ is the supercritical velocity,g the acceleration due to the gravity andz ₀ the supercritical water depth.It is found that hydraulic jumps can be divided into two groups broadly at the supercritical Froude number Fr2.5. In case of the first group, no surface roller is formed and only a little amount of air is entrained into the jump. A series of swelling-depression sequences are formed on the surface along the channel axis, and thus the bed shear stress experiences a sinusoidal variation. On the other hand, in case of the second group a definite roller is formed in the jump region and a considerable amount of air is entrained into the roller. The surface height increases rapidly from the toe line to end line. As the result, the bed shear stress decreases rapidly with increasing the distance from the toe line, but the decay rate is decreased with increasing the supercritical Froude number.It is concluded that in hydraulic jumps the surface height, velocity and bed shear stress are dependent on all of the three spatial coordinates. For example, the bed shear stress is greatly decreased with increasing the transverse distance from the channel axis.     

Pressure fluctuations measurement and analysis in hydraulic jumps using a microcomputer

Abstract

This paper presents a microcomputer based data acquisition system and an analysis of pressure fluctuations in the range of weak hydraulic jumps. Statistical analysis of data shows that the distribution of pressure fluctuations follows the Gaussian probability density function. The maximum pressure fluctuation amplitude is found to be in the range of 2.5 to 3.5 times the standard deviation. The spectral analysis of data indicates that large eddies in the low frequency range of 0 to 20 Hz are predominant over the small eddies in the high frequencies. This result is also predicted by the empirical correlation coefficient. Based on cross correlation analysis, the relationship between areal pressure fluctuations and point pressure fluctuations is developed and can be used to prevent over‐designing the strength of the stilling basins. It shows that the areal pressure fluctuations decrease as the jump longitudinal dimension increases.     

Polyethylene glycol compounds for functional hydraulic fluids

New compounds of phthalate; -phosphate, -borate, -phosphate-borate; polyethylene glycol and 2-(2-butoxy ethoxy ethanol) were synthesized, with the aim of producing new high performance functional hydraulic fluids. The synthesis process involves two stages: first, the synthesis of base compounds which are products of the reactions of phthalic acid anhydride, polyethylene glycol and acid [phosphoric, boric, phosphoric-boric and/or phthalic]. Then, the reaction of phthalic acid with 2-(2-butoxy ethoxy) ethanol. The base compounds so prepared, together with diluents and additives were formulated for the second stage of the synthesis process. The effectiveness of standard specification tests was investigated. Property improvement may be attained by the incorporation of phthalic acid anhydride in the reaction. Our new functional hydraulic fluids can be produced more economically and can monitor any pollution caused by fluid operations in the petroleum industry. Electronic Publication     

Kinematic instabilities in two-layer eccentric annular flows,part 1: Newtonian fluids

Primary-cementing displacement flows occur in long narrow eccentric annuli during the construction of oil and gas wells. A common problem is that the displacing fluid fingers up the upper wide side of the annulus, leaving behind a “mud channel” of displaced fluid on the lower narrow side of the annulus. Tehrani et al. report that the interface between displacing fluid and mud channel can in certain circumstances become unstable, and a similar phenomenon has been observed in our ongoing experiments. Here an explanation for these instabilities is provided via analysis of the stability of two-layer eccentric annular Hele-Shaw flows, using a transient version of the usual Hele-Shaw approach, in which fluid acceleration terms are retained. Two Newtonian fluids are considered, as a simplification of the general case in which the fluids are shear-thinning yield-stress fluids. It is shown that negative azimuthal buoyancy gradients are in general stabilizing in inclined wells, but that buoyancy may also have a destabilizing effect via axial buoyancy forces that influence the base-flow interfacial velocity. In a variety of special cases where buoyancy is not dominant, it is found that instability is suppressed by a positive product of interfacial velocity difference and reduced Reynolds-number difference between fluids. Even a small positive azimuthal buoyancy gradient, (heavy fluid over light fluid), can be stabilized in this way. Eccentricity of the annulus seems to amplify the effect of buoyancy on stability or instability, e.g. stably stratified fluid layers become more stable as the eccentricity is increased.     

