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 Journal of Engineering MathematicsJournal Prestige (SJR): 0.418 Citation Impact (citeScore): 1Number of Followers: 2      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1573-2703 - ISSN (Online) 0022-0833 Published by Springer-Verlag  [2658 journals]
• Resonant mode scanning to compute the spectrum of capillary surfaces with
dynamic wetting effects

Abstract: A capillary surface bound by a solid rectangular channel exhibits dynamic wetting effects characterized by a constitutive law relating the dynamic contact-angle to the contact-line speed through the contact-line mobility $$\Lambda$$ parameter. Limiting cases correspond to the free ( $$\Lambda =0$$ ) and pinned ( $$\Lambda =\infty$$ ) contact-line. Viscous potential flow is used to derive the governing integrodifferential equation from a boundary integral approach. The spectrum is determined from a boundary value problem where the eigenvalue parameter appears in the boundary condition. Here we introduce a new frequency scan approach to compute the spectrum, whereby we scan the complex frequency plane and compute the system response from which we identify the complex resonant frequency. Damping effects due to viscosity and Davis dissipation from finite $$\Lambda$$ do not attenuate signal response, but rather shift the response poles into the complex plane. Our new approach is verified against an analytical solution in the appropriate limit. We identify the critical mobility that maximizes Davis dissipation and the critical Ohnesorge number (viscosity) where the transition from underdamped to overdamped oscillations occurs, as it depends upon the static contact-angle $$\alpha$$ . Our approach is applied to a rectangular channel, but is suitable for a myriad of geometric supports.
PubDate: 2021-07-18

• Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw
channels

Abstract: The steady propagation of air bubbles through a Hele-Shaw channel with either a rectangular or partially occluded cross section is known to exhibit solution multiplicity for steadily propagating bubbles, along with complicated transient behaviour where the bubble may visit several edge states or even change topology several times, before typically reaching its final propagation mode. Many of these phenomena can be observed both in experimental realisations and in numerical simulations based on simple Darcy models of flow and bubble propagation in a Hele-Shaw cell. In this paper, we investigate the corresponding problem for the propagation of a viscous drop (with viscosity $$\nu$$ relative to the surrounding fluid) using a Darcy model. We explore the effect of drop viscosity on the steady solution structure for drops in rectangular channels or with imposed height variations. Under the Darcy model in a uniform channel, steady solutions for bubbles map directly on to those for drops with any internal viscosity $$\nu \ne 1$$ . Hence, the solution multiplicity predicted for bubbles also occurs for drops, although for $$\nu >1$$ , the interface shape is reversed with inflection points appearing at the rear rather than the front of the drop. The equivalence between bubbles and drops breaks down for transient behaviour, at the introduction of any height variation, for multiple bodies of different viscosity ratios and for more detailed models which produce a more complicated flow in the interior of the drop. We show that the introduction of topography variations affects bubbles and drops differently, with very viscous drops preferentially moving towards more constricted regions of the channel. Both bubbles and drops can undergo transient behaviour which involves breakup into two almost equal bodies, which then symmetry break before either recombining or separating indefinitely.
PubDate: 2021-07-18

• On asymptotic theory of plane separated flows

Abstract: A review of investigations of plane steady external flows of incompressible fluid with developed separation zones at high Reynolds numbers is presented.
PubDate: 2021-07-18

• Approximate phase speed of Lamb waves in a composite plate reinforced with
strong fibres

Abstract: In the study of wave propagation through plates and laminates, there are no complete three-dimensional solutions especially if the plate or plies are made of an anisotropic material. In such cases, laminate theories are approximate and often give poor results in comparison with exact solutions. When laminae or plate is highly anisotropic, the small parameter, which is appropriate is usually the modulus ratio, that is, the ratio of some elastic moduli. In the study of wave propagation, in some of the ranges of a wave number, a geometrical parameter, non-dimensional wave number, may be dominant in dispersion relations. The interaction between these two parameters, the modulus ratio and the geometric parameter, is a complicated one and it is not obvious which parameter will dominate in the given circumstances. The influence of these two parameters on dynamical behaviour of the plate, especially in the dispersion relation is examined in the present paper. To consider the singular perturbation problem, where at least one boundary layer arises, it is usual to apply the method of asymptotic expansions. In that case, a matched asymptotic expansion is suitable for the consideration. To satisfy well the corresponding equations, it was necessary to develop outer and inner expansions which are valid in the so-called outer and inner regions, respectively. Then, matching conditions have been derived, which lead to establishing an asymptotic expansion which is valid uniformly in the whole domain.
PubDate: 2021-07-18

