The effect of viscosity on the waves due to a moving oscillatory surface pressure

Summary It is known that when a pressure distributionp ₀(x) exp (i t moves with a constant velocityu ₀ parallel to on the surface (y=0) of an inviscid liquid, (i) the steady-state wave amplitude does not die out at large distances from the source for (u ₀/g1/4, and(ii) the solution becomes singular for u ₀/g=1/4.Here is shown that the consideration of even a slight viscosity of the liquid leads to a wave decay with distance for all (u ₀/g), the solution remaining non-singular for (u ₀/g=1/4. Furthermore, for a finitely distributedp ₀(x), the wave amplitude is always non-zero, unlike in the case of an inviscid fluid, where no energy is transmitted for an infinite number of values of a parameter involving ,u ₀ and the length of the pressure strip.

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Zusammenfassung Es ist bekannt, daß, wenn eine periodische Druckverteilungp ₀(x) exp (i t) sich mit einer konstanten Geschwindigkeitu ₀ inx-Richtung über die Oberflächey=0 einer reibungslosen Flüssigkeit bewegt, daß dann (1) die Amplitude des stationären Zustandes in großen Entfernungen von der Quelle nicht abklingt für (u ₀/g)1/4, und (2) die Lösung wird singulär für (u ₀/g)=1/4.Hier wird gezeigt, daß die Berücksichtigung einer geringen Zähigkeit in der Flüssigkeit zu einem Abklingen der Wellen führt für alle (u ₀/g)=1/4. Zudem ist für eine Verteilung vonp ₀(x) über eine endliche Länge die Wellenamplitude immer endlich, im Gegensatz zum reibungsfreien Fall, in dem keine Energie übertragen wird für gewisse Werte eines Parameters, der ,u ₀ und die Länge des Druckstreifens enthält.

     

A model for conjunctive use of ground and surface water with opportunity costs

In this study, we develop a multi-stage decision model for the conjunctive use of ground and surface water with an artificial recharge. We assume a certain supply and a random demand. We explicitly integrate opportunity costs for the unsatisfied demand. We also incorporate in the model the importance weight attributed by the decision-makers to the final groundwater level at the end of the planning horizon. We show, under some mild assumptions, that the problem can be formulated as a convex program with linear constraints. We illustrate the methodology through a hypothetical example, and discuss the optimal decision policy and its sensitivity to a number of factors.     

Computational topology-based characterization of pore space changes due to chemical dissolution of rocks

In this paper, we present an algorithm for the numerical simulation of reactive transport at the pore scale to facilitate observation of pore space and rock matrix evolution. Moreover, simulation at the pore scale opens up the possibility of estimating changes in the transport properties of rocks, such as permeability and tortuosity. To quantitatively analyze pore space evolution, we developed a numerical algorithm that can be used to construct persistence diagrams of the connectivity components for pore space and the rock matrix, which characterize the topology evolution during rock matrix dissolution. Introducing the “bottle-neck” metric in the space of the persistence diagrams, we cluster the numerical experiments in terms of similarities in topology evolution. We demonstrate that the application of this metric to the persistence diagrams allowed us to distinguish topologically different dissolution scenarios, for instance, the formation of a dissolution front near the inlet, homogeneous dissolution of the matrix inside the core sample, and formation of wormholes. We illustrate that the differences in topology evolution lead to cross-correlations among the transport properties of rocks (porosity-permeability-tortuosity).     

Work-roll surface degradation due to spalling of oxide layer

Surface damage of work-rolls in hot strip rolling mills is often associated with the local removal of the oxide layer developing in service. A simple thermomechanical stress analysis in combination with an estimated buckling stress shows that the size of oxide free spots on the roll surface is roughly the same as the lateral distance between the cracks in the observed crack pattern. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluation of pressure levels in pipelines due to solar heat gains

A detailed numerical model based on a lumped capacity analysis, is developed for assessing potential pressure increases in pipelines due to solar energy gains. Test computations for gasoline and water show that for operational conditions of no pipeline flow, pressures reach levels for which structural damage may occur.     

