Showing papers in "Journal of Engineering Mathematics in 2016"

Oblique water wave scattering by two unequal vertical barriers

Abstract: Scattering of an obliquely incident wave train by two non-identical thin vertical barriers either partially immersed or fully submerged in infinitely deep water was studied by employing Havelock’s expansion of water wave potential and reducing the problem ultimately to the solution of a pair of vector integral equations of the first kind. A one-term Galerkin approximation in terms of a known exact solution of the integral equation corresponding to a single vertical barrier is used to obtain very accurate numerical estimates for the reflection and transmission coefficients. The reflection coefficient is depicted graphically for two different arrangements of the vertical barriers. It is observed that total reflection is possible for some discrete values of the wavenumber only when the barriers are identical, either partially immersed or completely submerged. As the separation length between the two vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. Also, as the separation length becomes very small, the known results for a single barrier are obtained for normal incidence of the wave train.

A robust quantitative comparison criterion of two signals based on the Sobolev norm of their difference

Abstract: We present a method for the quantitative comparison of two surfaces, applicable to temporal and/or spatial extent in one or two dimensions. Often surface comparisons are simply overlaid graphs of results from different methodologies that are qualitative at best; it is the purpose of this work to facilitate quantitative evaluation. The surfaces can be analytical, numerical, and/or experimental, and the result returned by the method, termed surface similarity parameter or normalized error, has been normalized so that its value lies between zero and one. When the parameter has a value of zero, the surfaces are in perfect agreement, whereas a value of one is indicative of perfect disagreement. To provide insight regarding the magnitude of the parameter, several canonical cases are presented, followed by results from breaking water wave experimental measurements with numerical simulations, and by a comparison of a prescribed, periodic, square-wave surface profile with the subsequent manufactured surface.

Viscous fingering in yield stress fluids: a numerical study

Abstract: The effect of yield stress is numerically investigated on the viscous fingering phenomenon in a rectangular Hele–Shaw cell. It is assumed that the displacing fluid is Newtonian, while the displaced fluid is assumed to obey the bi-viscous Bingham model. The lubrication approximation together with the creeping-flow assumption is used to simplify the governing equations. The equations so obtained are made two-dimensional using the gap-averaged variables. The initially flat interface between the two (immiscible) fluids is perturbed by a waveform perturbation of arbitrary amplitude/wavelength to see how it grows in the course of time. Having treated the interfacial tension like a body force, the governing equations are solved using the finite-volume method to obtain the pressure and velocity fields. The volume-of-fluid method is then used for interface tracking. Separate effects of the Bingham number, the aspect ratio, the perturbation parameters (amplitude/wavelength), and the inlet velocity are examined on the steady finger width and the morphology of the fingers (i.e., tip-splitting and/or side-branching). It is shown that the shape of the fingers is dramatically affected by the fluid’s yield stress. It is also shown that a partial slip has a stabilizing effect on the viscous fingering phenomenon for yield-stress fluids.

Nonlocal dynamic response of embedded single-layered graphene sheet via analytical approach

Abstract: In the present research, static bending and dynamic responses of simply supported single-layered graphene sheet (SLGS) embedded in an elastic medium under uniform and sinusoidal loads are analytically investigated. The surrounding medium is simulated by visco-Pasternak model in which the damping and shearing effects are considered. Third-order shear deformation theory (TDST) is utilized because of its more accuracy relative to other plate theories. In order to consider size effects, nonlocal elasticity theory is employed. Applying Hamilton’s principle, governing equations of the SLGS are obtained and solved using Fourier series–Laplace transforms method. Finally, the detailed parametric study is conducted to scrutinize the influences of small-scale parameter, elastic medium, length-to-thickness ratio and aspect ratio of nanoplate on the static and dynamic behaviours of SLGS. Results indicated that the surrounding medium has a significant effect on the static and dynamic response, so that, increasing shear constant and damping coefficients cause to decrease the deflection of SLGS, considerably. The result of this study can be useful to control and improve the performance of this kind of nano-mechanical systems.

Determination of forcing functions in the wave equation. Part II: the time-dependent case

Abstract: An unknown time-dependent force function in the wave equation is investigated in this study. This is a natural continuation of Part I [J Eng Math 2015, this volume], where the space-dependent force identification has been considered. Additional data are given by a space integral average measurement of the displacement. This linear inverse problem has a unique solution, but it is still ill-posed since small errors in the input data cause large errors in the output solution. Consequently, when the input data are contaminated with noise, we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.