• Nonlinear rate-dependent spectral constitutive equation for viscoelastic
solids with residual stresses

Abstract: A spectral constitutive equation for finite strain viscoelastic bodies with residual stresses is developed using spectral invariants, where each spectral invariant has a clear physical meaning. A prototype constitutive equation containing single-variable functions is presented; a function of a single invariant with a clear physical interpretation is easily manageable and is experimentally attractive. The effects of residual stress and viscosity are studied via the results of some boundary value problems, and some of these results are compared with experimental data.
PubDate: 2021-07-18

• Dynamic simulations of an encapsulated microbubble translating in a tube
at low capillary and Reynolds numbers

Abstract: The dynamic translation of a micron-sized encapsulated bubble is investigated numerically inside a horizontal tube where liquid flows under constant pressure drop, when the effect of gravity is neglected. The coating of the bubble is treated as an infinitesimally thin viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed to solve the axisymmetric flow configuration combined with the spine or elliptic mesh generation techniques for updating the mesh. The microbubble is initially elongated and the Reynolds number of the flow is relatively small, i.e. $$\hbox {Re}< 5$$ . Benchmark simulations for long free bubbles robustly recover the scaling of the film thickness with the 2/3 power of the capillary number based on surface tension. In the case of encapsulated bubbles, for a sufficiently small capillary number and after a short transient period, the bubble acquires a Bretherton type shape that slowly expands in order to accommodate changes in the liquid pressure. The speed of translation is nearly constant, close to the mean velocity of the flow, and does not depend on surface tension, shell elasticity or bending resistance. Fluid motion in the thin film “contact” region that forms in the gap between the tube and the shell is seen to be a stable flow arrangement that entails a mixture of pressure driven and shear driven flow, with the latter greatly affected by the area dilatation modulus via the tangential stress balance. By introducing a modified capillary number based on the area dilatational modulus, rather than surface tension, it is seen that the dimensionless film thickness that occupies the region between the bubble and the tube wall increases with the 1/3 power of the modified capillary number with increasing area dilatation. Simulations when surface tension is absent indicate that tangential shear generated due to variation of the membrane stress in the transition region that joins the bulk of the flow configuration with the contact region, leads to film thinning via the 5/7 power of the modified capillary number. Variations in the transverse shear of the viscoelastic shell generate large lubrication overpressures in the thin film region between the tube and the shell that are exerted radially on the shell and are conjectured to be responsible for the onset of 3d buckled shapes. The latter are often observed experimentally in similar flow configurations of capsules and are characterized by wrinkles that develop in the azimuthal direction around the shell equator.
PubDate: 2021-06-30

• A numerical study on non-Fourier heat conduction model of phase change
problem with variable internal heat generation

Abstract: In this study, the authors proposed one-dimensional non-Fourier heat conduction model applied to phase change problem in the presence of variable internal heat generation and this has been performed by finite element Legendre wavelet Galerkin method (FELWGM). We derived the stability analysis of the non-Fourier heat conduction model in our present case. The finite difference technique has been used to change the non-Fourier heat conduction model into an initial value problem of vector-matrix form and then we applied Legendre wavelet Galerkin method for the numerical solution of the present problem. The location of moving interface is analytically obtained under the steady-state condition. The effectiveness of the proposed numerical technique is verified through the experimental value of parameters which indicate promising results. In addition, the effect of Stefan numbers, internal heat generation, and its linear coefficient on the location of moving interface are discussed in detail and represented graphically.
PubDate: 2021-06-24

• Introducing pre-impact air-cushioning effects into the Wagner model of
impact theory

Abstract: In this analysis, we consider the effects of non-quiescent initial conditions driven by pre-impact air–water interactions on the classical Wagner model of impact theory. We consider the problem of a rigid, solid impactor moving vertically towards a liquid pool. Prior to impact, viscous forces in the air act to deform the liquid free surface, inducing a flow in the pool. These interactions are then incorporated as initial conditions in the post-impact analysis. We derive expressions for the size of the effective contact set, the leading-order pressure and force on the impactor, and the speed and thickness of the jet at its base. In all cases, we show that the effect of the pre-impact behaviour is to cushion the impactor, reducing the size of the effective contact set and, hence, the force on the impactor. Small- and large-time asymptotic solutions are derived for general power-law impactors, and we show that the effects of the air die away as the impact progresses, so that we approach the classical Wagner solution.
PubDate: 2021-06-22

