Showing papers in "Journal of Engineering Mathematics in 2004"

Slamming in marine applications

Abstract: Practical slamming problems for ships and ocean structures are briefly described. Theoretical status and future challenges for water entry on an initially calm free surface, wetdeck slamming, green water and sloshing are presented. It is emphasized that slamming should be considered in the framework of structural dynamics response and integrated with the global flow analysis around a ship or ocean structure or with violent fluid motion inside a tank. Two-phase flow can give important loading and needs to be better understood. Slamming on a VLFS with shallow draft is dealt with in detail.

Initial water impact of a wedge at vertical and oblique angles

Abstract: This paper examines initial asymmetric wedge-impact flows with horizontal as well as vertical impact velocity. The method of two-dimensional vortex distributions is employed to model the initial-boundary-value problem. The numerical analysis involves discretization of the body surface and an iterative solution technique. Experimental drop tests of a prismatic wedge were performed to gain understanding and provide data for comparison of initial water impact when asymmetry and horizontal impact velocity are present. The experimental investigation of initial flow separation off the wedge vertex (i.e., keel) during impact is described. Initial separation-ventilation of the flow from the vertex due to asymmetric impact or horizontal-vertical impact velocity is examined in relation to the present theory. Agreement between the data and the numerical predictions was demonstrated for small degrees of asymmetry and small ratios of horizontal to vertical impact velocity. The initial flow detachment from the vertex also revealed interesting hydrodynamic characteristics.

Jets in bubbles

Abstract: Different types of jet formation in collapsing cavitation and gas bubbles near a rigid boundary are explored by using an advanced boundary-integral technique which incorporates the transition from simply connected to multiply connected bubbles (i.e. toroidal bubbles). Physical interpretation and understanding is facilitated by the calculation of the evolving bubble shape, fluid velocities and pressures, the partitioning of kinetic, potential and gravitational energies, the circulation around the bubble and the Kelvin impulse associated with both the complete bubble and the high-speed liquid jet. In the most vigorous jet formation examples considered it is found that upto 31% of the total energy and upto 53% of the Kelvin impulse is associated with the jet. Practical implications of this study beyond the usual damage mechanisms imply that the level of bubble compression will be signiffcantly lessened leading to lower bubble gas temperatures and thence the corresponding change in the chemical reactivity of its contents or the emission of light. Calculations also suggest interesting phenomena around a standoff distance of 1⋅2 maximum bubble radii where the circulation around the bubble and the kinetic energy of the jet appear to have maximum values. The practical implications and experimental confirmation of this are yet to be explored.

Mud removal and cement placement during primary cementing of an oil well - Part 2; steady-state displacements

Abstract: Uncontrolled flows of reservoir fluids behind the casing are relatively common in primary cementing and can lead to any of the following: blowout, leakage at surface, destruction of subsurface ecology, potential contamination of freshwater, delayed or prevented abandonment, as well as loss of revenue due to reduced reservoir pressures. One significant potential cause is ineffective mud removal during primary cementing. Ideally, the drilling mud is displaced all around the annulus and the displacement front advances steadily up the well at the pumping velocity. This paper addresses the question of whether or not such steady-state displacements can be found for a given set of process parameters.

Buckling of fiber-reinforced viscoelastic composite plates using various plate theories

Abstract: This paper studies the quasi-static stability analysis of fiber-reinforced viscoelastic composite plates subjected to in-plane edge load systems. The study is based on a unified shear-deformable plate theory. This theory enables the trial and testing of different through-thickness transverse shear-strain distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. Using the method of effective moduli solves the equations governing the stability of simply supported fiber-reinforced viscoelastic composite plates. The solution concerns the determination of the critical in-plane edge loads associated with the asymptotic instability of plates. In a study of this problem the general quasi-static stability solutions are compared with those based on the classical, first-order and sinusoidal transverse shear-deformation theories. Numerical applications using higher-order shear-deformation theory are presented and comparisons with the results of other theories are formulated.

