Hyperbolic equation characteristics

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

This shows that the usual ideas associated with propagating waves in electromagnetism or fluid dynamics do not describe the behaviors found here. These differences could be expected because of the mathematical structure of reaction-diffusion equations, which owing to their parabolic character propagate information with infinite velocity. On the contrary, in the case of classical wave equations or hyperbolic equations there is a well-defined domain of influence and a characteristic velocity of propagation of information. ... 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

The above phenomena me physically miomalous and can be remedied through the introduction of a hyperbolic equation based on a relaxation model for heat conduction, which accounts for a finite thermal propagation speed. Recently, considerable interest has been generated toward the hyperbolic heat conduction (HHC) equation and its potential applications in engineering and technology. A comprehensive survey of the relevant literature is available in reference [6]. Some researchers dealt with wave characteristics and finite propagation speed in transient heat transfer conduction [3], [7], [8], [9] and [10]. Several analytical and numerical solutions of the HHC equation have been presented in the literature. 

The speed is the speed with which a point with ionic fraction x in solution moves. In general, Eq. (6.3.19) is a nonlinear hyperbolic equation possessing real characteristic solutions. Note that the characteristic direction is... 

The transient continuity equations and the combined momentum equation constitute a set of hyperbolic equations. The formulation is well-posed provided the equations possess real characteristics. The conditions of well-posedness of averaged two-fluid models were extensively discussed in the literature (e.g., Lyczkowski et al. [106], Ramshaw and Trapp [107], Banerjee and Chan [56], Drew [108], Jones and Prosperetti [109], Prosperetti and Jones [110], Moe [111]). The condition under which the characteristic roots of Equations 1, 2, 7 are real reads, (derived in 43 for C, = 0) ... 

High-order convection/advection schemes are widely used in meteorological applications solving hyperbolic equations. For example, in European weather forecast models the explicit non-flux-based modified methods of characteristics have been very popular as they are very fast. Typical examples of this type of schemes are the semi-Lagrangian advection schemes of Bates and McDonald [11], McDonald [149] and McDonald [150]. These methods have an unrestricted time step advantage, but... 

Streamline Methods For some years, originating with [69] there has been interest in applying the method of characteristics to the solution of the hyperbolic equation. Such methods do not possess a local mass conservation... 

Two real characteristic lines exist therefore, there exist distinct regions of space-time that are influenced, or not influenced, by each point R Examples of hyperbolic equations are, of course, the 1-D convection equation, and also the wave equation... 

The most effective techniques for hyperbolic partial differential equations are based on the method of characteristics [19] and an extensive treatment of this method may be found in the literature of compressible fluid flow and plasticity fields. 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

Briefly the idea behind this method is to delineate families of curves in the x-t plane, called characteristic curves, along which the partial differential equations [(123) and (128)] become a system of ordinary differential equations which could then be integrated with greater ease. However, only hyperbolic partial differential equations possess two families of characteristics curves required by the method. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

This is a first-order differential equation that has hyperbolic characteristics. 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

For steady-state analysis (i.e., no time variation) the coupled system is essentially elliptic, with some hyperbolic characteristics. The continuity equation alone is clearly hyperbolic, having only first-order derivatives. That is, it carries information about velocity from an inlet boundary, across a domain, to an outlet boundary. By itself, the continuity equation has no way to communicate information at the at the outlet boundary back into the domain. Based on the second-derivative viscous terms, the momentum equation is elliptic in velocity. However, because it is first order in pressure, there is also a hyperbolic character to the momentum equation. Moreover the convective terms have a hyperbolic character. There are situations, for example in high-speed flow, where the viscous terms diminish or even vanish in importance. As this happens, and the second-derivative terms become insignificant relative to the first-derivative terms, the systems changes characteristics to hyperbolic. 

In the liner approximation, we see thus that the NHIM is made of periodic/ quasi-periodic orbits, organized in the usual tori characteristic of the integrable systems. Because the NHIM is normally hyperbolic, each point of the sphere has stable/unstable manifolds attached to it. This situation is exactly parallel to the one described earlier for PODS. The equation for it is... 

By considering the combined variable z = x — xj2, we remove the mixed partial differential term from Eq. (4.293). The transformation obtained is the hyperbolic partial differential equation (4.294). This equation represents a new form of the stochastic model of the deep bed filtration and has the characteristic univocity conditions given by relations (4.295) and (4.296). The univocity conditions show that the suspension is only fed at times higher than zero. Indeed, here, we have a constant probability for the input of the microparticles ... 

The partial differential equations system for steady flows of Maxwell type (i.e., with = 0) is of composite type, neither elliptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics axe present ... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

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