Weak Surface Tension Research Articles

Drawing of fibres composed of shear-thinning or shear-thickening fluid with internal holes

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$ th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

Lump solutions of the fractional Kadomtsev–Petviashvili equation

Of concern is the fractional Kadomtsev–Petviashvili (fKP) equation and its lump solution. As in the classical Kadomtsev–Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation [9] nor for the fKP-I when α≤45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\le \\frac{4}{5}$$\\end{document} [26]. Furthermore, we show that for any α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

Bifurcations and Exact Traveling Wave Solutions of the Generalized Serre–Green–Naghdi System with Weak Coriolis Effect and Surface Tension

For the generalized Serre–Green–Naghdi system with weak Coriolis effect and surface tension, by using the dynamical system methods and singular traveling wave theory developed by Li and Chen [2007] to its associate traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solutions, periodic wave solutions, peakons, periodic peakons as well as compacton solution families) are obtained. Exact explicit parametric representations are given.

Interaction structures of multi localized waves within the Kadomtsev–Petviashvili I equation

This paper investigates a (2+1)-dimensional Kadomtsev–Petviashvili I (KP I) equation, which describes the long waves in water with weak surface tension. The KP I equation can be reduced to a (1+1)-dimensional system consisting of a generalized multi component nonlinear Schrödinger (NLS) equation and a modified Korteweg–de Vries (mKdV) equation by using constraints the potential function of the KP I equation to its squared eigenfunction. So, any solution of the (1+1)-dimensional integrable system gives rise to a localized solution of the KP I equation. Then, according to the Lax pairs of the above (1+1)-dimensional system, the generalized (m,N−m)-fold Darboux transformation is adopted to explore various solutions. These solutions include single structure solutions, such as lump-chains, (N-1)-th degenerate lump-chains and Nth-order lump, and especially interaction solutions of arbitrary structures formed on the basis of single structure solutions. As a consequence, the corresponding single structure and mixed interaction solutions for the KP I equation can be obtained by solving the (1+1)-dimensional system with generalized DT. Moreover, all these structure solutions are discussed analytically and shown graphically. Finally, we also sum up various mathematical features of mixed localized wave solutions. The results obtained in this paper may be helpful to understand the interaction phenomena of localized nonlinear waves in KP I equation.

Improvement of wetting and necking of nickel-based superalloys fabricated by sequential dual-laser powder bed fusion via particle-scale computational fluid dynamics

The stability of additively manufactured products is mainly affected by their pore defects and surface quality and critical in their applications in various engineering fields. Based on this understanding, a three-dimensional high-fidelity particle-scale computational fluid dynamics (CFD) model for a sequential dual-laser powder bed fusion (SDL_PBF) process was developed in this paper. Using the model, the mechanism of improving the product quality of nickel-based superalloy GH4169 by dual laser sequential scanning strategies was particularly discussed. The effects of laser powers and offsets of sequential dual lasers on the molten pool wetting, pores, and necking were also specially studied, in which the process parameter window was determined through orthogonal experiments. The CFD simulations showed that, when the offsets of the sequential dual lasers are about 3.7 times of the laser radius, the wetting of the molten pool is the best in our test, because the molten pool flows around, resulting in a larger overlapping area. The SDL_PBF with a suitable rear laser source can eliminate the pores and the entrained air formed by the collapse of the recoil depression, which in turn is caused by the front laser. This is because the recoil depression generated by the rear laser can break down the pores caused by the incomplete melting of the front laser, so that the gas in the pores communicates with the air outside, and the liquid in the molten pool is replenished, thereby eliminating the pores and reducing the porosity of the product, which is consistent with the experimental results. In addition, the simulation results showed that the molten pool can flow stably and quickly due to the combined action of strong Marangoni force and steam recoil force as well as weak surface tension in the case of the SDL_PBF with appropriate process parameters, resulting in greatly improved surface quality. Therefore, this work can provide a way for improving multi-laser additive manufacturing of high-performance products. • A particle scale CFD model of SDL_PBF was developed for the first time. • The optimized process window was obtained through orthogonal experiment. • The mechanism of improving product quality was explored at mesoscopic scale.

Existence of Davey--Stewartson Type Solitary Waves for the Fully Dispersive Kadomtsev--Petviashvilii equation

We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero. Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation.

Solitary waves in a Whitham equation with small surface tension

AbstractUsing a nonlocal version of the center‐manifold theorem and a normal form reduction, we prove the existence of small‐amplitude generalized solitary‐wave solutions and modulated solitary‐wave solutions to the steady gravity‐capillary Whitham equation with weak surface tension. Through the application of the center‐manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four‐dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity‐capillary water wave problem, the associated linear operator is seen to undergo either a reversible bifurcation or a reversible bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second‐order or third‐order terms. These truncated systems relate directly to systems obtained in the study of the full gravity‐capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity‐capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity‐capillary Whitham equation and the full two‐dimensional gravity‐capillary water wave problem.

An Existence Theory for Gravity\u2013Capillary Solitary Water Waves

In the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

Conductive Polymer Spray Ionization Mass Spectrometry for Biofluid Analysis.