The effect of periodic surface pressure on a rectangular elastic plate floating on shallow water

The steady-state behaviour of a floating elastic plate of bounded dimensions acted upon by a localized external load is investigated using linear shallow water theory. In the case of a plate of arbitrary shape, the problem reduces to solving a system of boundary-value integral equations supplemented with differential relations for the free edge of the plate. Using a rectangular plate as the example, the effect of the frequency of the periodic actions and the positions at which they are applied on the amplitudes of the normal flexures of the plate and of the directional pattern of surface waves far from the plate is investigated. It is shown that waveguide properties occur for an elongated plate.     

Analytical solutions of channel and duct flows due to general pressure gradients

Pursued herein are the closed-form solutions of the Navier–Stokes equations for both planar channel and circular duct flows, influenced by either periodic or aperiodic pressure gradients, of which the amplitudes are sufficiently low to yield laminar incompressible flows. The analyses conducted for the unsteady flows parallel to the walls lead to the analytical solutions that encompass not only the long-time oscillations by periodic pressure gradients, but also the transient start-up flows commencing from zero velocity due to arbitrary aperiodic pressure gradients. With the standard methods employed, the present solutions generalizing the classic ones are written in the forms rendering the explicit dependence on the pressure gradient, and are numerically validated by the existing solutions of simple sinusoidal oscillations and a flow involving an aperiodic impulsive pressure gradient. By virtue of their functional forms, the present solutions can be applied with any pressure gradients, even if the gradients are not in closed forms.     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.     

An analytical solution for displacements due to reservoir compaction under arbitrary pressure changes

Production of fluids in reservoirs, such as oil, gas and water, leads to changes in the stress and strain fields, generating compaction in the reservoir and subsidence of the ground surface. Some of the severe consequences of these displacements are rock fracture, induced seismicity, fault reactivation, wellbore failure, and reduction in the storage capacity of aquifer systems. Analytic methods based on the nucleus of strain approach with simplifying assumptions can be useful to estimate compaction and subsidence, and to indicate when more detailed analyses are required. However, these methods may present problems by the evaluation of the displacement response inside and in the vicinity of the reservoir due to the presence of singularities in the solution. In this work, an analytical displacement solution is obtained for reservoirs of arbitrary shape in a linear elastic semi-infinite medium under arbitrary distribution of pressure changes. For this, the reservoir is discretized in parallelepiped cells. A three-dimensional integration of the solution is carried out over each cell for an infinitesimal reservoir based on the nucleus of strain approach. The final displacement field is given by the superposition of the solutions of individual cells. The three-dimensional integration eliminates the singularities and allows the evaluation of displacements throughout the entire model, including the reservoir. The solutions presented can be easily implemented in a computational procedure, maintaining the low effort of similar methods. Finite element models are employed to verify the accuracy of the proposed solution.     

Movement of drops on a solid surface due to a contact angle gradient

Experimental results on the motion of liquid drops on horizontal surfaces resulting from a contact angle gradient are presented. Silicon surfaces were modified using dodecyltrichlorosilane to generate the gradient. Water drops with initial diameters of 0.31 ‐ 0.5 mm (15 ‐ 65 nl) were placed on the surface, their movement videotaped, and subsequently analyzed. To characterize the gradient surface the static contact angle was measured along the surface.     

Mixing and charge transfer in a nanofluidic system due to a patterned surface

In this work, electroosmotic flow mixing enhancement is investigated in a sufficiently long nano-channel with a rectangular nonconducting obstacle mounted on the lower wall with heterogeneous surface electrical potential. A linear pressure drop is observed above the region of patch potential to form a recirculating zone. Formation of vortical flow due to the combined effect of enhanced surface area, heterogeneity in surface potential and external electric field is discussed. A numerical method is adopted based on a finite volume approach to solve the governing coupled Nernst–Planck, Navier–Stokes and Maxwell’s equations to describe ion transport, electrical potential distribution and their impact on flow field. It is observed that an increment in patch potential strength and block height increases the species concentration gradient and surface area, respectively, which enhances the diffusive flux resulting in better mixing performance.     

