Interface jump condition

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. 

During interphase mass transfer, concentration gradients will be set up across the interface. The concentration variations in the bulk phases x and y will be described by differential equations whereas at the interface /, we will have jump conditions or boundary conditions. Standart (1964) and Slattery (1981) give detailed discussions of these relations for the transport of mass, momentum, energy, and entropy. It will not be possible to give here the complete derivations and the reader is, therefore, referred to these sources. A masterly treatment of this subject is also available in the article by Truesdell and Toupin (1960), which must be compulsory reading for a serious researcher in transport phenomena. 

The interface boundary conditions, i.e., the momentum jump conditions, are expressed as (i.e., no surface tension gradients are considered) ... 

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[ ... 

Furthermore, adopting the macro scale viewpoint several alternative modeling approaches are proposed in the literature for deriving the jump conditions. Moreover, coinciding results are achieved from independent thermodynamical and mechanical derivations, since the surface tension is equivalently defined from energy and force considerations [150]. The relevant concepts can be briefly summarized as follows. In a series of papers a general balance principle in which the interface is represented by a 2D dividing surface of no thickness... 

The balance principle applied to mass, momentum and energy for an interface is expressed by the jump condition ... 

The local jump conditions are time averaged in a similar manner. To derive the averaged form of the generic condition we multiply (3.109) with jv nTJ take the sum over all the interface occurrences obtained within the entire averaging time interval e [t-T/2 t+T/2], and thereafter multiply the resulting relation with the fixed reciprocal averaging time period T. The averaged jump condition becomes ... 

Eliminating the unknown interface quantities using the equilibrium relation and the rates of movement of the two media relative to the interface, the component mass jump condition can be used to calculate the position of the interphase. 

Here, etj is the ij component of the rate-of-strain tensor. Because the leading-order approximation of the shape is a sphere, these conditions with /0 = 0 are just the exact interface boundary conditions applied at the surface of a spherical drop. The only possible confusion with these conditions is that all terms appear to be 0(1) except for the Ca l term in (7-210). It should be noted, however, that this is just the dimensionless form of the capillary pressure jump for a spherical drop, i.e.,... 

The principal intention of a consistent numerical boundary approach for the manipulation of demanding media interfaces is the satisfaction of the proper physical jump conditions (far field, impedance, or metal). Their role is deemed crucial, since they can guarantee the stability of the time-domain simulation and assure the desired order of convergence over the entire domain. Evidence of this is the evolution of various state-of-the-art algorithms, whose foremost properties and characteristics are described in the following paragraphs. 

Extension to the 2-D TM case opts for the vector form of 3Q/31= R 3Q/Bx + ZdQJdy with Q = HxHyEz T. If the material interface is again placed at i = L, constitutive parameter matrices R and Z receive the appropriate values at media A and B, as in the 1-D problem. Herein, derivative matching is performed for dEJdx and ()Hy/()x with the ensuing jump conditions across the interface... 

As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

The jump condition [V] = 0 means that V is continuous across the cavity interface E, i.e. — V(- = 0 on E, where the subscript e and i indicate positions in the external and internal space with respect to E, respectively. The second jump condition [diV] = 0 regards the gradient of the potential,... 

In equations 1.2a-1.3b, we have used c i to represent the bulk concentration of species A (moles per unit volume), and c to represent the surface concentration of species A (moles per unit area). The nomenclature for the homogeneous reaction rate, and heterogeneous reaction rate, Rj, follows the same pattern. The surface concentration is sometimes referred to as the adsorbed concentration or the surface excess concentration, and the derivation (Whitaker, 1992) of the jump condition essentially consists of a shell balance around the interfacial region. The jump condition can also be thought of as a surface transport equation (Slattery, 1990) and it forms the basis for various mass transfer boundary conditions that apply at a phase interface. 

There are many problems for which we wish to know the concentration, Cj, and the normal component of the molar flux of species A at a phase interface. The normal component of the molar flux at an interface will be related to the adsorption process and the heterogeneous reaction by means of the jump condition given by equation 1.36 and relations of the type given by equation 1.37, and this flux will be influenced by the convective, and diffusive, fluxes. 

Under the restrictions outlined above, upstream (1) and downstream (2) states are related by the jump conditions across an interface ... 

In the continuum model (Kn<0.001), the following assumptions can be made (1) a linear relation between stress and strain, (2) no-slip boundary condition at the fluid-soUd interface, (3) linear relation between heat flux and temperature and (4) no-jump condition of temperature at the fluid-solid interface. If the mean free path is not much smaller than the characteristic length, the flow is not near equilibrium and the above assumptions are no longer valid. 

Tangential stress condition. The stress at any point on the interface in a direction tangential to the interface jumps as we cross from one phase to the other by an amount equal to the force exerted by surface tension gradients (Marangoni forces). 

Matched asymptotic expansion techniques yield a jump condition on the normal derivative of p across the interface and a value for u - see [10] for details. 

The numerical methods employed here are based on the VOF method described in [14]. It solves the balance Eqs. (1.1) and (1.2) taking into account the interfacial jump conditions (1.11) and (1.12). The principle idea of the method is to capture the interface position implicitly by means of a phase indicator function/ =/(f, x) with / = 1 in the dispersed phase and/ = 0 in the continuous phase. Due to the absence of phase change, the transport of/is governed by the advection equation... 

The pressure distribution inside the collision complex at t = 0.2 ms is shown in Fig. 1.23. The maximum pressure appears inside the low viscous liquid next to the inner interface. Three positions exhibiting a pressure jump can be observed, two at the outer surface of the droplet and one at the inner interface, where surface tension is not present. The momentum jump condition at the interface is given by Eq. (1.12). Extracting the normal component of the jump condition in Eq. (1.12) yields... 

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. 

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