We present a conductive polymer spray ionization (CPSI) method for the direct mass spectrometric analysis of hydrophilic drugs, saccharides, peptides, and proteins in biofluids. Carbon nanotubes (CNTs) were introduced into poly(methyl methacrylate) (PMMA) to fabricate a conductive composite substrate CNT/PMMA in the shape of a triangle (8 mm wide and 10 mm long) with its apex pointed toward the inlet of a mass spectrometer. In comparison with a traditional paper spray substrate, the conductive polymer absorbs less hydrophilic compounds owing to its hydrophobic nature. When aqueous biofluid samples are loaded, they also exhibit less diffusion on this nonporous surface. Only 1.0-2.0 μL solvent suffices to extract the components in a dried biofluid spot and to form charged microdroplets (4.5 kV high voltage applied). Furthermore, the hydrophobic polymer surface only needs to overcome weak surface tension to emit charged microdroplets, so that the signal has a typical duration of 7.5 min. For sunitinib, acarbose, melamine, and angiotensin II, the ion intensity of the target compound from the conductive polymer support is significantly higher than paper spray, typically by a factor of 20 to 100. These results suggest that the CNT/PMMA conductive polymer spray has great potential in the analysis of hydrophilic drugs, saccharides, peptides, and proteins in biofluids.

The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow

We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is well-posed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$, while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$. The physical source of the ill-posedness of the principal problem [IBVP], when $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$, is revealed to be due to the leading-order problem in the innermost region localised close to the initial contact point being in the form of a local Rayleigh–Taylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$, the inclusion of weak surface tension effects will restore well-posedness of the problem [IBVP] which will, however, remain temporally unstable.

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

Influence of viscosity and surface tension of fluid on the motion of bubbles

Boundary integral simulation has been conducted to study the motion and deformation of bubbles with weak viscous and surface tension effects in fluid. Both normal and tangential stress boundary conditions are satisfied and the weak viscous effects are confined to the thin boundary layers around bubble surfaces, which is also known as boundary layer theory of bubble. By using this method, the influence of viscosity and surface tension of fluid on the motion of bubbles has been studied. Both axisymmetric and three-dimensional numerical results are compared with analytical results of Rayleigh-Plesset equation. Good agreement between them is achieved, which validates the numerical model. On this basis, interaction model between two vertically placed bubbles is established, by taking the surface tension, gravity, and viscous effects into consideration. Variations of physical quantities including bubble deformation, jet velocity, and energy of fluid are studied. Last but not least, the influence of viscosity and surface tension on the motion of a spherical bubble is investigated. It is found that viscous effects of fluid depress the pulsation of bubble and part of fluid energy is transformed into viscous dissipation energy. As a result, the development of bubble jet, the radius of the bubble, and the jet velocity are reduced gradually. On the other hand, the surface tension of fluid does not change the range of the bubble pulsation but reduces the period of the bubble pulsation and enhances the potential energy of the bubble. This model and numerical results aim to provide some references for bubble dynamics in bioengineering, chemical engineering, naval architecture, and ocean engineering, etc.

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.

Modulational instability in the Whitham equation with surface tension and vorticity

We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity, we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions to the physical problem.

Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 &lt;µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King &amp; Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King &amp; Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

Bubble rise dynamics in a viscoplastic material

The axisymmetric dynamics of a bubble rising in a Bingham fluid under the action of buoyancy is studied. The Volume-of-Fluid (VOF) method is used to solve the equations of mass and momentum conservation, coupled to an equation for the volume fraction of the Bingham fluid. A regularised constitutive model is used for the description of the viscoplastic behavior of the material. The numerical results demonstrate that the rise dynamics are complex for large yield stresses, and for weak surface tension. Under these conditions, for which the bubble is highly deformable, the rise is unsteady and is punctuated by periods of rapid acceleration which separate stages of quasi-steady motion. During the acceleration periods, the bubble aspect ratio exhibits oscillations about unity, whose amplitude and wavelength increase with increasing yield stress and decreasing surface tension. These oscillations are accompanied by the alternating formation and destruction of unyielded zones at the bubble equator, as the bubble appears to “swim” upwards.

Calculating models on surface tension of RE2O3–MgO–SiO2 (RE=La, Nd, Sm, Gd and Y) melts

A thermodynamic model was developed for determining the surface tension of RE2O3–MgO–SiO2 (RE=La, Nd, Sm, Gd and Y) melts considering the ionic radii of the components and Butler's equation. The temperature and composition dependence of the surface tensions in molten RE2O3–MgO–SiO2 slag systems was reproduced by the present model using surface tensions and molar volumes of pure oxides, as well as the anionic and cationic radii of the melt components. The iso-surface tension lines of La2O3–MgO–SiO2 slag melt at 1873 K were calculated and the effects of slag composition on the surface tension were also investigated. The surface tensions of La2O3, Gd2O3, Nd2O3 and Y2O3 at 1873 K were evaluated as 686, 677, 664 and 541 mN/m, respectively. The surface tension of pure rare earth oxide melts linearly decreases with increasing cationic field strength, except for Y2O3 oxide, while Y2O3 has a much weaker surface tension. The evaluated results of the surface tension show good agreements with literature data, and the mean deviation of the present model is found to be 1.05% at 1873 K.

On the existence and conditional energetic stability of solitary water waves with weak surface tension

An existence and stability theory for solitary water waves with weak surface tension has recently been given by Buffoni (2005, 2009) [2,3]. The theory, which is variational in nature, relies upon the assumption that the infimum of the variational functional is strictly subhomogeneous with respect to a small parameter. In this Note we rigorously establish the relevant strict-subhomogeneity property and thus complete Buffoni's theory.

Existence and stability of regularized shock solutions, with applications to rimming flows

This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ɛ, then taking the limit ɛ → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equations (e.g. waves of depression in gas dynamics), we show that it is also helpful for stability analysis. The general approach is illustrated by applying it to rimming flows, i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder, with or without surface tension (which plays the role of the regularizing effect). In the latter case, the spectrum of available linear eigenmodes appears to be continuous, but in the former, it is discrete and, most importantly, remains discrete in the limit of infinitesimally weak surface tension. The regularized (discrete) spectrum is fully determined by the point where the velocity of small perturbations vanishes, with the rest of the domain, including the shock region, being unimportant.