The Dynamic Boundary Condition

The third type of boundary condition at the surface S involving the bulk-phase velocities is known as the dynamic condition. It specifies a relationship between the tangential components of velocity, [u - (u n)n] and [u - (u n)n]. However, unlike the kinematic and thermal boundary conditions, there is no fundamental macroscopic principle on which to base this relationship. The most common assumption is that the tangential velocities are continuous across S, i.e., 

This is the most common form of the no-slip condition. 

It is generally accepted, based on empirical evidence, that the no-slip condition applies under almost all circumstances for small-molecule (Newtonian) fluids at either solid surfaces or at a fluid-fluid interface and also applies under many circumstances for complex liquids, such as polymer solutions or melts. This assertion is based primarily on comparisons of predictions from solutions of the equations of motion, which incorporate the no-slip condition, and experimental data - we shall discuss one example of a problem for which this kind of comparison has been done in the next chapter. Here, we simply note that these comparisons with experiments are often between macroscopic quantities - such as overall 

Here n is the unit normal to the boundary, u and T are the (continuum) velocity and stress, and P is an empirical parameter known as the slip coefficient. The Navier-slip condition says, simply, that there is a degree of slip at a solid boundary that depends on the magnitude of the tangential stress. We note, however, that it is generally accepted that the slip coefficient is usually very small, and then the no-slip condition (2 123) appears as an excellent approximation to (2-124) for all except regions of very high tangential stress. 

For a Newtonian fluid, an equivalent statement of the Navier-slip condition is 

---

In summary, we have so far seen that there are two types of boundary conditions that apply at any solid surface or fluid interface the kinematic condition, (2-117), deriving from mass conservation and the dynamic boundary condition, normally in the form of (2-122), but sometimes also in the form of a Navier-slip condition, (2-124) or (2-125). When the boundary surface is a solid wall, then u is known and the conditions (2-117) and (2-122) provide a sufficient number of boundary conditions, along with conditions at other boundaries, to completely determine a solution to the equations of motion and continuity when the fluid can be treated as Newtonian. 

The excess pressure is then found from the dynamic boundary condition Eq. (10.4.26). Using the unsteady Bernoulli equation + d(f)ldt - 0, we can eliminate the unknown constant A to give the eigenvalue relation... 

The kinematics and dynamics boundary conditions at the interfaces close the hydrodynamic problem (l)-(2). On the solid-liquid boundary the non-slip boundary conditions are applied -the liquid velocity close to the particle boundary is equal to the velocity of particle motion. In the case of pure liquid phases the non-slip boundary condition is replaced by the dynamic boundary condition. The tangential hydrodynamic forces of the contiguous bulk phases, nx(P+Pb) n, are equal from both sides of the interface, where n is the unit normal of the mathematical dividing surface. The capillary pressure compensates the difference between the... 

The covariance of the interface can be related to the other quantities in the mixture by using the dynamic boundary condition on the interface. This condition gives for the Fourier-Stieltjes component of the fluctuation in the pressure of the mixture,... 

The dynamic boundary condition that governs the forces balance at the interface, is ... 

One can use the dynamic boundary conditions of Eqs. (12)-(13) without considering the influence of the osmotic pressure and the Gibbs adsorption (the case of significant contribution of these values to the interfacial balance of forces is analyzed in Refs. [4,19]) ... 

Of course, the distinction between reactive- and bound-state wave functions becomes blurred when one considers very long-lived reactive resonances, of the sort considered in Section IV.B, which contain Feynman paths that loop many times around the CL Such a resonance, which will have a very narrow energy width, will behave almost like a bound-state wave function when mapped onto the double space, since e will be almost equal to Fo - The effect of the GP boundary condition would be therefore simply to shift the energies and permitted nodal structures of the resonances, as in a bound-state function. For short-lived resonances, however, Te and To will differ, since they will describe the different decay dynamics produced by the even and odd n Feynman paths separating them will therefore reveal how this dynamics is changed by the GP. The same is true for resonances which are long lived, but which are trapped in a region of space that does not encircle the Cl, so that the decay dynamics involves just a few Feynman loops around the CL... 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