A theory of pad conditioning for chemical-mechanical polishing

Abstract: Statistical models are presented to describe the evolution of the surface roughness of polishing pads during the pad-conditioning process in chemical-mechanical polishing. The models describe the evolution of the surface-height probability-density function of solid pads during fixed height or fixed cut-rate conditioning. An integral equation is derived for the effect of conditioning on a foamed pad in terms of a model for a solid pad. The models that combine wear and conditioning are then discussed for both solid and foamed pads. Models include the dependence of the surface roughness on the shape and density of the cutting tips used in the conditioner and on other operating parameters. Good agreement is found between the model, Monte Carlo simulations and with experimental data.

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

Abstract: The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.

Mixing measures for a two-dimensional chaotic Stokes flow

Abstract: The effectiveness of a large number of protocols for mixing in a two-dimensional chaotic Stokes flow, according to a variety of measures, is investigated. The degree to which the various mixing measures are correlated is computed, and while no single protocol simultaneously optimises all measures, it is found that a small subset of the protocols perform well against most measures. However, it is difficult to elicit general rules for selecting effective protocols: for example, superficially similar protocols are found to exhibit considerably different mixing capabilities. The results presented here suggest that the selection of effective protocols by `sieving' (i.e., by successively eliminating candidate protocols that fail increasingly discerning mixing measures) may be ineffective in practice.

Generation of interfacial instabilities in charged electrified viscous liquid films

Abstract: This study is concerned with the stability of a two-dimensional incompressible conducting liquid film surrounded by a passive conducting medium, when an electric field is applied in a direction parallel to the initially flat bounding fluid interfaces. Currents generate charges at the bounding interfaces which in turn affect the stress balances there. In the absence of an electric field, the viscous liquid film is stable (instability can be induced by the inclusion of van der Waals forces for ultra thin films). A complete model is presented, at arbitrary Reynolds number, which accounts for conductivity and permittivity contrasts between the fluid and surrounding medium, as well as surface tension. The linear stability of the system is considered for arbitrary Reynolds numbers and it is shown that the stable film can become unstable if, (i) σ R ɛ p >1, or (ii) σ R ɛ p <1 and (σ R −1)(1−ɛ p )<0, where σ R is the ratio of outer to inner conductivity and e p is the ratio of inner to outer permittivity. Instability is possible only if the electric field is non-zero and the scalings near bifurcation points that can be used to construct nonlinear theories are calculated. Several asymptotic limits are also considered including zero Reynolds numbers and short or long waves. The instability criteria given above are constructed explicitly in the case of Stokes flow.

Wave dynamics on a thin-liquid film falling down a heated wall

Abstract: The dynamics of a thin liquid film falling down a uniformly heated wall is studied. The model introduced by Kalliadasis et al. [J. Fluid Mech. 475 (2003) 377] for the same problem is revisited and its deficiencies, namely the prediction of a critical Reynolds number with 20% error, cured. For the energy equation a high-order Galerkin projection in terms of polynomial test functions is developed. It is shown that not only does this more refined formulation correct the critical Reynolds number, but it also gives, with an appropriate expansion close to criticality, the long-wave theory. Bifurcation diagrams for permanent solitary waves are constructed and compared with the solution branches obtained from different models. It is shown that, in all cases, the long-wave theory exhibits limit points and branch multiplicity, while the other models predict the continuing existence of solitary waves. Time-dependent computations show that the free surface and interfacial temperature approach a train of coherent structures that resemble the infinite-domain stationary solitary pulses.

