Jump condition formulation

In the jump-condition formulation the physical problem is generally decomposed into k bulk phase domains where the continuity and momentum equations for isothermal incompressible flows holds, and at the interface between these domains boundary conditions are specified using the interface jump conditions. That is, across the interface some quantities are required to be continuous, while others are required to have specific jumps. The discontinuous (singular) momentum jump condition can be derived by use of the surface divergence theorem (see e.g., [63] p 51 [26]). A rigorous derivation of the jump balances for the multi-fluid model is given in sect 3.3. 

Free-surface flow is a limiting case of flow with interfaces, in which the treatment of one of the phases is simplified. For instance, for some cases of gas-liquid flow, we may consider the pressure pgas in the gas to depend only on time and not on space and the viscous stresses in the gas to be negligible. For such flows the jump condition formulation must be used, since the bulk momentum equation breaks down. The jump conditions become boundary conditions on the border of the liquid domain [245, 183[ ... 

The derivation of this relation can apparently be found in Chang et al [32], in which it was also shown (using rather complex mathematics) that this formulation admits solutions that are consistent with the boundary condition applied in the jump-condition formulation. 

The derivation of this relation can apparently be found in Chang et al. [35], in which it was also shown (using rather complex mathematics) that this formulation admits solutions that are consistent with the boundary condition applied in the jump-condition formulation.In a similar manner as for the VOF technique, the evolution of the interface is determined solving a transport equation for the level set function. That is, instead of solving Eq. (3.5), the following equation is solved [192] ... 

The formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been... 

To delineate more thoroughly the modifications that have been introduced, it seems desirable to discuss briefly the content chapter by chapter. The basic formulations in Chapter 1 and the derivation in Chapter 2 of jump conditions across combustion waves are largely unchanged from the first edition. Therefore, the following discussion focuses on the remaining chapters. 

At catalytic surfaces chemical reaction engineers often prefer to use a heat jump formulation which contains an explicit term for the heat generation due to the heterogeneous chemical reactions. The reformulation of the surface enthalpy term in the above enthalpy jump condition follows the same principles as explained in sect 1.2.4 deriving the single phase temperature equations from the enthalpy equation. 

Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems Local Instant Formulation. Int J Multiphase Flow 1 395-409... 

The accurate modeling of multiphase flows requires that the phase interaction terms given by the jump conditions and the turbulent like effects are correctly parameterized. This problem is very involved since not all of the phase interaction and phase change terms are independent. Most of the existing expressions are of empirical nature, thus experimental data are needed in order to develop and validate laws. Guiding principles for formulating closure laws are given by [31, 3, 35, 36, 37] ... 

Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems Local Instant Formulation. Int J Multiphase Flow 1 395-409 Delhaye JM, Achard JL (1977) On the averaging operators introduced in two-phase flow. In Banerjee S, Weaver JR (eds) Transient Two-phase Flow. Proc. CSNI Specialists Meeting, Toronto, 3.-4. august... 

It is quite clear that for mitochondria, where is localized within the vesicle, a jump-like pH increase can lead to only one elementary act, that indeed was observed in [182]. In this case the component of the ATP-synthase is faced inside the vesicle and, thus, according to Fig. 5.27, the state 5 state 1 transition becomes impossible. Only a single act of the AH ionization would take place after the pH jump due to a passive leakage of protons via the mitochondrial membrane. Otherwise, for submitochondrial particles turned inside-out [21] a multiple ATP synthesis after the pH increase is achieved by means of the reiteration of the proton cycle. According to the model considered above, the transmembrane electrochemical gradient of protons plays the role of an extensive but not intensive factor [190,191]. Its increase (or decrease) changes the number of elementary acts of ATP synthesis which can be performed, until the condition formulated above for the elementary act ceases to be fulfilled. 

In this section the basis elements of the volume of fluid (VOF) method are described. In general the VOF model is composed of a set of continuity and momentum equations, as well as a transport equation for the evolution of a phase indicator function which is used to determine the location and orientation of the interface. We distinguish between the jump condition—and the whole field formulations of the method, in which both forms are based on a macroscopic view defining the interface as a 2D surface. The jump condition form is especially convenient for free-surface flow simulations, whereas the whole field formulation is commonly used for interfacial flow calculations in which the internal flow of all the phases are of interest. 

Delhaye JM (1974) Jump conditions and entropy sources in two-phase systems local instant formulation. Int J Multiph Flow 1 395 09... 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

The proper dynamical formulation of a Levy flight on the semi-infinite interval with an absorbing boundary condition at x = 0, and thus the determination of the first passage time density, has to ensure that in terms of the random walk picture jumps across the sink are forbidden. This objective can be consistently achieved by setting/(x, t) = 0 on the left semi-axis, i.e., actually removing the particle when it crosses the point x = 0. This procedure formally corresponds to the modified dynamical equation... 

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