Finite Wave Speed Research Articles

Inferring the effect of interventions on COVID-19 transmission networks

Countries around the world implement nonpharmaceutical interventions (NPIs) to mitigate the spread of COVID-19. Design of efficient NPIs requires identification of the structure of the disease transmission network. We here identify the key parameters of the COVID-19 transmission network for time periods before, during, and after the application of strict NPIs for the first wave of COVID-19 infections in Germany combining Bayesian parameter inference with an agent-based epidemiological model. We assume a Watts–Strogatz small-world network which allows to distinguish contacts within clustered cliques and unclustered, random contacts in the population, which have been shown to be crucial in sustaining the epidemic. In contrast to other works, which use coarse-grained network structures from anonymized data, like cell phone data, we consider the contacts of individual agents explicitly. We show that NPIs drastically reduced random contacts in the transmission network, increased network clustering, and resulted in a previously unappreciated transition from an exponential to a constant regime of new cases. In this regime, the disease spreads like a wave with a finite wave speed that depends on the number of contacts in a nonlinear fashion, which we can predict by mean field theory.

Energy decay for damped Shear beam model and new facts related to the classical Timoshenko system

In this paper, we consider a beam model known as Shear beam model (no rotary inertia). We assure, based on behavior of the wave speeds, that the Shear beam model corresponds to a part of the classical Timoshenko beam model which is governed only by one wave speed. Unlike the classical dissipative Timoshenko type system with viscous damping acting on transverse displacement, we prove that the corresponding dissipative Shear beam model has an energy exponential decay regardless any relationship between coefficients of the system. This happens because such model has only one finite wave speed for all wave numbers.

Memory-dependent generalized thermoelasticity with finite wave speeds

The present article deals with the question: how time dependent non-locality influence the domain of dependency of thermal signals in solids? In order to establish the hypothesis “domain of influence” in an isotropic medium, based on the temperature-rate dependent thermoelasticity (TRDTE) theory, a thermodynamic consistent model is formulated by incorporating memory-dependent derivative (MDD). Firstly, the domain of influence theorem is established followed by a domain of dependence inequality and it certifies that for a finite time t>0 a solution of a given thermo-dynamical problem, corresponding to the data defined in a bounded support, vanishes outside a suitably defined bounded domain D*(t). It physically interprets that an initial perturbation of a bounded thermo-elastic domain generates thermal signals which for any t>0 can not occupy the whole space, i.e. it propagates with a finite speed. Finally, a two-dimensional thermoelastic problem in the isotropic medium is considered for a supporting visualization of the domain of influence theorem of the memory-dependent generalized thermoelasticity theory.

Transient response of the hygro-thermo-elastic field in hollow cylinders under heat-moisture shock via hyperbolic non-Fourier heat conduction and non-Fick diffusion

The classical Fourier heat conduction theory as well as the widely used Fick’s diffusion law predicts an infinite wave velocity. This is physically unrealistic. By generalizing the classical Fourier’s and Fick’s laws, this paper presents a hyperbolic diffusion law to apply heat and moisture coupling. The transient response of the hygro-thermo-elastic field in infinitely long hollow cylinders subjected to sudden heat and moisture shock on the inner and outer surfaces is studied. With the aid of the Laplace transform and the decoupling technique, the closed-form solutions of temperature, moisture, elastic displacement and stresses are determined respectively. The analytical results show that the thermal and moisture relaxation time effect between temperature and moisture is significant for composites. Compared with the classic counterpart, the finite hygrothermal wave speed of the pipe is achieved and decreases with the relaxation time rising. The temperature, moisture, elastic displacement and stresses are calculated. Numerical results are displayed graphically to show the influence of the phase lag of heat/moisture flux on the response of the hygro-thermo-elastic fields. Non-Fourier and non-Fick effects are remarkable between the classic model and hyperbolic hygrothermal coupling model. Some drawbacks induced by the classical Fourier’s and Fick’s laws are averted.