Characterization of natural sand proppant used in hydraulic fracturing fluids

Natural sand was the first material introduced as a proppant to hydraulic fracturing operations to keep induced rock fractures both open and conductive. It has remained the most commonly used proppant for this process, since its economic advantages are obvious. In this context, the growth of the unconventional gas industry in Algeria has raised many environmental and economical concerns in terms of local sand valorization for a possible use as proppant agents. This paper deals with the physical analysis of 10 sand samples of two types: quarry sand samples and Aeolian dune sand samples, from different geographic locations in Algeria according to ISO 13503-2 which provides standard testing procedures for evaluating certain physical properties of proppants used in hydraulic fracturing and gravel packing operations. These assessments should enable to compare the physical characteristics of the various tested sands and to select the useful ones to evaluate their interaction with one chemical (SP breaker) in fracturing fluid system by sorption under several operating conditions (temperature, proppant dosage, SP breaker concentration).     

Electrochemical determination of water in environmental hydraulic fluids using the karl Fischer reaction

Different procedures based on the Karl Fischer reaction were investigated with respect to their applicability for water determinations in environmental hydraulic fluids: (i) continuous coulometry using a recently described diaphragm-free cell; (ii) on-line stripping of water at elevated temperature using either continuous coulometry or direct potentiometry for detection of the liberated water. Except for one of the oils, Statoil PA, which is a poly(α-olefin) with certain polymers added, no significant difference was found among coulometry using an optimized imidazole-buffered methanolic reagent containing 75% (v/v) chloroform, the two different stripping techniques (working in the temperature interval 100-110 °C), and the commercially available Hydranal Coulomat AG-H. The high stability and sensitivity of the coulometric technique described made it possible to work with sample amounts in the low milligram-range, and this is shown to increase the reliability of the coulometric method as compared to normally used procedures.     

On the equilibrium of fluids in accelerating frames

Summary The problem of the (relative) equilibrium of liquids in accelerating frames is examined in its entirety, and the conditions under which this equilibrium can be realized are determined in a rigorous manner. Also the question of the existence of a free surface of a liquid in relative equilibrium is answered, and expressions for the pressure, as a function of position and time, are given.Moreover, generalized buoyancy is calculated for the most important cases of equilibrium. Two examples of practical interest are given.     

On the residual opening of hydraulic fractures

Hydraulic stimulation technologies are widely applied across resource and power generation industries to increase the productivity of oil/gas or hot water reservoirs. These technologies utilise pressurised water, which is applied inside the well to initiate and drive fractures as well as to open a network of existing natural fractures. To prevent the opened fractures from complete closure during production stage, small particles (proppants) are normally injected with the pressurised fluid. These particles are subjected to confining stresses when the fluid pressure is removed, which leads to a partial closure of the stimulated fractures. The residual fracture openings are the main outcome of such hydraulic stimulations as these openings significantly affect the permeability of the reservoirs and, subsequently, the well productivity. Past research was largely focused on the assessment of conditions and characteristics of fluid driven fractures as well as proppant placement techniques. Surprisingly, not much work was devoted to the assessment of the residual fracture profiles. In this work we develop a simplified non-linear mathematical model of residual closure of a plane crack filled with deformable particles and subjected to a remote compressive stress. It is demonstrated that the closure profile is significantly influenced by the distribution and compressibility of the particles, which are often ignored in the current evaluations of well productivity.     

