Well posed boundary condition

Unfortunately, for a given domain of interest boundary conditions can be chosen that over- or underspecify the problem. An example of an over-specified problem is a constant area duct, with a fixed fluid velocity at the inlet and but different at the outlet. Naturally, for an incompressible fluid both conditions cannot be physically satisfied in the absence of a mass source. In the same sense, a closed box with only heat flux boundary conditions is under specified since the temperature level is not constrained, and therefore unpredictable. It becomes clear that defining well posed boundary conditions is quite important for the proper solution of a flow problem. An easy way to check for well posed boundary conditions is to ask yourself Could the chosen configuration be physically recreated in the laboratory . 

Another characteristic of turbulent flows is unpredictability, that is the high sensitivity of the solution to very small perturbations that are always present in real physical systems or numerical simulations. This unpredictability, also known as dynamical chaos, is a well known feature of much simpler low-dimensional nonlinear dynamical systems. Although in a strict mathematical sense a unique solution of the Navier-Stokes equation always exists for well-posed initial conditions (at least for large finite times), in practice the details of the forcing and boundary conditions are only known within some approximations and thus the solution in the turbulent regime repre-... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

One of the basic elements of the computational algorithm is the determination of dependent variables at the inlet/outlet boundaries of a computational domain representing a finite length combustor. The essence of the problem lies in the fact that the nonstationary flow field has to be considered throughout a whole (unbounded) physical space, and only in this case the problem is mathematically well-posed. When solving a specific problem numerically, one has to consider a computational domain of a finite size, in which boundary conditions at artificial boundaries are to be imposed. 

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

It is well known that, for the Navier-Stokes equations, the prescription of the velocity field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoeleistic fluids have memory the flow inside the domain depends on the deformations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases,. ..). The number... 

The problem consists in finding the appropriate conditions to prescribe for the extrastress tensor at the inflow boundary x = 0), so that the steady problem is well-posed. The method of analysis is a variant of the algorithm leading to Theorem 3.1. 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

Like all other models, LES requires the specification of proper boundary and initial conditions in order to fully determine the system and obtain a mathematically well-posed problem. However, this concept deviates from the more familiar average models in that the boundary conditions apparently rep>-resent the whole fluid domain beyond the computational domain. Therefore, to specify the solution completely, these conditions must apply to all of the space-time modes it comprises. 

As a specific realization of these ideas, consider the celebrated problem of the one-dimensional particle in a box already introduced earlier in the chapter. Our ambition is to examine this problem from the perspective of the finite element machinery introduced above. The problem is posed as follows. A particle is confined to the region between 0 and a with the proviso that within the well the potential V (x) vanishes. In addition, we assert the boundary condition that the wave function vanishes at the boundaries (i.e. V (0) = rfr (a) = 0). In this case, the Schrodinger equation is... 

The condition (5-224) provides a third boundary condition through matching with the core solution for the radial velocity. Although (5-224)-(5-226) is a well-posed problem, we shall not solve it here. The solution for F0 was numerical, and thus the present problem must also be solved numerically. The most efficient approach is to solve the problems for F0 and F together as this avoids storing the solutions for F0 and/or interpolating to evaluate the coefficients in (5-225). 

Together with the governing equation and boundary condition, (9-129), the matching condition (9-157) yields a well-posed problem that could be solved, in principle, to obtain... 

The governing conservation equations developed here lead to a well-posed system of partial differential equations, on specification of the appropriate boundary conditions. It can be noted here that boundary conditions corresponding to DNA hybridization are already incorporated through specification of the source term / , (Eq. 21) for control volumes adjacent to the channel-fluid interface and need not be duplicated in prescription of boundary conditions. Other pertinent boundary conditions are summarized in T able 1. 

The remaining boundary conditions are the phase equilibrium relationships at the interface given by the applieable phase diagram. It may be convenient in calculations to describe the eoexistence curve and tie-hnes by empirical equations such as those proposed by Hand (1930). Whatever the details of the method used, the problem is well posed cmee initial compositions, diffusion coefficients, and equihbrimn data are speeified. The solution yields the concentration profiles in both phases. 

It is commonly believed that a correct mathematical presentation of physical situations ought to result in properly posed problems. In two-phase flow problems, however, the existence of an assumed physical situation, e.g., stratified wavy flow configuration, is not certain under all operational conditions. Therefore, ill-posedness in some domains of the parameters space does not necessarily imply that the formulation is globally incorrect. Moreover, the boundary of the well-posed domain may have physical significance since it signals the existence of additional physical features which the original model neglects. When these features become consequential, one expects a different physical behavior, such as transition to a different flow pattern, and a different model is required to simulate this transition. 

The well-posedness boundary (ZRC) (included in Figures 10, 11, 13) represents the limit of operational conditions (U, U, ) for which the governing set of continuity and momentum equations is still well-posed with respect to all wave modes. Hence, it is considered as an upper bound for the stratified-wavy flow pattern. Indeed, the data of stratified-wavy/annular transition follows the ZRC curve in the region of H < 0.5. 

Having introduced the principles of special relativity in classical mechanics and electrodynamics as well as the foundations of quantum theory, we now discuss their unification in the relativistic, quantum mechanical description of the motion of a free electron. One might start right away with an appropriate ansatz for the basic equation of motion with arbitrary parameters to be chosen to fulfill boundary conditions posed by special relativity, which would lead us to the Dirac equation in standard notation. However, we proceed stepwise and derive the Klein-Gordon equation first so that the subsequent steps leading to Dirac s equation for a freely moving electron can be better understood. 

A simple scheme. Now suppose that a numerical solution for the pressure field is available, for example, the finite difference solutions presented later in Chapter 7. The solution, for instance, may contain the effects of arbitrary aquifier and solid wall no-flow boundary conditions we also suppose that this pressure solution contains the effects of multiple production and injection wells. How do we pose the streamline tracing problem using T without dealing with multivalued functions The solution is obvious subtract out multivalued effects and treat the remaining single-valued formulation using standard methods. Let us assume that there exist N wells located at the coordinates (Xn,Yn), having... 

The simplest borehole flow invasion problem can be posed using the radial flow model of Chapter 6, and, in particular, applying pressure-pressure boundary conditions at the well and farfield boundaries. Because our mud is assumed to be lossy, we can ignore the presence of cake buildup many shallow wells are, in fact, circulated with water or brine as the drilling fluid. 

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