Abstract: We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a temperature jump boundary. We also derive the solution of the unsteady Grad equations in one dimension with a time-dependent point heat source term and the Grad analogue of the rotlet, a well-known singularity of Stokes flow which corresponds to a point torque. PubDate: 2021-05-12

Abstract: An asymptotic approach of analysis is used to analyze the antisymmetric anti-plane shear dispersion of an elastic inhomogeneous five-layered plate in the presence of material contrasts. The resultant exact dispersion relation, overall cut-off frequency and the low-frequency range are determined. Two different asymptotic contrasting material setups for layered plates are employed with regards to the optimal shortened polynomial dispersion relation in the context of the structure under consideration. The asymptotic behaviors of the displacements and stresses in the respective layers of the plate are also examined. Finally, we provided some concluding remarks. PubDate: 2021-05-12

Abstract: Energy stability theory is applied to the study of the nonlinear stability of natural convection in an inclined fluid layer having a uniform internal heat source (sink), with the boundaries of the layer maintained at constant temperatures. The stability limit is found in terms of the thermal Rayleigh number \(R_{1}\) and the internal Rayleigh number \(R_{2}\) . The region of stability is found in \(R_{1}\) – \(R_{2}\) plane where the base state is stable against arbitrary perturbations. The Prandtl number Pr of the fluid and the angle of inclination of the fluid layer play an important role in determining the stability region. PubDate: 2021-04-22

Abstract: In this note we examine fluid violent kinematics in a plunging breaker. The fluid motion is computed in the frame of the potential flow theory. The fluid kinematics is basically generated by forcing the motion of a rectangular tank, starting from rest. When a sufficient level of energy is injected in the fluid, the free surface has highly nonlinear behavior, here a plunging breaker. In the vicinity of the main tip crest, a sharp corner (cusp) appears along the surface of the barrel. The appearance of this critical jet is described and discussed in terms of the spatial and temporal variations of the pressure field. In the present case a local pressure maximum is captured that follows a continuous decreasing pressure gradient in a region of positive Gaussian curvature of the pressure. It is also shown that, at the free surface before the appearance of the critical jet, there is a strong correlation between the change of sign of the Gaussian curvature of the pressure on the one hand and the radius of curvature of the free surface profile on the other hand. PubDate: 2021-04-22

Abstract: This article considers a thin-walled hollow cylinder, which is composed of a fibrous and swellable hyperelastic material. The fibers are arranged in two families and they are taken to be parallel within each fiber family. The two fiber families are also assumed to be mechanically equivalent and symmetrically disposed in the ground substance material. At each instant of the homogeneous swelling, the material is taken to be incompressible. This article studies the interplay of swelling, fiber orientation, and the mechanical properties of the constituents on the initiation as well as on the axial propagation of bulging. PubDate: 2021-04-22

Abstract: Two-dimensional boundary layer flows in quiet disturbance environments are known to become unstable to Tollmien–Schlichting waves. The experimental work of Liepmann et al. (J Fluid Mech 118:187–200, 1982), Liepmann and Nosenchuck (J Fluid Mech 118:201–204, 1982) showed how it is possible to control and reduce unstable Tollmien–Schlichting wave amplitudes using unsteady surface heating. We consider the problem of an oncoming planar compressible subsonic boundary layer flow with a three-dimensional vibrator mounted on a flat plate, and with surface heating present. It is shown using asymptotic methods based on triple-deck theory that it is possible to choose an unsteady surface heating distribution to cancel out the response due to the vibrator. An approximation based on the exact formula is used successfully in numerical computations to confirm the findings. The results presented here are a generalisation of the analogous results for the two-dimensional problem in Brennan et al. (J Fluid Mech 909:A16-1, 2020). PubDate: 2021-04-22