• A thin double layer analysis of asymmetric rectified electric fields
(AREFs)

Abstract: We use perturbation methods to analyze the “asymmetric rectified electric field (AREF)” generated when an oscillating voltage is applied across a model electrochemical cell consisting of a binary, asymmetric electrolyte bounded by planar, parallel, blocking electrodes. The AREF refers to the steady component of the electric potential gradient within the electrolyte, as discovered via numerics by Hashemi Amrei et. al. (Phys Rev Lett 121(18):185504). We adopt the Poisson–Nernst–Planck framework for ion transport in dilute electrolytes, taking into account unequal ionic diffusivities. We consider the mathematically singular, and practically relevant, limit of thin Debye layers, $$1/(\kappa L) = \epsilon \rightarrow 0$$ , where $$\kappa ^{-1}$$ is the Debye length, and L is the length of the half-cell. The dynamics of the electric potential and ionic strength in the “bulk” electrolyte (i.e., outside the Debye layers) are solved subject to effective boundary conditions obtained from consideration of the Debye-scale transport. We specifically analyze the case when the applied voltage has a frequency comparable to the inverse bulk ion diffusion time scale, $$\omega = {\mathcal {O}}(D_A/L^2)$$ , where $$D_A = 2D_+ D_-/(D_+ + D_-)$$ is the ambipolar diffusivity, and $$D_{\pm }$$ are the ionic diffusivities. In this regime, the AREF extends throughout the bulk of the cell, varying on a lengthscale proportional to $$\sqrt{D_A /\omega }$$ , and has a magnitude of $${\mathcal {O}}(\epsilon ^2 k_B T /(L e))$$ to leading order in $$\epsilon$$ . Here, $$k_B$$ is the Boltzmann constant, T is temperature, and e is the charge on a proton. We obtain an analytical approximation for the AREF at weak voltages, $$V_0 \ll k_B T/e$$ , where $$V_0$$ is the amplitude of the voltage, for which the AREF is $${\mathcal {O}}(\epsilon ^2 V_0^2 e /(k_B T L))$$ . Our asymptotic scheme is also used to calculate a numerical approximation to the AREF that is valid up to logarithmically large voltages, $$V_0 = {\mathcal {O}}((k_B T/e)\ln (1/\epsilon ))$$ . The existence of an AREF implies that a charged colloidal particle undergoes net electrophoretic motion under the applied oscillatory voltage. Additionally, a gradient in the bulk ionic strength, caused by the difference in ionic diffusivities, leads to rectified diffusiophoretic particle motion. Here, we predict the electrophoretic and diffusiophoretic velocities for a rigid, spherical, colloidal particle. The diffusiophoretic velocity is comparable in magnitude to the electrophoretic velocity, and can thus affect particle motion in an AREF significantly.
PubDate: 2021-06-17
DOI: 10.1007/s10665-021-10139-x

• Robust simulation of mineral precipitation–dissolution problems with
variable mineral surface area

Abstract: We consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new model.
PubDate: 2021-06-17
DOI: 10.1007/s10665-021-10132-4

• A simplified model for hydroelasticity of containerships

Abstract: In this work, a semi-analytic method has been developed to perform the hydroelasticity analysis of containerships. For the solution of the hydrodynamic problem, a time-domain method is developed based on impulse response function (IRF); however, for the solution of the structural responses, modal superposition technique is used assuming the ship is based on Euler–Bernoulli beam theory. The time-domain amplitude of the displacements and velocities corresponding to several modes is then determined using a semi-analytic approach using Duhamel integral technique. In this paper, the effect of structural flexibility in the calculation of structural displacement, shear force, and bending moment is studied. To check the efficiency and correctness of the proposed semi-analytic method, initially, the computed results are compared with published and experimental results for two container ships with different lengths. In the second phase, a comparative study has been made to check the effect of several physical and geometric parameters such as ship length, vessel speed, and wavelength to ship length ratio. It is seen from the comparative study that ship length, Froude number, wave to ship length ratio, etc. have a significant effect in the calculations of global bending moment, shear force. From the computed results, it may be concluded that the proposed semi-analytic approach is capable of generating results within an acceptable range of engineering accuracy with negligible computational effort, and thus, it can be a very useful tool for preliminary design load for larger vessels.
PubDate: 2021-06-17
DOI: 10.1007/s10665-021-10142-2

• The generation of Tollmien–Schlichting waves by free-stream
turbulence in transonic flow