Three-Dimensional Non-Fourier Heat Transfer Analysis of Multilayer Functionally Graded Graphene Platelets Reinforced Composite Truncated Conical Shells

The three-dimensional transient heat transfer analysis of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) truncated conical shells subjected to asymmetric thermal shock is presented. In order to include the effect of finite heat wave speed, the non-Fourier heat conduction law is employed. The multilayer FG-GPLRC truncated conical shells are composed of perfectly bonded co-axial GPLRC shell layers with uniformly and randomly distributed graphene platelets. Since the weight fractions of the two adjacent layers should be different to create FG-GPLRC shells, the material properties have layerwise variations in the thickness direction. Therefore, a transformed differential quadrature method (TDQM) is used to discretize the governing equations in the spatial domain. Then, a multi-step time integration scheme based on the non-uniform rational B-spline (NURBS) is applied to obtain the temperature distribution in the temporal domain. The approach is verified by showing its fast rate of convergence and doing some comparison studies. Afterward, the effects of the different parameters on the temperature distribution of the FG-GPLRC truncated conical shells are explored and discussed. It is expected that the TDQM- and NURBS-based multi-step method to be used on different engineering problems in future investigations.

Axially-distributed bubble-bubble interaction under a coustic excitation in pipeline

Acoustically-excited bubble dynamics is the foundation of pipeline bubble detection based on acoustic technology. Due to the existence of multiple bubbles in pipeline flow, the Bjerknes forces among arbitrary bubbles under acoustic excitation may enforce bubble-bubble interaction and then change the features of bubble dynamics. Based on traditional free bubble’s Rayleigh-Plesset (R-P) model, this paper tries to establish bubble-bubble interaction model in consideration of the second Bjerknes force and bubble distribution in the pipeline axial direction. Meanwhile, the influence of finite wave speed in compressible fluid is considered. The proposed model is numerically calculated by the fourth-order Runge-Kutta method. Firstly, the differences in bubble feature between the free bubble’s R-P model and bubble-bubble interaction model are compared under excitation with different frequencies and amplitudes. Results show that the differences in bubble dynamics are minor when the bubble’s distance is large enough. When the bubble’s distance is fixed, the differences are significant on condition that the frequency of acoustic excitation is nearly the resonant frequency of bubbles. Secondly, through establishing compressible model and incompressible fluid model, we compare the differences between the two models. Numerical calculations show that the second Bjerknes force under the compressible assumption acts as an external force and forces the bubble to vibrate. On the other hand, the second Bjerknes force under the incompressible assumption changes the dynamics of bubble-bubble interaction as well as the resonant features. Finally, we study the effect of bubble-bubble distance and bubble’s axial position on bubble vibration characteristics. The bubble-bubble distance affects the second Bjerknes force and may lead the bubbles to vibrate nonlinearly. The bubble’s axial position changes the phase of external acoustic force and leads to the difference in initial vibration feature. When this difference is coupled with the second Bjerknes force, the bubble-bubble interaction may be changed even into nonlinear vibration, leading the bubble’s oscillation spectrum to differ from linear vibrations significantly. These results demonstrate that the resonant state of a small bubble may be converted into nonlinear vibration state if the second Bjerknes force is present. On the other hand, the resonant state of a large bubble can keep linear vibration when the second Bjerknes force is not obvious.

Dynamic model of fractional thermoelasticity due to ramp-type heating with two relaxation times

This is an attempt to design a fractional heat conduction model in a bounded cylindrical region exposed to axisymmetric ramp-type heat flux and discuss its thermal behaviour. This model is drafted using the classical theory of fractional thermoelasticity due to the involvement of two relaxation times to the heat conduction equation and the equations of motion. In order to achieve finite thermal wave speed, the heat conduction equation is derived from the viewpoint of Maxwell–Cattaneo law in the context of fractional derivative. Analytical results for the distribution of temperature, displacement and thermal stresses are obtained using integral transforms in the Laplace domain. The Gaver–Stehfest method has been used to invert the results of Laplace domain. The convergence of infinite series solutions has been discussed. As a specific case this model has been applied to a thick circular plate subjected to the axisymmetric ramp-type heat flux. The results obtained for the thermal variations are validated by comparing to coupled and generalized theory of thermoelasticity. These comparisons are shown graphically.