Flow of surfactant-laden thin films down an inclined plane

Abstract: A theory is formulated to describe the dynamics of a thin film flowing down an inclined plane laden with insoluble surfactant, present in dilute concentrations. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant monolayer concentration. The contact line singularity is relieved by assuming the presence of a thin precursor layer ahead of the advancing film. Base flow solutions for a flow of constant flux are examined over various inclination angle, precursor-layer thickness, Peclet number, and capillary parameter ranges. Application of a transient growth analysis highlights the presence of an instability and the vulnerability of the flow to transverse disturbances of intermediate wavenumber. Our results reveal that several key features of the much-studied uncontaminated film flow, including stability, are modified qualitatively by the inclusion of surfactant.

Oblique slamming, planing and skimming

Abstract: This paper describes how tangential impact velocities can be incorporated into well-known impact theories in deep and shallow water. Taking the deep and shallow flows in turn, it is shown how to link the normal impact Wagner and Korobkin theories to the tangential impact theories of planing and skimming, respectively. An instability is revealed that limits the configurations that can be analysed in the deep water case when the angle of impact is comparable to the deadrise angle. Most of the discussion is confined to two-dimensional flow but a model is also proposed that may describe the spray sheets that can be generated in three-dimensional skimming.

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

Abstract: The effect of inertia on the Yih–Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr–Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.

Integral equation methods for Stokes flow in doubly-periodic domains

Abstract: A fast integral-equation technique is presented for the calculation of Stokes flow in doubly-periodic domains. While existing integral formulations typically rely on a Fourier series to compute the governing Greens' function, here a method of images is developed which is faster, more flexible, and easily incorporated into the fast multipole method. Accurate solutions can be obtained with obstacles of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. The performance of the method is illustrated with several numerical examples.

Analytical solutions for unsteady free convection in porous media

Abstract: Analytic solutions for two of the similarity cases identified by Johnson and Cheng (1978) for the unsteady free-convection boundary-layer flow over an impermeable vertical flat plate adjacent to a fluid saturated porous medium are given in the present paper. These are the solutions corresponding to an exponential (e sup a 2 t sup) and a power-law (t m) variation of the surface temperature, respectively. They represent exact solutions for doubly infinite plates and approximate solutions for semi-infinite plates. In the latter cases their validity is restricted to the so-called `conduction regime' of the flow. It is shown that in the power law case, physical solutions only exist in the range m>−1 of the temperature exponent and they can be expressed in terms of Kummer's confluent hypergeometric functions. For m ≥ 0 exponentially decaying unique solutions were found, while in the range −1

Stability of Equilibria in a Four-dimensional Nonlinear Model of a Hydraulic Servomechanism

Abstract: Starting from plysical laws a four-dimensional nonlinear model for mecano-hydraulic servomechanisms is deduced. The stability of its equilibria is analysed using a theorem of Lyapunov and Malkin to handle the critical case due to the presence of zero in the spectrum of the matrix of the linear part around equilibria. Stability diagrams are drawn and simulation results are presented through phase diagrams.

Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions

Abstract: It is well-known that a standard lubrication analysis of the equations of motion in thin liquid films coating the inside surface of a rotating horizontal cylinder leads, under creeping-flow conditions, to a cubic equation for the film thickness profile which, depending on the fluid properties of the liquid, the speed of rotation and the fill fraction F, has either (a) a continuous, symmetric (homogeneous) solution; (b) a solution containing a shock; or (c) no solution below a certain speed. By means of an asymptotic analysis of the recently proposed “modified lubrication equation” (MLE) [M. Tirumkudulu and A. Acrivos, Phys. Fluid 13 (2000) 14–19], it is shown that the solutions of the cubic equation referred to above correctly describe the film-thickness profiles although, when shocks are involved, under exceedingly restrictive conditions, typically F~ 10−3 or less. In addition, using the MLE, the linear stability of these film profiles is investigated and it is shown that: the “homogeneous” profiles are neutrally stable if surface-tension effects are neglected but, if the latter are retained, the films are asymptotically stable to two-dimensional disturbances and unstable to axial disturbances; on the other hand, the non-homogeneous profiles are always asymptotically stable, thus confirming results given earlier [T.B. Benjamin, W.G. Pritchard, and S.J. Tavener (preprint, 1993)] on the basis of the standard lubrication analysis.