Oblique wave interaction with porous, flexible barriers in a two-layer fluid

Abstract: Oblique wave scattering by multiple porous, flexible barriers is studied in a two-layer fluid in both the cases of surface-piercing and bottom-standing partial barriers in water of finite depth. The mathematical problem is handled for a solution using a generalized orthogonal relation suitable for a two-layer fluid along with the least square approximation method for single and double barriers. Various characteristics of the eigensystem, including convergence of the eigenfunction expansions for the velocity potentials associated with surface gravity waves in two-layer fluid, are derived. Wave scattering by multiple barriers is studied using a wide-spacing approximation method and compared with the solution obtained through the least square approximation method in the case of double barriers. The effectiveness of the barrier system on wave scattering is analyzed in different cases by analyzing the scattering coefficients in surface and internal modes, surface and interface wave elevations, deflection of the flexible barriers under wave action, and wave loads on the barriers. It is observed that multiple zeros in wave reflections occur for waves in surface and internal modes for various values of nondimensional barrier spacing and an oblique angle of incidence. Further, the condition for Bragg resonance is derived in the case of multiple barriers in a two-layer fluid. In the case of wave scattering by double barriers, for certain combinations of barrier spacing and porous-effect parameter, optimum wave forces are exerted for the same angle of incidence. The findings of the present study are likely to play a significant role in the protection of marine facilities from wave action.

Effect of anisotropic permeability on fluid flow through composite porous channel

Abstract: This paper presents a theoretical model for a fully developed flow through a composite porous channel. The channel consists of a fluid layer sandwiched between two porous layers. A generalized Brinkman-extended Darcy model for the porous layers and Navier–Stokes equations for the fluid layer are used to investigate the flow in detail. The continuity of stress and velocity are used at the interface and no slip at the impermeable walls. We assume that the porous layers are anisotropic. Accordingly, $$\varphi $$ is taken as the angle between the horizontal direction and the principal axes with permeability $$K_{2}$$ or $$K_{4}$$ for the lower or upper porous layers considering the anisotropic nature of the porous medium. It is shown that the anisotropic permeability and orientation angle $$\varphi $$ have a strong effect on the fluid flow and skin friction. We present some important findings based on the response to variations in the anisotropy.

Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs

Abstract: A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and nonintrusive methods based on generalized polynomial chaos (PC). These methods allow us to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution at different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. Finally, we present a synthetic experiment studying the variance-based sensitivity of the wave load on an offshore structure to a number of input uncertainties. In the numerical examples presented the PC methods exhibit fast convergence, suggesting that the problem is amenable to analysis using such methods.

Dynamics of a high speed coned thrust bearing with a Navier slip boundary condition

Abstract: A modified Reynolds equation for flow dynamically represented as incompressible is used to model the dynamics of a thin film bearing with slip flow and a rapidly rotating coned rotor Previous studies including a Navier slip length shear condition on the bearing faces are extended to investigate applications with a coned bearing gap A modified Reynolds equation for the film flow is coupled, through the pressure exerted by the fluid film, to the dynamic motion of the stator Introducing a new variable leads to explicit analytical expressions for the pressure field and force on the stator with the equation for the time-dependent face clearance transformed to a nonlinear second-order non-autonomous ordinary differential equation The face clearance for periodic axial motion of the coned rotor is obtained using a stroboscopic map solver; a focus is investigating bearing behaviour under extreme conditions The coupled fluid flow and unsteady bearing dynamics are examined for a range of configurations to evaluate potential face contact over a range of bearing surface conditions

Shear horizontal surface acoustic waves in functionally graded magneto-electro-elastic half-space

Abstract: The propagation of shear horizontal surface acoustic waves (SHSAWs) in an inhomogeneous magneto-electro-elastic (MEE) half-space with 6-mm symmetry is studied. By virtue of both the direct approach and Stroh-formalism, the dispersion relations corresponding to two general cases of material properties variation are obtained. In the first case, it is assumed that all material properties involving the MEE properties and density vary similarly in depth, whereas, the second case considers identical variation for the MEE properties, which differs from the variation of the density. The non-dispersive SHSAW velocities pertinent to the homogeneous MEE media are obtained under eight different surface electromagnetic boundary conditions as the limiting cases of the current study. The dispersion curves corresponding to eleven inhomogeneity profiles of practical importance are presented in an effective dimensionless format, and the effects of different types of inhomogeneity functions describing the composition of the functionally graded magneto-electro-elastic (FGMEE) half-space on the dispersion relation are discussed.

