System Of Quasilinear Hyperbolic Equations Research Articles

The generalized Chester-Whitham invariant

The problem of interaction of strong and weak discontinuities is solved in the general case for a system of quasilinear hyperbolic equations with two independent variables. It is proved that the product of the left eigenvector of the system by a derivative of the vector function in a strong discontinuity direction remains constant during the interaction. Examples of using this fact in solving the problems of gasdynamics are presented.

Thermomechanical coupled waves in a nonlinear medium

Several technological situations at moderate and high temperatures as well as many physical experiments at low temperatures show the necessity of taking into account the wave structure of heat transport. Motivated by experimental evidence a thermo-mechanical framework for a deformable heat-conducting continuum has been developed. In this framework, a history effect is present, it is introduced via a proportionality law between the heat flux vector and the gradient of a scalar thermal variable. The theory leads to a modified model of thermoelasticity with an extra thermal stress effect and wave-type heat conduction. The model is governed by a system of quasi-linear hyperbolic equations. All essential material functions are examined, and the impact of their nonlinearity on the solution to initial-boundary value problems is studied. A numerical scheme is developed to solve initial-boundary value problems on finite domains by a modification of Lax–Wendroff’s scheme. It turns out that the previously studied case of NaF crystals considered as rigid heat conductors was quantitatively quite acceptable: while it does not show any elastic waves, it still gives correct speeds and amplitudes of the pure thermal waves. However, this feature of upward compatibility of theories is lost for materials with higher thermal expansion coefficients. Hence, for new applications, it is of the essence to develop numerical methods for the fully coupled theory in 2D and 3D cases.

Non-adiabatic elongational flows of viscoelastic melts

Elongational flows of viscoelastic melts are frequently encountered in manufacturing processes in the textile industry. The most common stretching flow is melt-spinning. In this process, a polymeric melt is withdrawn from a reservoir, axially stretched, and simultaneously cooled down until the melt freezes.¶This paper addresses the fundamental question of existence of solutions for the system of quasilinear hyperbolic equations governing the melt-spinning process of a Giesekus fluid and a Phan-Thien--Tanner fluid. The problem is originally posed as a free boundary problem. It will be shown that the free boundary problem can be reformulated as an equivalent boundary-initial value problem for which we prove the (local in time) existence of classical solutions.

Dynamic analysis of orthotropic shells by the grid-characteristic method

The objective of this paper is to produce the dynamic analysis of orthotropic shells by a grid-characteristic method. The main advantages of composite structures are high specific strength, stiffness and anisotropic properties which can be utilised to optimise mechanical behaviour. At the same time the increase in stiffness and decrease in mass emphasise vibrating problems, thus the prediction of composite structure dynamic behaviour is a very topical subject. The analysis produced in this work is supported by the development of the explicit finite-difference method based on the utilisation of quasi-linear hyperbolic system of equations and combined with the special splitting procedure. The method has been implemented to the dynamic analysis of composite cylindrical shell. Numerical results of the dynamic response analysis demonstrated high efficiency and computational stability of the algorithm.

Hyperbolic framework for thermoelastic materials

A THERMODYNAMIC framework of a deformable continuum is developed in which the conservative state variable vector is enlarged by adding the spatial gradient of a scalar thermal internal state variable responsible for the description of thermal history effects. The theory leads to a modified model of thermoelasticity with internal state variables and with a wave-type heat conduction governed by a system of quasi-linear hyperbolic equations. In a general non-deformable case, the observed material properties such as specific heat, quasi-equilibrium thermal conductivity and speed of thermal (the so-called second sound) waves, all regarded as functions of , lead to a particular specification of all material functions and the evolution equation for the scalar internal state variable. The short review of the heat pulse experiment is made. Main assumptions of the present approach are formulated and some arguments referred to the rate-type description are presented. A set of remarks and comparisons with another modification of the Fourier law characterizes the model. An analysis of hyperbolicity of thermoelasticity by propagation conditions of weak discontinuity waves is performed.

Existence, uniqueness, and dependence on a parameter of solutions of differential-functional equations with ordinary and partial derivatives

For a system of quasilinear hyperbolic equations with a system of differential equations with lag, we prove theorems on the existence and uniqueness of a solution of the Cauchy problem and its continuous dependence on the initial conditions.

