A simple 1D model of inviscid fluid–solid interaction

ABSTRACTThe exact solutions for the Riemann problem concerning the one‐dimensional triple‐pressure Euler equations with the Coulomb‐type frictional term are displayed in perfectly explicit forms, where both the rarefaction and shock waves are presented in parabolic shapes with equal curvature under the action of the Coulomb‐type frictional term. Specifically, the curved delta shock wave is formed by sending the limit of Riemann solution comprised of double shock waves, and the vacuum state is also grown up by taking the limit of Riemann solution comprised of double rarefaction waves when all the three perturbation parameters are dropped to zero, where the remarkable concentration and cavitation phenomena can be closely observed and explored. Besides, the numerical simulations in correspondence are also offered to validate our results.

In this paper, the rate-type material model, whose solution consists of shock and rarefaction waves as derived in Radha et al. [Riemann problem for rate-type materials with nonconstant initial conditions. Math Method Appl Sci. 2021;44:13866–13880], is considered to discuss the shock wave interactions that occurred in the double Riemann problem with non-constant initial data. It is observed that the solution to the double Riemann problem is ultimately (after interaction of shock waves) given through either a 1-shock wave and a 2-rarefaction wave or 1-shock wave and 2-shock wave or 1-rarefaction wave and 2-shock wave subject to the initial data.

We present a numerical method, based on the Dafermos regularization,for computing a one-parameter family of Riemann solutions of a system ofconservation laws. The family is obtained by varying either the left orright state of the Riemann problem. The Riemann solutions are requiredto have shock waves that satisfy the viscous profile criterion prescribedby the physical model. The system is not required to satisfy strict hyperbolicity or genuine nonlinearity; the left and right states need not be close; and the Riemann solutions may contain an arbitrary number of waves, including composite waves and nonclassical shock waves. The method uses standard continuation softwareto solve a boundary-value problem in which the left and right states ofthe Riemann problem appear as parameters. Because the continuation methodcan proceed around limit point bifurcations, it can sucessfully computemultiple solutions of a particular Riemann problem, including ones thatcorrespond to unstable asymptotic states of the viscous conservation laws.

The Riemann problem for the Umami Chaplygin Aw-Rascle model of traffic flow with Coulomb-like friction is solved by introducing the reasonable flux-function. Three kinds of non-self-similar Riemann solutions involving delta-shock are given exactly by the means of variable substitution method. The serious traffic congestion may be explained by the delta-shock, for which, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Since the friction term is involved, all waves containing the rarefaction wave, contact discontinuity, shock wave, and delta-shock are bent into parabolic curves. By analyzing the limiting behavior of the solutions, it is discovered that as the Umami Chaplygin gas pressure and flux-function vanish simultaneously, the Riemann solution containing the rarefaction wave and the contact discontinuity tends to the vacuum solution of the zero-pressure Euler equations with friction, and the solution including the shock wave and the contact discontinuity tends to the delta-shock solution of the zero-pressure Euler equations with friction.

The Riemann problem for two-dimensional gas dynamics with isentropic or polytropic gas is considered. The initial data is constant in each quadrant and chosen so that only a rarefaction wave, shock wave, or slip line connects two neighboring constant initial states. With this restriction sixteen (respectively, fifteen) genuinely different wave combinations for isentropic (respectively, polytropic) gas exist. For each configuration the numerical solution is analyzed and illustrated by contour plots. Additionally, the required relations for the initial data and the symmetry properties of the solutions are given. The chosen calculations correspond closely to the cases studied by T. Zhang and Y. Zheng [SIAM J. Math. Anal., 21 (1990), pp. 593–630], so that the analytical theory can be directly compared to our numerical study.

The unsteady axisymmetric flow forming in the process of starting an impulse wind tunnel equipped with a throttled prechamber was numerically simulated. A regime with the operating flow Mach number M = 8 was under consideration. Numerical simulation on the basis of Reynolds-averaged Navier–Stokes code and the k-ω SST turbulence model was performed. The starting process was simulated as a sudden rupture of a diaphragm separating the prechamber volume with high pressure and temperature from low pressure part of the facility including a nozzle, an operating section and an exhaust vacuum tank.. The diaphragm rupture initiates a gasdynamic process associated with the Riemann problem when the primary shock wave arises which moves through the nozzle to the operating section. Computed data show another “inverse” shock wave emerging in the nozzle. A starting “shock-wave package” forms between these shock waves, it includes a flow portion of high temperature and pressure. The specificity is discussed of identified the characteristics and flow patterns forming during the process of moving the shock-wave package.