Abstract: The high-speed impact of a droplet onto a flexible substrate is a highly non-linear process of practical importance, which poses formidable modelling challenges in the context of fluid–structure interaction. We present two approaches aimed at investigating the canonical system of a droplet impacting onto a rigid plate supported by a spring and a dashpot: matched asymptotic expansions and direct numerical simulation (DNS). In the former, we derive a generalisation of inviscid Wagner theory to approximate the flow behaviour during the early stages of the impact. In the latter, we perform detailed DNS designed to validate the analytical framework, as well as provide insight into later times beyond the reach of the proposed analytical model. Drawing from both methods, we observe the strong influence that the mass of the plate, resistance of the dashpot, and stiffness of the spring have on the motion of the solid, which undergo forced damped oscillations. Furthermore, we examine how the plate motion affects the dynamics of the droplet, predominantly through altering its internal hydrodynamic pressure distribution. We build on the interplay between these techniques, demonstrating that a hybrid approach leads to improved model and computational development, as well as result interpretation, across multiple length and time scales. PubDate: 2021-04-22

Abstract: The study here is concerned with a thin solid body passing through a boundary layer or channel flow and interacting with the flow. Relevant new features from modelling, analysis and computation are presented along with comparisons. Three scenarios of such fluid-body interactive evolution in two-dimensional settings are considered in turn, namely a long body translating upstream or downstream, a long body with little or no translation and a short body with or without translation. The main progress and findings concern predictions of the time taken by the body to traverse the flow and impact upon the underlying wall, the delicate behaviour at the onset of impact, the dependence on parameters such as the initial conditions and the mass and shape of the body, and the influence of streamwise translation of the body in the surrounding fluid flow. PubDate: 2021-04-22

Abstract: Long wavelength instabilities of boundary layers caused by centrifugal effects or wall roughness are investigated. The wall roughness is modelled by small amplitude surface waviness. The instability is described in the nonparallel regime where it develops on the same length scale as the unperturbed flow. It is shown that instabilities initiated by disturbances close to the leading edge initially deform rapidly into algebraically growing eigensolutions but then deform into exponentially growing disturbances. The disturbances ultimately develop in a quasi-parallel manner and then pass successively through the high Görtler number, or equivalent large roughness parameter, regimes first described by Denier et al. (Nasa Contractor ICASE Report 90-31, 1990; Philos Trans R Soc A 334:51–85, 1991). It is shown that the mode which develops downstream is the most rapidly growing one available and not the second most unstable mode as claimed in a recent paper. PubDate: 2021-04-22

Abstract: Detailed fibre architecture plays a crucial role in myocardial mechanics both passively and actively. Strong interest has been attracted over decades in mathematical modelling of fibrous tissue (arterial wall, myocardium, etc.) by taking into account realistic fibre structures, i.e. from perfectly aligned one family of fibres, to two families of fibres, and to dispersed fibres described by probability distribution functions. It is widely accepted that the fibres, i.e. collage, cannot bear the load when compressed, thus it is necessary to exclude compressed fibres when computing the stress in fibrous tissue. In this study, we have focused on mathematical modelling of fibre dispersion in myocardial mechanics, and studied how different fibre dispersions affect cardiac pump function. The fibre dispersion in myocardium is characterized by a non-rotationally symmetric distribution using a \(\pi \) -periodic Von Mises distribution based on recent experimental studies. In order to exclude compressed fibres for passive response, we adopted the discrete fibre dispersion model for approximating a continuous fibre distribution with finite fibre bundles, and then the general structural tensor was employed for describing dispersed active tension. We first studied the numerical accuracy of the integration of fibre contributions using the discrete fibre dispersion approach, then compared different mechanical responses in a uniaxially stretched myocardial sample with varied fibre dispersions. We finally studied the cardiac pump functions from diastole to systole in two heart models, a rabbit bi-ventricle model and a human left ventricle model. Our results show that the discrete fibre model is preferred for excluding compressed fibres because of its high computational efficiency. Both the diastolic filling and the systolic contraction will be affected by dispersed fibres depending on the in-plane and out-of-plane dispersion degrees, especially in systolic contraction. The in-plane dispersion seems affecting myocardial mechanics more than the out-of-plane dispersion. Despite different effects in the rabbit and human models caused by the fibre dispersion, large differences in pump function exist when fibres are highly dispersed at in-plane and out-of-plane. Our results highlight the necessity of using dispersed fibre models when modelling myocardial mechanics, especially when fibres are largely dispersed under pathological conditions, such as fibrosis. PubDate: 2021-04-20