Transfers between libration point orbits of Sun–Earth and Earth–Moon systems by using invariant manifolds

Abstract: Single-impulse and low-thrust low-energy transfers from a Lyapunov orbit around $$L_2$$ point of the Sun–Earth system to the periodic orbits around the equilibrium points $$L_i\; (i=3,4,5)$$ of the Earth–Moon system are investigated In our computation, the series expansions of invariant manifolds are used to generate the departure state, the series expansions of periodic orbits around the triangular libration points, and Lissajous orbits around the collinear libration points are used to generate the target orbits In order to construct the first guesses for single-impulse and low-thrust low-energy transfers, we establish the corresponding optimization problems under some suitable assumptions and solve them by means of an improved cooperative evolutionary algorithm (ICEA) For computing optimal transfers, the nonlinear programming problems are established by direct description and solved by a local optimization method with the initial-guess transfers computed by ICEA Numerical results show that the low-thrust transfers outperform the corresponding single-impulse transfers in terms of propellant mass

Cavity formation on the surface of a body entering water with deceleration

Abstract: The two-dimensional water entry of a rigid symmetric body with account for cavity formation on the body surface is studied. Initially the liquid is at rest and occupies the lower half plane. The rigid symmetric body touches the liquid free surface at a single point and then starts suddenly to penetrate the liquid vertically with a time-varying speed. We study the effect of the body deceleration on the pressure distribution in the flow region. It is shown that, in addition to the high pressures expected from the theory of impact, the pressure on the body surface can later decrease to sub-atmospheric levels. The creation of a cavity due to such low pressures is considered. The cavity starts at the lowest point of the body and spreads along the body surface forming a thin space between a new free surface and the body. Within the linearised hydrodynamic problem, the positions of the two turnover points at the periphery of the wetted area are determined by Wagner’s condition. The ends of the cavity’s free surface are modelled by the Brillouin–Villat condition. The pressure in the cavity is assumed to be a prescribed constant, which is a parameter of the model. The hydrodynamic problem is reduced to a system of integral and differential equations with respect to several functions of time. Results are presented for constant deceleration of two body shapes: a parabola and a wedge. The general formulation made also embraces conditions where the body is free to decelerate under the total fluid force. Contrasts are drawn between results from the present model and a simpler model in which the cavity formation is suppressed. It is shown that the expansion of the cavity can be significantly slower than the expansion of the corresponding zone of sub-atmospheric pressure in the simpler model. For forced motion and cavity pressure close to atmospheric, the cavity grows until almost complete detachment of the fluid from the body. In the problem of free motion of the body, cavitation with vapour pressure in the cavity is achievable only for extremely large impact velocities.

A mollification regularization method for identifying the time-dependent heat source problem

Abstract: We study in this paper the problem of determining time-dependent source functions in a parabolic equation with data given at some fixed locations in the domain. To solve this ill-posed inverse problem, we develop a mollification regularization method with a Gaussian kernel. We derive an a priori error estimate between the exact solution and its regularized approximation. Moreover, we propose an a posteriori parameter choice strategy for the selection of the regularization strength and derive an error estimate associated with the strategy. Numerical results are presented to illustrate the accuracy and efficiency of our method.

Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator

Abstract: We investigate the connection between the linear harmonic oscillator equation and some classes of second-order nonlinear ordinary differential equations of Lienard and generalized Lienard type, which physically describe important oscillator systems. By means of a method inspired by quantum mechanics, and which consists of the deformation of the phase space coordinates of the harmonic oscillator, we generalize the equation of motion of the classical linear harmonic oscillator to several classes of strongly nonlinear differential equations. The first integrals, and a number of exact solutions of the corresponding equations, are explicitly obtained. The devised method can be further generalized to derive explicit general solutions of nonlinear second-order differential equations unrelated to the harmonic oscillator. Applications of the obtained results for the study of the traveling wave solutions of the reaction–convection–diffusion equations, and of the large amplitude-free vibrations of a uniform cantilever beam, are also presented.