Abstract: This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of $$O(\mathrm{{Re}}^{-{1}/{2}})$$ , and a vorticity deformation layer is also present with thickness $$O(\mathrm{{Re}}^{-{1}/{4}})$$ . The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness.
PubDate: 2021-06-17
DOI: 10.1007/s10665-021-10138-y

• Instabilities at a sheared interface over a liquid laden with soluble
surfactant

Abstract: The linear stability of a semi-infinite fluid undergoing a shearing motion over a fluid layer that is laden with soluble surfactant and that is bounded below by a plane wall is investigated under conditions of Stokes flow. While it is known that this configuration is unstable in the presence of an insoluble surfactant, it is shown via a linear stability analysis that surfactant solubility has a stabilising effect on the flow. As the solubility increases, large-wavelength perturbations are stabilised first, leaving open the possibility of mid-wave instability for moderate surfactant solubilities, and the flow is fully stabilised when the solubility exceeds a threshold value. The predictions of the linear stability analysis are supported by an energy budget analysis which is also used to determine the key physical effects responsible for the (de)stabilisation. Asymptotic expansions performed for long-wavelength perturbations turn out to be non-uniform in the insoluble surfactant limit. In keeping with the findings for insoluble surfactant obtained by Pozrikidis & Hill (IMA J Appl Math 76:859–875, 2011), the presence of the wall is found to be a crucial factor in the instability.
PubDate: 2021-06-17
DOI: 10.1007/s10665-021-10140-4

• Linear and energy stable schemes for the Swift–Hohenberg equation with
quadratic-cubic nonlinearity based on a modified scalar auxiliary variable
approach

Abstract: In this study, we develop linear and energy stable numerical schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity. A modified scalar auxiliary variable (SAV) approach is used to construct the temporally first- and second-order accurate discretizations. Different from the classical SAV approach, the proposed schemes permit us to solve the governing equations in a step-by-step manner, i.e., the calculation of inner product is not needed. We analytically prove the energy stability. We solve the resulting system of discrete equations using the linear multigrid method. We perform various numerical examples to show the accuracy and energy stability of the proposed method. The pattern formations in two- and three-dimensional spaces are also simulated.
PubDate: 2021-06-02
DOI: 10.1007/s10665-021-10122-6

• Receptivity of inviscid modes in supersonic boundary layers to wall
perturbations

Abstract: The present paper investigates the receptivity of inviscid first and second modes in a supersonic boundary layer to time-periodic wall disturbances in the form of local blowing/suction, streamwise velocity perturbation and temperature perturbation, all introduced via a small forcing slot on the flat plate. The receptivity is studied using direct numerical simulations (DNS), finite- and high-Reynolds-number approaches, which complement each other. The finite-Reynolds-number formulation predicts the receptivity as accurately as DNS, but does not give much insight to the detailed excitation process, nor can it explain the significantly weaker receptivity efficiency of the streamwise velocity and temperature perturbations relative to the blowing/suction. In order to shed light on these issues, an asymptotic analysis was performed in the limit of large Reynolds number. It shows that the receptivity to all three forms of wall perturbations is reduced to the same mathematical form: the Rayleigh equation subject to an equivalent suction/blowing velocity, which can be expressed explicitly in terms of the physical wall perturbations. Estimates of the magnitude of the excited eigenmode can be made a priori for each case. Furthermore, the receptivity efficiencies for the streamwise velocity and temperature perturbations are quantitatively related to that for the blowing/suction by simple ratios, which are of $$O(R^{-1/2})$$ and have simple expressions, where R is the Reynolds number based on the boundary-layer thickness at the centre of the forcing slot. The simple leading-order asymptotic theory predicts the instability and receptivity characteristics accurately for sufficiently large Reynolds numbers (about $$10^4$$ ), but appreciable error exists for moderate Reynolds numbers. An improved asymptotic theory is developed by using the appropriate impedance condition that accounts for the $$O(R^{-1/2})$$ transverse velocity induced by the viscous motion in the Stokes layer adjacent to the wall. The improved theory predicts both the instability and receptivity at moderate Reynolds numbers ( $$R=O(10^3)$$ ) with satisfactory accuracy. In particular, it captures well the finite-Reynolds-number effects, including the Reynolds-number dependence of the receptivity and the strong excitation occurring near the so-called synchronisation point.
PubDate: 2021-06-02
DOI: 10.1007/s10665-021-10124-4