Surface-tension-driven dewetting of Newtonian and power-law fluids

Abstract: The dewetting over a planar substrate of a thin layer of highly viscous fluid under the action of surface tension is considered, with a doubly-nonlinear fourth-order degenerate parabolic equation governing the flow of a power-law fluid. Asymptotic methods are applied to analyse the motion in the shear-thinning, shear-thickening and Newtonian cases, the last of these corresponding mathematically to a critical value of the relevant exponent. In particular, the role played by the local behaviour in the neighbourhood of the contact line is analysed and the dependence of the one-dimensional large-time dewetting behaviour on the fluid’s constitutive properties characterised. Stability issues are also touched upon.

Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity

Abstract: Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented.

Interaction of free-surface waves with floating flexible strips

Abstract: The method developed by the author to derive a set of algebaic equations to solve the interaction of free-surface waves with a single floating rigid or flexible two-dimensional platform with small draft is extended to the case that the platform consists of strips with different constant flexural rigidity and mass. The method is based on the application of Green's theorem, with a specific choice of the Green function to arrive at a differential-integral equation along the platform. This equation can be solved exactly by means of superposition of exponential functions, a standard method to solve a set of linear differential equations. After integration with respect to the space coordinate the residue theorem leads to both the dispersion relation along each individual strip and an algebraic equation for the coefficients. Due to very fast convergence with respect to the number of coefficients taken into account the series are truncated. Depending on the water-depth, in each series three to ten terms are taken into account. Results are shown for a structure consisting of several strips that are tightly connected and for disjoint strips. In the latter case the computation of the water level between the strips is also computed. The water level and the reflection and transmission coefficients are not unknowns in the algebraic equation, but are computed afterwards by means of Green's theorem.

Effect of inertia on the Marangoni instability of two-layer channel flow, Part I: numerical simulations

Abstract: A numerical method is developed for simulating the flow of two superposed liquid layers in a two-dimensional channel confined between two parallel plane walls, in the presence of an insoluble surfactant. The algorithm combines Peskin’s immersed-interface method with the diffuse-interface approximation, wherein the step discontinuity in the fluid properties is replaced by a transition zone defined in terms of a mollifying function. A finite-difference method is implemented for integrating the generalized Navier–Stokes equation incorporating the jump in the interfacial traction, and a finite-volume method is implemented for solving the surfactant transport equation over the evolving interface. The accuracy of the overall scheme is confirmed by successfully comparing the numerical results with the predictions of linear stability analysis and numerical simulations based on a boundary-element method for Stokes flow. Results for selected case studies suggest that inertial effects have a mild effect on the growth rate of the surfactant-induced Marangoni instability.

Green's functions for transversely isotropic piezoelectric multilayered half-spaces

Abstract: This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.

A numerical model for the jet flow generated by water impact

Abstract: In this paper a numerical model is developed aimed at describing the jet flow caused by water impact. The study, carried out in the framework of a potential-flow assumption, exploits the shallowness of the jet region to significantly simplify the local representation of the velocity field. This numerical model is incorporated into a fully nonlinear boundary-element solver that describes the flow generated by the water entry of two-dimensional bodies. Attention is focused on the evaluation of the capability of the model to provide accurate free-surface shape and pressure distribution along the wetted part of the body contour, with particular regard to the jet region. After a careful verification, the proposed model is validated through comparisons with the similarity solution of the wedge impact with constant entry velocity. This similarity solution is derived with the help of an iterative procedure which solves the governing boundary-value problem written in self-similar variables.