This book is devoted to the multiplicative differential calculus. Its seven pedagogically organized chapters summarize the most recent contributions in this area, concluding with a section of practical problems to be assigned or for self-study. Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. From 1967 till 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. It is also called an alternative or non-Newtonian calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics, finance, biology, and engineering. Multiplicative Differential Calculus is written to be of interest to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It is primarily a textbook at the senior undergraduate and beginning graduate level and may be used for a course on differential calculus. It is also for students studying engineering and science.   Authors Svetlin G. Georgiev is a mathematician who has worked in various areas of the study. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales. He is also the author of Dynamic Geometry of Time Scales (CRC Press). He is a co-author of Conformable Dynamic Equations on Time Scales, with Douglas R. Anderson (CRC Press). Khaled Zennir earned his PhD in mathematics from Sidi Bel Abbès University, Algeria. He earned his highest diploma in Habilitation in Mathematics from Constantine University, Algeria. He is currently Assistant Professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and long-time behavior. The authors have also published: Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDE; Boundary Value Problems on Time Scales, Volume 1 and Volume II, all with CRC Press.

Global analysis of SIS epidemic models with variable total population size

The nature of the shock wave, governed by Burgers equation, is presented from viewpoint of saddle-node bifurcations. First, the inviscid Burgers equation is studied in detail, and the solution of the system with a certain initial condition is obtained. The solution in vector form is reduced into a Map in order to investigate the stability and bifurcation in the system. It has been proved that there exists a thin spatial zone where a saddle-node bifurcation occurs as time goes on, and the velocity of the fluid behaves as jumping, namely, the characteristic of shock wave. Further, the period-doubling bifurcation is captured, that means there exist multiple states as time increases, and the complicated spatio-temporal pattern is formatted. In addition to above, the viscous Burgers equation is further studied to extend to dissipative systems. By traveling wave transformation, the governing equation is reduced into an ordinary differential equation. Then, the instability or bifurcation condition is obtained, and it is proved that there are three singular points in the system as the bifurcation condition is satisfied. More, the velocity distribution is obtained with a certain initial condition. The results show that the discontinuity resulting from saddle-node bifurcation is removed with the introduction of viscousity, and another kind of velocity change with strong gradient is obtained. However, the change of velocity is continuous with sharp slopes. As a conclusion, all results can provide a fundamental understanding of the nonlinear phenomena relevant to shock wave and other complicated nonlinear phenomena, from viewpoint of nonlinear dynamics.

SUMMARY The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how "numerical" basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

The objective of this study is to model the complex behavior of non-lane based heterogeneous traffic which is predominantly occupied by vehicles with varying physical and dynamical characteristics and their staggered car following behavior. This study proposes a new continuum model by considering the properties of two-sided lateral gap in a non-lane based heterogeneous traffic stream. The model is built on a strong empirical background and modified car following theory. In particular, the model consists of density dependent disturbance propagation speed, viscosity term and a frictional clearance term to describe the complex vehicular interactions in a traffic stream. Viscosity term in the model represents the driver’s reaction to a sudden change in density and further, it helps in smoothening out the shock fronts. Frictional clearance term in the model considers the effect of slow moving vehicles on the dynamics of fast moving vehicles. The results of mathematical and numerical analysis show that the proposed model satisfies all the qualitative and physical properties of the real world system. Moreover, two sided lateral gap in the model improves the stability region of traffic flow and increases the capacity and density of traffic stream. In addition, it is able to dissipate the traffic perturbation rapidly when compared to one-sided lateral gap model. It is anticipated that the new model provides the basis for evaluating alternative transport policy measures and helps in analyzing the system performance in the future.

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

SUMMARY We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time steps.

Generalized stokes’ flow and radiative heat transfer model of a non-Newtonian fluid in a darcy porous medium subject to Navier’s slip conditions on the penetrable porous boundary: Group theoretical and compatibility analysis

Analytic calculation of the energy characteristics of slip line fields in plane plastic deformation problems