Love wave propagation in a fiber-reinforced medium sandwiched between an isotropic layer and gravitating half-space

Abstract: This paper is devoted to study of the dispersion equation of Love waves in a fiber-reinforced medium sandwiched between an isotropic layer and elastic half-space under the influence of gravity. The equations of motion have been discussed in each media. The frequency equation of a Love wave was obtained using the separation of variables method and Whittaker’s function expansion under a suitable assumption. The boundary conditions were introduced at interfaces of the upper layer, intermediate medium, and half-space. The dispersion equation was derived in closed form by means of Biot’s gravity parameter. The particular cases have been derived in the absence of reinforcement and gravitational force of the reinforced layer and half-space, respectively. Numerical solutions were discussed graphically to show the nature of wave propagation. Dimensionless phase velocity was obtained against non-dimensional wave number for different values of reinforced parameters, Biot’s gravity parameter, and thickness ratio of the upper layer and intermediate layer.

From individual behaviour to an evaluation of the collective evolution of crowds along footbridges

Abstract: This paper proposes a crowd dynamic macroscopic model grounded on microscopic phenomenological observations which are upscaled by means of a formal mathematical procedure. The actual applicability of the model to real-world problems is tested by considering the pedestrian traffic along footbridges, of interest for Structural and Transportation Engineering. The genuinely macroscopic quantitative description of the crowd flow directly matches the engineering need of bulk results. However, three issues beyond the sole modelling are of primary importance: the pedestrian inflow conditions, the numerical approximation of the equations for non trivial footbridge geometries and the calibration of the free parameters of the model on the basis of in situ measurements currently available. These issues are discussed, and a solution strategy is proposed.

Nonlinear dynamic responses of viscoelastic fiber–metal-laminated beams under the thermal shock

Abstract: The present study is concerned with the nonlinear dynamic responses for the viscoelastic fiber–metal-laminated beams subjected to thermal shock First, the one-dimensional heat conduction equation with variable coefficients in the direction of thickness is established, and this equation is solved by differential quadrature method (DQM) and the fourth-order Runge–Kutta method An effective numerical approach is presented to solve this kind of problem The fiber layer is considered to be the standard linear material Based on von Karman geometric nonlinear theory and Timoshenko beam hypothesis, using Hamilton’s principle, the governing equations of dynamic for the fiber–metal-laminated beam under thermal shock are derived The dynamic equations in terms of the displacements are discretized in spatial domain by adopting DQM and discretized in time domain by Newmark method synthetically Then the Newton iteration method is used to solve the nonlinear algebraic equations at every grid of the time domain Eventually, the temperature field in the beam and the dynamic displacement fields, and the stress responses of the beam are obtained In numerical examples, the influences of temperature, geometric nonlinearity, and material parameters on the dynamic responses of the beam are discussed

Ordinary differential equation algorithms for a frequency-domain water wave Green’s function

Abstract: Our study investigated an algorithm for a second-order ordinary differential equation for the frequency-domain Green’s function of free-surface waves in water of infinite depth. A power-series method is introduced if the wave frequency $$\omega <1$$ ; otherwise, if $$\omega \ge 1$$ , then the trigonometrically fitted block Numerov-type method (TBNM) is employed. The calculation precision of the power-series method and the TBNM reached $$10^{-7}$$ and $$10^{-6}$$ , respectively. The two methods have a high calculation efficiency compared with calculating the Green’s function using the series expansion representation approach. The calculation speed for these two methods is 15 times faster using the same computing codes.

Regularized model of post-touchdown configurations in electrostatic MEMS: bistability analysis

Abstract: This paper considers the problem of stiction in electrostatic–elastic deflections whereby elastic surfaces adhere to one another after coming into physical contact under attracting Coulomb interactions. This phenomenon is studied in a family of recently derived models which account for forces which become important when the elastic surfaces are in close proximity. The presence of bistability in these models results in hysteresis, or nonreversibility, which accounts for the difficulty in achieving separation after an initial contact event. We use singular perturbation techniques to derive explicit formula for the critical parameters over which bistability occurs and discuss new operational modes which arise when bistability is present.

Developing an Aw–Rascle model of traffic flow

Abstract: A traffic flow model is established based on the “car following” principle with a maximal constraint on the density–velocity relationship. The model develops the Aw–Rascle model and amends some “nonphysical” features. Moreover, we construct the solutions of the Riemann problem for the model. The Riemann solutions provide a more reasonable invariant region and show the phase-transition phenomena.