Abstract: A fiber-reinforced material comprised of a soft polymeric matrix reinforced with polymeric filaments is often modeled as an equivalent anisotropic nonlinearly elastic solid. Although the response of a single constituent polymeric material can be modeled by nonlinear thermo-elasticity over a large range of deformations and temperatures, there can be conditions requiring a theory that extends the range of application to account for other features, such as nonlinear viscoelasticity and an evolving microstructure due to a combination of mechanical and nonmechanical factors. In a multi-constituent fiber-reinforced material these effects can be expected to occur with different initial triggering and ongoing potency in the separate polymer matrix and fiber constituents. This paper summarizes a number of constitutive models for fiber-reinforced materials that include these features, discusses the connection of these models to a nonlinearly elastic scaffold, provides a framework for the incorporation of these features into the constitutive theory for an equivalent general simple solid, and shows how certain terms in the mathematical structure can be associated with the matrix constituent while other terms can associated with the fibrous constituent. PubDate: 2021-04-07

Abstract: Piezoelectric materials have a wide range of industrial applications in different branches of engineering due to their electromechanical coupling. So, investigating their responses to either mechanical or electric loadings helps engineers for efficient design of smart systems. However, most of the studies have assessed the well-known 6 mm piezoelectric materials or piezoceramics and few papers have studied other piezoelectric crystals despite of their application in industry. In this paper, fundamental solutions of a trigonal piezoelectric half-plane belonging to 3m crystal class is obtained. The governing differential equations are derived and solved analytically using potential method. It is shown that the solution for the 3m material can be degenerated to 6 mm solution as a special case. The contour lines were depicted for two practical piezoelectric materials belonging to 3m and 6 mm crystal classes including lithium niobate and PZT-4 and they were compared to each other. The numerical results showed that the response of the trigonal material is asymmetric due to anisotropy and the effect of anisotropy on some responses is considerable causing totally different behavior from 6 mm piezoelectric material. PubDate: 2021-03-26

Abstract: The flow of Bingham materials inside rectangular channels is characterized by a plug region located in the central region of the conduit and four dead zones in the vicinity of the corners of the geometry (i.e., the unyielded regions) that grow as the Bingham number is increased. In this paper, a semianalytical technique is presented for solving the flow of Bingham yield-stress materials inside straight rectangular ducts. To capture the topological shapes of the yielded and unyielded regions, a “squircle” mapping approach is proposed and the interfaces between the yielded and unyielded regions are captured based on minimizing the variational formulation for the velocity principle. The nonlinear governing equations are consequently solved using the Ritz method. To find the optimized solution, a genetic algorithm method is implemented, applying bounded constraints and a sufficient number of populations. The presented results are benchmarked against numerical works available in literature, revealing that the presented solution can accurately capture the positions of both the plug and dead regions. The critical Bingham number at which the plug region meets the dead regions is reported for different aspect ratios. The velocity profiles and the friction factor of flow with different Bingham numbers and aspect ratios are investigated in detail. Following this, in the limiting case in which the Bingham number is sufficiently small, an alternative, simpler analytical approach using elliptical mapping is also proposed. The authors believe that the proposed method could be employed as a useful tool to obtain fast, accurate approximate analytical solutions for similar classes of problem. PubDate: 2021-03-26

Abstract: The paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains. PubDate: 2021-03-26