Weakly coupled systems of quasilinear hyperbolic equations

We obtain some general comparison results for entropy solutions of weakly coupled systems of conservation laws with source. For some particular couplings, by using these comparison methods, we show the global existence of the solutions to the Cauchy problem.

On a local in time solvability of the Neumann problem of quasilinear hyperbolic parabolic coupled systems

AbstractWe prove a local in time existence theorem of classical solutions to some coupled system of quasilinear hyperbolic equations and quasilinear parabolic equations with Neumann boundary condition. This coupled system contains a non‐linear thermoelastic equation as an important physical example.

Solutions Containing a Large Parameter of a Quasi-Linear Hyperbolic System of Equations and their Nonlinear Geometric Optics Approximation

It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data ${\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf {R}}^n}, \eta \ne 0$. Here ${r_1}(\eta )$ is a characteristic vector, and ${a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $n = 2$ or $3$ and that ${a_0}(\sigma ,x)$ have the vanishing mean with respect to $\sigma$, it is shown that a genuine solution exists in a time interval independent of $\lambda$, and that the formal solution is asymptotic to the genuine solution as $\lambda \to \infty$.

Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation

It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data λ − 1 a 0 ( λ x ∙ η , x ) r 1 ( η ) , λ > 0 , x , η ∈ R n , η ≠ 0 {\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf {R}}^n}, \eta \ne 0 . Here r 1 ( η ) {r_1}(\eta ) is a characteristic vector, and a 0 ( σ , x ) {a_0}(\sigma ,x) is a smooth scalar function of compact support. Under the additional requirements that n = 2 n = 2 or 3 3 and that a 0 ( σ , x ) {a_0}(\sigma ,x) have the vanishing mean with respect to σ \sigma , it is shown that a genuine solution exists in a time interval independent of λ \lambda , and that the formal solution is asymptotic to the genuine solution as λ → ∞ \lambda \to \infty .

One system of quasilinear hyperbolic equations: Uniqueness and global solvability of the Cauchy problem

In a coordinate patch, the harmonic maps from the two-dimensional Minkowski space into a complete Riemannian manifold satisfy a nonlinear system of hyperbolic partial differential equations. The Cauchy problem for such systems was studied by Gu Chao-Hao [ Comm. Pure Appl. Math. 33 (1980) , 727–737] and O. A. Ladyzhenskaya and V. I. Shubov [ Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 110 (1981) , 81–94 [in Russian]]. In this paper we investigate uniqueness and global existence of C 2 solutions of the Cauchy problem for one generalization of the previous hyperbolic system, when the map takes its values in R m endowed with the Riemannian metric.

Formation of shock discontinuities for a class of nonlinear transmission lines

We consider the problem of nonexistence of smooth globally defined solutions to a quasilinear hyperbolic system of equations; these equations are shown to model the behavior of a distributed parameter nonlinear transmission line, with voltage dependent capacitance and nonzero resistance and leakage conductance. Specific conditions are exhibited under which shocks form if the gradients of the initial data are, pointwise, sufficiently large. The analysis is based upon a study of the behavior of appropriate set of Riemann Invariants along their respective characteristic curves.

A global mathematical model of the cerebral circulation in man

A mathematical model of the cerebral circulation has been formulated. It was based on non-linear equations of pulsatile fluid flow in distensible conduits and applied to a network simulating the entire cerebral vasculature, from the carotid and vertebral arteries to the sinuses and the jugular veins. The quasilinear hyperbolic system of equations was numerically solved using the two-step Lax-Wendroff scheme. The model's results were in good agreement with pressure and flow data recorded in humans during rest. The model was also applied to the study of autoregulation during arterial hypotension. A close relationship between cerebral blood flow (CBF) and capillary pressure was obtained. At arterial pressure of 80 mmHg, the vasodilation of the pial arteries was unable to maintain CBF at its control value. At the lower limit of autoregulation (60 mm Hg), CBF was maintained with a 25% increase of zero transmural pressure diameter of nearly the whole arterial network.

Nonequilibrium gas-condensate flow in a laval nozzle

A scheme is proposed which simplifies the algorithm and reduces the labor involved in solving the system of quasi-linear hyperbolic equations describing supersonic nonequilibrium two-phase flow in an axisymmetric nozzle.

Global Solutions for Certain Systems of Quasi-Linear Hyperbolic Equations

Global Solutions for Certain Systems of Quasi-Linear Hyperbolic Equations