Derivation and numerical validation of a homogenized isothermal Li-ion battery model

Abstract: Lithium-ion batteries are multiscale systems with processes occurring at different scales. We start from a mathematical model derived on the microscale level of a single battery cell [Latz et al., NMA’10, 2011, pp. 329–337] where we can resolve the porous structure of the electrodes. Direct numerical simulations on this scale lead to a huge number of degrees of freedom and, consequently, very high computational costs. From an application perspective, it is often sufficient to predict the macroscopic properties of the electrodes. Therefore, we derive homogenized macroscopic equations with effective transport coefficients for the concentration of lithium ions in the electrolyte, potential in the electrolyte, and potential in the solid particles. These upscaled equations are coupled via the Butler–Volmer interface conditions to a microscale equation for the concentration of lithium ions in the electrode particles. We follow the idea developed by Ciucci and Lai [Transport Porous Med 88(2):249–270, 2011] and extend their analysis by computing the asymptotic order of the interface exchange current densities, which is an important factor in the homogenization study of the problem. We also perform a numerical homogenization and run extensive numerical simulations in order to validate the derived upscaled model. The numerical experiments show very good agreement between the homogenized model and the microscale one. Hence we are able to predict the macroscopic properties of the porous electrodes and capture the small-scale effects on the large scales without fully resolving all the microscale features.

Solution of Multi-Component, Two-Phase Riemann Problems with Constant Pressure Boundaries

Abstract: Riemann problems for two associated hyperbolic systems of conservation laws are considered. The Riemann problem for constant flow velocity finds existing solutions in the literature. Here, it is proved that the associated Riemann problem with the alternative assumption of constant pressure boundaries can be calculated from the constant velocity solution. This introduces the total velocity as an unknown function of time, which is explicitly determined in an algorithmic fashion.

Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse

Abstract: We consider computation of the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem. Using the normalization condition, we can compute the derivatives of the differentiable eigenvalues of the quadratic eigenvalue problem. Using the constrained generalized inverse, we present an efficient algorithm to compute the particular solutions to the governing equation of the derivatives of eigenvectors. The proposed method is suitable for the computation of the eigenpair derivatives of a symmetric quadratic eigenvalue problem when the first-order derivatives of eigenvalues are distinct or repeated. A numerical example is included to illustrate the validity of the proposed method.

A study on the onset of thermally modulated Darcy–Bénard convection

Abstract: A stability analysis of linearized Rayleigh–Benard convection in a densely packed porous layer was performed using a matrix differential operator theory. The boundary temperatures were assumed to vary periodically with time in a sinusoidal manner. The correction in the critical Darcy–Rayleigh number was computed and depicted graphically. It was shown that the phase difference between the boundary temperatures rather than the frequency of modulated temperatures decides the nature of influence of modulation on the onset of convection. Conclusions were drawn regarding the possible transitions from harmonic to subharmonic solutions. The results on the onset of thermally modulated convection in a rectangular porous enclosure were obtained using those on the modulated Darcy–Benard convection.

Mixed convection in a Falkner-Skan system

Abstract: The effects of mixed convection on the classical Falkner–Skan similarity solutions are considered, now involving a mixed convection parameter $$\lambda $$ as well as the exponent m associated with the outer flow. The forced convection solutions indicate a singularity in the temperature field as $$m \rightarrow 0.070722$$ . Numerical solutions for $$m>0$$ show the existence of a critical value $$\lambda _\mathrm{c}$$ with $$\lambda _\mathrm{c}<0$$ and solutions only for $$\lambda \ge \lambda _\mathrm{c}$$ . The nature of the solution for $$\lambda \gg 1$$ is investigated. For $$m=0$$ , there are solutions for all $$\lambda <0$$ , opposing flow, and only for a finite range of $$\lambda $$ in aiding flow with the asymptotic solution as $$\lambda \rightarrow -\infty $$ also being considered. Solutions for $$m<0$$ are obtained in the cases when there is a solution to the Falkner–Skan system and for a value of m when no solution to this system exists. In the former case, two completely separate parts to the solution are seen, whereas in the latter case, a solution exists only in aiding flow for a limited range of $$\lambda $$ . The variation of solution with the exponent m is also treated for both aiding and opposing flows. In both cases, a solution is seen to exist for all $$m>0$$ , which, however, is limited to a relatively small range of m when $$m<0$$ .