Supplementary remark to ‘Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds’ [Applied Mathematics and Computation 346 (2019) 531–544

The authors regrettably find that some related important information still should be included into Remark 2.4 in the published paper (Applied Mathematics and Computation 346 (2019) 531–544), so we amend it in this note.

An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by Sainte-Marie et al. in [1]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE.We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with a posteriori subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm.

Retrogressive failure of a static granular layer on an inclined plane

When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

Dependence of Rayleigh integral predictions on the ground motion model for acoustic signals produced by underground explosions

Seismoacoustic signatures are produced by above- and below-ground explosions and are often observed at local, regional, and global distances. In the case of an underground explosion, seismic waves propagate to the surface and produce acoustic signatures via pumping of the atmosphere by the ground motion that can be predicted using a Rayleigh integral analysis. Acoustic signature predictions will be discussed and compared with observations from the Source Physics Experiment (SPE) in two scenarios. First, a rigid piston model of the ground motion will be highlighted as a first order model. Second, a more realistic model treating the ground as an elastic medium with finite compressional wave speed will be developed and discussed to demonstrate how such a model changes predicted acoustic signals at local distances. Acoustic signals predicted using each methodology will be compared with observations from SPE to identify how characteristics of the acoustic signal can be leveraged to improve characterization of the underground explosive source.

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.

Hyperbolic heat conduction and thermoelastic solution of functionally graded CNT reinforced cylindrical panel subjected to heat pulse

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton Raphson method is used to solve the non-Fourier heat conduction equations to obtain temperature, displacements, and stresses in a cylindrical panel made of functionally graded, carbon nanotubes (CNTs) reinforced composite. The heat conduction of a domain made of nanocomposites subjected to heat generation is simulated with a finite heat wave speed. Furthermore, based on the three-dimensional elasticity theory, the thermoelastic analysis of the nanocomposite cylindrical panel subjected to the transient heat conduction is presented. The dynamic Young's modulus of Single-Walled Carbon Nanotubes (SWCNT) can be expressed as a function of loading rate and environmental temperature. All material properties such as heat capacity (Cp), thermal relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of both temperature and CNT volume fraction. The hyperbolic heat conduction is solved to obtain temperature in the spatial and temporal domains. Then by implementing the obtained temperature in thermoelastic equations of the cylindrical panel, the displacements and stresses will be obtained at each time step. The proposed method marches in the time direction block by block. In each block, there are several time levels, and the numerical results at these time levels are obtained simultaneously. Through this way, the numerical solution at the (n + 1)th time level depends on the solutions at its previous levels from the 1st to the nth levels. The final results in the temporal domain are obtained using the Newton–Raphson method. Accuracy of the present solution is confirmed by comparing with some available results in the literature. A detailed numerical study is conducted to examine the effects of heat wave speed and heat flux, CNT volume fraction and the geometrical parameters on the deflection of the cylindrical panel.

Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

Fractional derivative is a widely accepted theory to describe physical phenomena and processes with memory effect that is defined in the form of convolution with power kernel. Due to the shortcomings of power law distribution, some derivatives with other kernels are proposed, including Caputo–Fabrizio derivative, Atangana–Baleanu derivative and so on. In this paper, in order to provide some flexible and more appropriate tools which can better describe cases of the dynamics with memory effects or of nonlocal phenomena, we derive the definition of general fractional derivatives with memory effects named GC derivative and GRL derivative from some basic principles. We demonstrate that the mathematical expression of Gurtin–Pipkin theory of heat conduction with finite wave speeds is a special example of GC/GRL derivative.

Thermoelastic Analysis of Multilayered FG Spherical Shells Based on Lord–Shulman Theory

The thermoelastic wave in multilayered spherical shells with functionally graded (FG) layers under thermal boundary conditions is studied. The Lord–Shulman generalized coupled thermoelasticity theory is applied to illustrate the effect of finite heat wave speed. The material properties are assumed to be temperature dependent, and consequently, the governing equations become nonlinear ones. The layerwise-differential quadrature method together with Newmark time integration scheme and Newton–Raphson method are employed to solve the governing equations. The fast rate of convergence of the method is illustrated, and its accuracy is assessed by comparing the results with various existing solutions in the open literature wherever possible. Afterward, the effects of different parameters and also the temperature dependence of material properties on the transient thermoelastic responses of the FG spherical shells are studied and discussed. It is found that the temperature dependence of material properties, thermo-mechanical coupling, thickness-to-outer radius ratio and FG layer layout significantly affect the thermo-mechanical behavior of the FG shells.