Thin-film flows near isolated humps and interior corners

Abstract: This paper concerns the surface-tension-driven flow of a thin layer of viscous liquid following a sudden change in the shape of its substrate. It is assumed that the substrate either develops an isolated hump or bends to create an interior corner. The flow is modelled using an evolution equation derived from lubrication theory, extended in the case of a corner with fully nonlinear expressions for interfacial curvature and volume conservation. Numerical simulations and large-time asymptotics are used to describe the evolution of the film. Over long times the film typically forms a quasi-static puddle adjacent to the hump (or in the corner) and a wave-like disturbance propagates into the far field. For sufficiently large humps and sharp corners, the film pinches off to form an effective contact line at the edge of the puddle, at which the film height tends to zero as time tends to infinity; as long as the film does not rupture (which it cannot in the mathematical framework adopted), the effective contact line drifts slowly away from the hump towards a limiting position dictated by the transient dynamics. Flows off humps with maxima less than a critical height have a qualitatively different structure, captured by one of two possible branches of similarity solutions of the thin-film equation, whereby pinch-off does not occur.

The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

Abstract: It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains.

A finite-element method for interfacial surfactant transport, with application to the flow-induced deformation of a viscous drop

Abstract: A Galerkin finite-element method is developed for solving the transport equation governing the evolution of the surface concentration of an insoluble surfactant over a stationary or evolving fluid interface. The numerical procedure is implemented on an unstructured three-dimensional surface grid consisting of six-node curved triangular elements. Numerical investigations show that the finite-element method is superior to a previously developed finite-volume method for both convection- and diffusion-dominated transport, and especially when the interfacial grid is coarse and steep gradients arise due to local accumulation. The numerical methods for surface transport are combined with a boundary-element method for Stokes flow, and dynamical simulations are performed to illustrate the possibly significant effect of the surface equation of state relating the surface tension to the surfactant concentration on the deformation of a viscous drop in simple shear flow.

Diffusion-limited evaporation of thin polar liquid films

Abstract: The stability of evaporating very thin films of a polar liquid is investigated. The microscopic interaction with the substrate and capillarity are taken into account in a lubrication equation. The stability of a flat interface is studied when evaporation is limited by the diffusion of the vapour in the gas phase. The evaporation rate is computed and evaporation is shown to be stabilizing. A stability phase diagram is obtained. A weakly nonlinear analysis leads to a film-thickness amplitude equation that is non local in space. Physical consequences of the results are eventually discussed.

Modelling of diffuse interfaces with temperature gradients

Abstract: The work is devoted to capillary phenomena in miscible liquids under the assumption that they have a constant and the same density. The model consists of the heat equation, diffusion equation, and the Navier-Stokes equations with the Korteweg stress. We study several configurations corresponding to the microgravity experiments planned for the International Space Station. The basic conclusion of the numerical simulations is that transient capillary phenomena in miscible liquids exist and can produce convective flows sufficiently strong to be observed experimentally. In particular, there exists a miscible analogue to the Marangoni convection where the temperature gradient is applied along the transition zone between two fluids. Convection also appears if, instead of the temperature gradient, the case where the width of the transition zone varies in space is considered. Finally, similar to the immiscible case, miscible drops move in a temperature gradient.

Surfactant-driven motion and splitting of droplets on a substrate

Abstract: A theoretical and computational model is presented to predict the motion of a small sessile liquid droplet, lying on a solid substrate including surfactant effects. The model, as formulated, consists of coupled partial differential equations in space and time, and several auxilliary relationships. The validity of the long-wave, or ‘lubrication’ approximation is assumed. It is shown that there are circumstances where surfactant injection or production will cause the droplet to split into two daughter droplets. It is conjectured that the results are relevant to basic mechanisms involved in biological cell division (cytokinesis). It is also demonstrated that motion of a droplet, analogous to the motility of a cell, can be produced by surfactant addition. Computed examples are given here, in both two and three space dimensions. Approximate energy requirements are also calculated for these processes. These are found to be suitably small.

A numerical Green's function for multiple cracks in anisotropic bodies

Abstract: The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary-integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.