Dispersion-enhanced solute transport in a cell-seeded hollow fibre membrane bioreactor

Abstract: We present a matched asymptotic analysis of the fluid flow and solute transport in a small aspect ratio hollow fibre membrane bioreactor. A two-dimensional domain is assumed for simplicity, enabling greater understanding of the typical behaviours of the system in a setup which is analytically tractable. The model permits analysis related to Taylor dispersion problems, and allows us to predict the dependence of the mean solute uptake and solute exposure time on key parameters such as the inlet fluid fluxes, porous membrane porosity and cell layer porosity and width, which could be controlled or measured experimentally.

Analytical model for the temperature field around a nonuniform three-dimensional moving heat source: friction stir welding modelling

Abstract: A transient semi-analytical solution is devised to find the temperature distribution around a nonuniform three-dimensional heat source in a finite rectangular plate with cooling surfaces and nonhomogenous boundary conditions. This solution can serve as a viable tool to find the temperature field and consequently provides useful insights into material modelling and response in a variety of industrial applications, including friction welding processes. As a numerical example, the established solution is applied to the friction stir welding as a solid-state welding method. The comparison between the obtained analytical results and the results found through FEM and experiment showed that the analytically obtained results and the numerical and experimental data are in good agreement.

Effect of balance weight on dynamic characteristics of a rotating wind turbine blade

Abstract: Iron pellets are often added in blades to maintain moment balance in the design process of a wind turbine. The balance weight will change the natural vibration characteristics of the wind turbine blade. Resonant frequencies may appear for this reason, so it is necessary to study the effects of balance weight on the dynamic characteristics of a blade. In this paper, the wind turbine blade, after the weight balance process, is modeled as an Euler–Bernoulli beam with lumped masses. A mathematical model for a rotating nonuniform blade with a lumped mass on an arbitrary section is established, while nonlinear partial differential equations governing the coupled extensional–bending–bending vibration are obtained by applying the Hamiltonian principle. The associated modal problem is obtained from the governing equations, and then the differential form of the modal problem is transformed to integral form based on Green’s functions (structural influence functions). A direct numerical approach is applied to calculate natural frequencies and vibrating modes. The effects of the mass and position of the balance weight and the rotating speed on the natural frequencies and mode shapes of the blade are discussed.

Modelling suspended sediment in environmental turbulent fluids

Abstract: Modelling sediment transport in environmental turbulent fluids is a challenge. This article develops a model of the longitudinal transport of suspended sediment in environmental fluid flows such as floods and tsunamis. The model is systematically derived from a three-dimensional turbulence model based on the Smagorinski large eddy closure. Embedding the physical dynamics into a family of problems and analysing the linear dynamics of the system, the centre manifold theory indicates the existence of a slow manifold parameterised by macroscale variables. Computer algebra then constructs the slow manifold in terms of fluid depth, depth-averaged longitudinal velocities, and suspended sediment concentration. The model includes the effects of sediment erosion, advection and dispersion and the interactions between the sediment and turbulent fluid flow. Vertical distributions of the velocity and concentration in steady flow agree with established experimental data. For a pilot study, numerical simulations of the suspended sediment under long waves show that the developed model predicts physically reasonable sediment flow interaction.

Falling film on a flexible wall in the presence of insoluble surfactant

Abstract: This work investigates the effect of an insoluble surfactant on the gravity-driven flow of a liquid film down a vertical flexible wall. The paper builds upon previous work [Matar et al., Phys Rev E 76(5):056301, 2007; Sisoev et al., Chem Eng Sci 65(2):950–961, 2010] to include the Marangoni effect attributable to the gradient of surfactant concentration on a free surface. Here we employ an integral method to derive a set of asymptotic evolution equations valid for a moderate flow rate, based on a long-wave approximation. A normal-mode approach is used to examine the linear stability of the system. Similar to the work presented by Matar et al., the results show that a flexible wall with weak damping acts to stabilize flow, while wall tension plays an unstable role. The insoluble surfactant, which acts to stabilize film flow, can reduce the effects of wall flexibility (wall damping and tension) on flow linear stability. The nonlinear evolution equations for the system are solved numerically for both a given initial perturbation wave packet and a periodic perturbation at the inlet boundary. The equations are mainly concerned with the evolution of the flow stability and wave interaction processes, during which solitary-like waveforms are observed. When wall damping is weak, it tends to deplete the ripples preceding the solitary-like humps. However, as wall damping increases in strength, the ripples intensify; a similar phenomenon is observed with an increase in wall tension. The surfactant, which reduces the amplitude and traveling speed of the solitary-like waveforms, acts to distinctly weaken the dispersion of the interfacial wave.