On the theory of turbulence for incompressible fluids

The theory of turbulence, based upon the Reynolds equations of mean motion and the dynamical equations of the velocity correlations of successive orders derived from the equations of the turbulent velocity fluctuation by using the condition of pseudo-similarity and the hypotheses on the viscous dissipation terms in the correlation equations, is developed by a method of successive approximation. As examples in the first order approximation, we have solved the turbulent flows through a channel, in a plane wake and in jets by using the equations of mean motion and of double correlation, while the terms in the triple correlation have been neglected. The agreements between the calculated values and the experiments are satisfactory. In the present paper, the equations of the triple and quadruple correlations in addition to those used in the first order approximation are solved for the plane turbulent wake in the second order approximation by the method of substitution, starting from the solution of the first order approximation. Agreements between theory and existing experiments for the triple velocity correlation are also satisfactory. The theory has also yielded the components of the quadruple correlation which can be tested by experiment. Presented at the Third Asian Congress of Fluid Mechanics on September 1, 1986, in Tokyo, Japan.     

On the two-dimensional regularized long-wave equation in fluids and plasmas

Summary.  To the two-dimensional regularized long-wave equation we perform computerized symbolic computation to obtain a new auto-B?cklund transformation and picture out some new bell-shaped and anti-bell-shaped solitary waves, which could help in the investigations on the Rossby waves in rotating fluids and the drift waves in plasmas. Received January 8, 2002; revised April 18, 2002 Published online: February 10, 2003 Acknowledgements We thank the editor Prof. N. Aksel and the referees for their valuable comments. This work has been supported by the Excellent Young Teachers Program of the Ministry of Education of China, by the National Key Basic Research Special Foundation (NKBRSF) of China under Grant No.␣G1999032701, by the China Talent Fund, by the National Natural Science Foundation of China under Grant No. 10272017, by the W. T. Wu Foundation on Mathematics Mechanization, by the Talent Construction Special Fund and Basic-Sciences Doctoral Education Fund of Beijing University of Aeronautics and Astronautics. BT also thanks the Enterprise Chair Professors Programme of the Bright Oceans Inter-Telecom Corporation and Beijing University of Posts and Telecommunications. YTG would like to acknowledge the Cheung Kong Scholars Programme of the Ministry of Education of China and Li Ka Shing Foundation of Hong Kong.     

On the undrained and drained hydraulic fracture splits

The simulation of hydraulic fracturing (HF) involves the solution of a hydro-mechanically coupled system. This article presents a new iterative sequential coupling algorithm, the undrained HF split, that improves the simulation of HFs in impermeable media. A poromechanics analogy is used to derive a stable split for the hydro-mechanically coupled system in which the mechanical subproblem is solved first. The proposed undrained HF split is applied to the simulation of cohesive HFs in an impermeable elastic medium. The cubic law is used as the constitutive model for simulating fluid flow in fractures. A minimum hydraulic aperture is assumed in the cohesive tip zone, where the mechanical aperture smoothly vanishes. While general in its nature, the undrained HF splitting scheme is employed within the context of a two-dimensional eXtended finite element model for the fractured solid, and a regular finite element model for fluid in the fracture. The undrained HF split is successfully used to simulate self-similar plane strain HFs as well as the propagation of HFs from a wellbore under anisotropic stress conditions. Fracture trajectories and local alteration of stress field are investigated. The solution of the undrained HF split converges to the same solution as the fully coupled model, whereas the commonly used P→W sequential algorithm, referred to in this article as the drained HF split, generates spurious oscillations and fails to converge in many problems. The undrained HF split is shown to be stable and robust in applications where the drained HF split is unstable.     

On some helical flows of Oldroyd-B fluids

Summary In this study the velocity fields and the associated tangential stresses corresponding to some helical flows of Oldroyd-B fluids between two infinite coaxial circular cylinders and within an infinite circular cylinder are determined in forms of series in terms of Bessel functions. At time t = 0 the fluid is at rest and the motion is produced by the combined action of rotating and sliding cylinders. The solutions that have been obtained satisfy the governing differential equations and all imposed initial and boundary conditions. For λ _r = 0, λ = 0 or λ _r = λ = 0 they reduce to the similar solutions for a Maxwell, second grade or Newtonian fluid, respectively. Finally, for comparison, the velocity profiles corresponding to the four models are plotted for different values of t.