Convergence of implicit difference scheme for the problem of conjunctive ground water and surface flow with an arbitrary channel cross section

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 27–46, 1990.     

Identification of ground water flow patterns using particle models

Particle models are often used to simulate transport processes in ground water. The ground water flow pattern is one of the driving parameters of the transport model. In this paper a parameter identification algorithm is developed for particle type models to identify the underlying flow pattern from concentration measurements. The estimation problem is solved with a gradient based algorithm. For each generated particle track, the adjoint track is determined to efficiently compute gradient of the criterion.     

Analysis of viscous flow due to a stretching sheet with surface slip and suction

The viscous flow due to a stretching sheet with slip and suction is studied. The Navier–Stokes equations admit exact similarity solutions. For two-dimensional stretching a closed-form solution is found and uniqueness is proved. For axisymmetric stretching both existence and uniqueness are shown. The boundary value problem is then integrated numerically.     

Nonlinear free surface condition due to second order diffraction by a pair of cylinders

An analysis of the non-homogeneous term involved in the free surface condition for second order wave diffraction on a pair of cylinders is presented. In the computations of the nonlinear loads on offshore structures the most challenging task is the computation of the free surface integral. The main contribution to this integrand is due to the nonhomogeneous term present in the free surface condition for second order scattered potential. In this paper, the free surface condition for the second order scattered potential is derived. Under the assumption of large spacing between the two cylinders, waves scattered by one cylinder may be replaced in the vicinity of the other cylinder by equivalent plane waves together with non-planner correction terms. Then solving a complex matrix equation, the first order scattered potential is derived and since the free surface term for second order scattered potential can be expressed in terms of the first order potentials, the free surface term can be obtained using the knowledge of first order potentials only.     

Application of Hankel transforms to boundary value problems of water flow due to a circular source

With the aid of Hankel transform technique, we obtain close-form solutions for discontinuous boundary-condition problems of water flow due to a circular source, which located on the upper surface of a confined aquifer. Owing to difficult evaluations of the original solutions that are in a form of an infinite range integral with a singular point and Bessel functions in integrands, we adopt two numerical algorisms to transform the original solutions as a series form for convenient practical applications. We apply the solutions in series form to numerical examples to analyze the characteristics of the flow in the confined aquifers subjected to pumping or recharge. By numerical examples, it indicates that: the drawdown will reduce with the increase of the layer thickness and the distance from the center of a circular source when pumping in a region with a finite thickness and a finite width; two algorisms for closed-form solutions of an infinite range integral have almost the same results, but the second algorism is superior for a faster convergence; in a semi-infinite confined aquifer, the drawdown due to a constant pumping rate Q and uplift due to recharge by a given hydraulic head s₀ will both decrease with the increase of K_r/K_v; however, the radius r₀ of the circular source has a reverse influence on the drawdown and the uplift, i.e., the drawdown decrease with the increase of r₀, while the uplift increase with r₀.     

Amplification of surface ground motion by an inclusion of arbitrary shape for harmonic surface line loading

The plane strain model for the Lamb's problem with an elastic inclusion of arbitrary shape embedded completely within an elastic half space is investigated by using an indirect boundary integral equation method for steady-state elastodynamics. The surface of the half space is subjected to vertical or horizontal harmonic line loads. The displacement field is evaluated throughout the elastic medium so that the continuity of the displacement and traction fields along the interface between the half space and the inclusion is satisfied in a least-square sense. The numerical results demonstrate that the presence of the inclusion may cause locally very large amplification of the surface ground motion and that the amplification pattern depends upon the frequency and the type of the input load, the impedance contrast between the half space and the inclusion, the type of the inclusion, and the location of the observation point at the surface of the half space.     

Boundary layer due to the axisymmetric spreading of material on the surface of a fluid

The effect of the axisymmetric spreading of a layer of material (oil or solid particles) on the surface of a viscous fluid is studied. Assuming high Reynolds numbers, the boundary layer equation is derived and solved for general power law surface velocities. The composite streamlines show sharp turns near the surface.