Finite volume solution of 2-D hyperbolic conduction in a heterogeneous medium

ABSTRACTFinite volume method that has been shown to be an effective tool for the study of non-Fourier conduction in one-dimensional heterogeneous material is extended to two-dimensional (2D) heterogeneous materials. The accuracy of 2D numerical formulation is examined by comparing the numerical results with the exact solutions for a homogeneous material with constant thermal properties. The effects of temperature-dependent properties due to both heating and cooling of the medium on the wave propagation are investigated. Transient energy balance shows significant effects of finite thermal wave speed on the transient behavior of internal and wave energy changes in the system. The effects of finite thermal wave speed on the entropy production for the cooling case are examined by invoking the second law of thermodynamics. The 2D transient temperature distributions in a heterogeneous medium show the expected effects of flux lagging constant on the thermal wave propagations through heterogeneous material.

Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds

ABSTRACTThis article is aimed at determining the thermoelastic displacement, stress, and temperature in a thick circular plate of finite thickness and infinite extent whose lower and upper surfaces are traction free, subjected to a given axisymmetric temperature distribution. The problem is formulated in the context of fractional order thermoelasticity theory with finite wave speeds. Integral transform technique is used to obtain the general solution in Laplace transform domain. Inversion of the Laplace transforms is done using a numerical scheme. A mathematical model is prepared for a copper material plate. Thermoelastic stresses, temperature and displacement are shown graphically and the effects of fractional-order parameters are discussed.

Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell

This problem deals with the thermo-elastic interaction due to step input of temperature on the stress free boundaries of a homogeneous visco-elastic orthotropic spherical shell in the context of a new consideration of heat conduction with fractional order generalized thermoelasticity. Using the Laplace transformation, the fundamental equations have been expressed in the form of a vector-matrix differential equation which is then solved by eigen value approach and operator theory analysis. The inversion of the transformed solution is carried out by applying a method of Bellman et al (1966). Numerical estimates for thermophysical quantities are obtained for copper like material for weak, normal and strong conductivity and have been depicted graphically to estimate the effects of the fractional order parameter. Comparisons of the results for different theories (TEWED (GN-III), three-phase-lag model) have also been presented and the effect of viscosity is also shown. When the material is isotropic and outer radius of the hollow sphere tends to infinity, the corresponding results agree with that of existing literature.

Thermomechanical coupling and second sound in superfluid flows

The present paper is concerned with the modelling of the influence of the thermomechanical coupling on the propagation of temperature waves (second sound) in superfluids. For an adequate heat transfer analysis in superfluids, finite thermal wave speed must be considered. Besides hyperbolic heat transfer, turbulent flows are generally observed despite the lost of internal friction. Due to the thermo-mechanical coupling, density waves may induce moving heat sources or sinks while temperature waves propagate at a different and independent speed. In particular, a rotational flow in this kind of fluid can strongly affect heat propagation. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. Such a phenomenological continuum approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand its influence on the superfluid flow.

Viscothermoelasticity with finite wave speeds: thermomechanical laws

The development of constitutive equations of linear viscothermoelasticity, free of the paradox of infinitely fast propagation of heat, is in the focus of this work. Two models from hyperbolic thermoelasticity—theory with one relaxation time and theory with two relaxation times—provide the stepping-stone. The setting for these models is offered, respectively, by the primitive thermodynamics of Edelen and the thermodynamic orthogonality of Ziegler, in both cases involving internal parameters. The telegraph-like heat conduction is governed in the first case by the Maxwell–Cattaneo model and in the second by the Fourier law. In the first case, temperature and its first derivative appear in the thermomechanical constitutive relation, whereas in the second case also the second derivative is present.