Abstract: The present article discusses the solute transport process in steady laminar blood flow through a non-Darcy porous medium, as a model for drug movement in blood vessels containing deposits. The Darcy–Brinkman–Forchheimer drag force formulation is adopted to mimic a sparsely packed porous domain, and the vessel is approximated as an impermeable cylindrical conduit. The conservation equations are implemented in an axisymmetric system (R, Z) with suitable boundary conditions, assuming constant tortuosity and porosity of the medium. Newtonian flow is assumed, which is physically realistic for large vessels at high shear rates. The velocity field is expanded asymptotically, and the concentration field decomposed. Advection and dispersion coefficient expressions are rigorously derived. Extensive visualization of the influence of effective Péclet number, Forchheimer number, reaction parameter on velocity, asymptotic dispersion coefficient, mean concentration, and transverse concentration at different axial locations and times is provided. Increasing reaction parameter and Forchheimer number both decrease the dispersion coefficient, although the latter exhibits a linear decay. The maximum mean concentration is enhanced with greater Forchheimer numbers, although the centre of the solute cloud is displaced in the backward direction. Peak mean concentration is suppressed with the reaction parameter, although the centroid of the solute cloud remains unchanged. Peak mean concentration deteriorates over time since the dispersion process is largely controlled by diffusion at the large time, and therefore the breakthrough curve is more dispersed. A similar trend is computed with increasing Péclet number (large Péclet numbers imply diffusion-controlled transport). The computations provide some insight into a drug (pharmacological agents) reacting linearly with blood. PubDate: 2021-03-23

Abstract: This paper investigates the interaction of water waves with a group of submerged porous reef balls, which are hemispheres with centers lying on a plane seabed. An analytical solution based on potential theory is developed. In the solving process, the series solutions of the velocity potentials in the exterior and internal fluid domains of the reef balls are obtained by means of multipole expansions and separation of variables, respectively. The unknown expansion coefficients in the velocity potentials are determined by matching the porous boundary condition on the surface of each reef ball, where the vital point is the shift of multipoles among different local spherical coordinate systems. The special case of multiple impermeable reef balls is also considered. The wave forces in the sway, surge, and heave directions acting on the reef balls as well as the surface elevation near the structures are calculated. The predictions of the analytical solution are in excellent agreement with the numerical results of an independently developed three-dimensional multidomain boundary-element method solution. Case studies show that the hydrodynamic interaction is obvious only when reef balls are very close. Moreover, the feasibility of the submerged porous breakwaters composed of a series of porous reef balls with appropriate arrangements is examined. PubDate: 2021-03-23

Abstract: The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear. PubDate: 2021-03-23

Abstract: This paper presents, for the first time, an analytical formulation to determine the transient response of an elastic beam possessing distributed inertia and connected to a coupling inertial resonator, represented by a gyroscopic spinner. The latter couples the transverse displacement components of the beam in the two perpendicular directions, thus producing roto-flexural vibrations. A detailed parametric study is presented that illustrates the effects of the beam’s distributed inertia and of the resonator’s characteristics. The limit case of massless beam is examined and it is shown that in some situations the distributed inertia in the beam should not be neglected. Analytical results are also validated by finite element computations. An illustration is also presented that demonstrates the effectiveness of using the considered inertial devices to mitigate hazardous vibrations in structural systems. It is envisaged that this paper may be useful in the analysis of flexural waveguides and metamaterials consisting of inertial elastic beam elements. PubDate: 2021-03-20

Abstract: This paper presents a Green’s function method for the modal analysis of structures with interval parameters. By introducing first order approximations, the governing equations of modal displacements for interval analysis are decomposed into two sets of governing equations for the midpoints and deviations of modal responses, respectively. Utilizing the similarity between these equations and the governing equations for deterministic static analysis, Green’s functions for the linear static problem are directly used in the modal analysis process. Different from the traditional finite element method which needs the element refinement to improve solution accuracy, the proposed Green’s function method needs just one element for each beam member to achieve accurate results. Numerical examples presented demonstrate the applicability of the proposed method, and reveal some regular patterns of the effects of interval parameters in structural modal analysis. PubDate: 2021-03-20

Abstract: Eccentric annular regions formed by two impermeable cylinders are considered. Flow of Newtonian fluid in the annular region in the presence of heat transfer is studied. The eccentric annulus is mapped to concentric annular region through conformal mapping. Analytical solution for both temperature and velocity is obtained and graphically depicted. An increase in eccentricity results in an increase in velocity but a decrease in temperature. PubDate: 2021-03-20