• Optimal control of the Sabatier process in microchannel reactors

Abstract: We consider the optimization of a chemical microchannel reactor by means of PDE-constrained optimization techniques, using the example of the Sabatier reaction. To model the chemically reacting flow in the microchannels, we introduce a three- and a one-dimensional model. As these are given by strongly coupled and highly nonlinear systems of partial differential equations (PDEs), we present our software package cashocs which implements the adjoint approach and facilitates the numerical solution of the subsequent optimization problems. We solve a parameter identification problem numerically to determine necessary kinetic parameters for the models from experimental data given in the literature. The obtained results show excellent agreement to the measurements. Finally, we present two optimization problems for optimizing the reactor’s product yield. First, we use a tracking-type cost functional to maximize the reactant conversion, keep the flow rate of the reactor fixed, and use its wall temperature as optimization variable. Second, we consider the wall temperature and the inlet gas velocity as optimization variables, use an objective functional for maximizing the flow rate in the reactor, and ensure the quality of the product by means of a state constraint. The results obtained from solving these problems numerically show great potential for improving the design of the microreactor.
PubDate: 2021-06-01
DOI: 10.1007/s10665-021-10134-2

• Transient-thermoelastic analysis of periodically rotated functionally

Abstract: A closed-form solution for transient thermal stress analysis of functionally graded hollow cylinder exposed to high-temperature difference is obtained under the influence of periodic rotation. All mechanical and thermal properties except the Poisson’s ratio are assumed to be graded in the radial direction as a power-law function. The transient heat conduction and equilibrium equations are solved on the Laplace domain by using Bessel functions and the Gauss quadrature integration procedure. The inverse transformation to the real space is achieved by using the modified Durbin method. The novelty of this study is to provide a general solution to the functionally graded cylinder under the effect of periodic rotation in a transient regime. The effects of periodic rotation and high-temperature difference on temperature and thermal stresses are investigated for a specific ceramic-metal mixture by using this solution. The solution presented in this study can be adopted simply by changing the coefficients of inhomogeneity in the power-law variation for any pair of materials.
PubDate: 2021-06-01
DOI: 10.1007/s10665-021-10141-3

• The triple-deck stage of marginal separation

Abstract: The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar–turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous–inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler–Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed.
PubDate: 2021-06-01
DOI: 10.1007/s10665-021-10125-3

• A smoothly curved body skimming on shallow water

Abstract: We investigate the dynamics of a smoothly curved convex body skimming on a layer of shallow water, with the phenomenon of skipping stones as a primary modelling motivation but also industrial applications in mind. The skimming process of such an object may consist of two successive stages: an impact stage and a conditional planing stage. The focus here is on explaining the change from one stage to the other, with the body movement responding freely to the fluid flow pressures and vice versa. We first introduce a water impact model and analyse the conditions under which a smooth body may rapidly transit to a planing motion via small time asymptotics. A planing model is then introduced, and in particular we investigate the effects of pressure conditions in the separation flow; we demonstrate under certain weak adverse pressure gradient conditions a smooth body’s planing motion at early times can be seen as undergoing three successive transitory phases.
PubDate: 2021-06-01
DOI: 10.1007/s10665-021-10130-6

• A finite deformation isogeometric finite element approach to
fibre-reinforced composites with fibre bending stiffness

Abstract: It is a common technique in many fields of engineering to reinforce materials with certain types of fibres in order to enhance the mechanical properties of the overall material. Specific simulation methods help to predict the behaviour of these composites in advance. In this regard, a widely established approach is the incorporation of the fibre direction vector as an additional argument of the energy function in order to capture the specific material properties in the fibre direction. While this model represents the transverse isotropy of a material, it cannot capture effects that result from a bending of the fibres and does not include any length scale that might allow the simulation of size effects. In this contribution, an enhanced approach is considered which relies on the introduction of higher-gradient contributions of the deformation map in the stored energy density function and which eventually allows accounting for fibre bending stiffness in simulations. The respective gradient fields are approximated by NURBS basis functions within an isogeometric finite element framework by taking advantage of their characteristic continuity properties. The isogeometric finite element approach that is presented in this contribution for fibre-reinforced composites with fibre bending stiffness accounts for finite deformations. It is shown that the proposed method is in accordance with semi-analytical solutions for a representative boundary value problem. In an additional example it is observed that the initial fibre orientation and the particular bending stiffness of the fibres influence the deformation as well as the stress response of the material.
PubDate: 2021-05-28
DOI: 10.1007/s10665-021-10117-3

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