This power law decay is captured in MPC dynamics simulations of the reacting system. The rate coefficient kf t) can be computed from — dnA t)/dt)/nA t), which can be determined directly from the simulation. Figure 18 plots kf t) versus t and confirms the power law decay arising from diffusive dynamics [17]. Comparison with the theoretical estimate shows that the diffusion equation approach with the radiation boundary condition provides a good approximation to the simulation results. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

Inner slip, between the solid wall and an adsorbed film, will also influence the surface-liquid boundary conditions and have important effects on stress propagation from the liquid to the solid substrate. Linked to this concept, especially on a biomolecular level, is the concept of stochastic coupling. At the molecular level, small fluctuations about the ensemble average could affect the interfacial dynamics and lead to large shifts in the detectable boundary condition. One of our main interests in this area is to study the relaxation time of interfacial bonds using slip models. Stochastic boundary conditions could also prove to be all but necessary in modeling the behavior and interactions of biomolecules at surfaces, especially with the proliferation of microfluidic chemical devices and the importance of studying small scales. 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

The uncorrelated particle distribution (4.1.12) is used, as standard initial conditions for the correlation dynamics. After the transient period the solution (for the stable regime) becomes independent on the initial conditions. For both the joint correlation functions boundary conditions at large distances X (oo, t) = Y(oo, t) = 1 has to be fulfilled due to the correlation weakening. The black sphere model imposes the additional boundary condition (5.1.39) for the correlation function Y(r,t). 

In general, the condition of systems H and N can be described by vectors xH t) = xlH,..., Xff and xN(t) = x, ..., x , respectively. The combined trajectory of these systems in n + m-dimensional space is described by the function rj t) = F(xh,xn) which is determined by solutions of the global model equations. The form of F is determined by knowledge of the laws of co-evolution, and therefore there is a possibility of investigations in different spheres of science. The available estimates of F (Krapivin, 1996) reveal a correlation between the notions survivability and sustainability. According to Ashby (1956), the dynamic system is alive within the time interval (ta, tb), if its determining phase coordinates are within admissible limits xlH>min N< x/N>max. And since systems H and N have a biological basis and limited resources, one of the indicated boundary conditions turns out to be unnecessary (i.e., for the components of vector... 

Consequently, this density operator must evolve with time. It is shown in Appendix I that, with the above boundary condition, the dynamics of this density operator is given by Eq. (1.15), that is,... 

A one-dimensional one-phase dispersion model subject to the Danckwerts boundary conditions has been used for a description of the dynamics of a nonisothermal nonadiabatic packed bed reactor. The dimensionless governing equations are ... 

Zaika Yu.V. (1996) The solvability of the equations for a model of gas transfer through membranes with dynamic boundary conditions, Computer Mathematics and Math. Physics 36(12), 1731-1741. 

Although each set of boundary conditions defines a unique trajectory, not all of the IF quantities at each of the end points can be controlled in an experiment (4,47,48) these quantities usually have random distributions (impact parameter, vibrational phase, etc.). Consequently, it is useful to choose the values of the uncontrollable boundary conditions randomly. For a sufficiently high number of the randomly chosen values (on a relevant interval), all boundary conditions are included and the resulting set of trajectories (related to an elementary process) represents a dynamical picture of the elementary process within the quasiclassical approach (6,44). 

The same Navier dynamic boundary condition Eq. (1) and the subsequent expression Eq. 3 for the extrapolation length b can also be written down for non-Newtonian and polymeric fluids, where r is the shear viscosity and 11 is the local viscosity at the interface. The expression Eq. (2b) for 3 is equally valid for poly-... 

The 7600 used is located at Lawrence Berkeley Laboratory, is approximately ten years old and has 65 K of 60 bit word fast memory (small core). Because CLAMPS has dynamic memory allocation, it is possible to fit a simulation in fast memory of up to about 2000 atoms as long as the potential tables are not too extensive. The compiler used was the standard CDC FTN 4.8, 0PT=2. The only difference between the CDC coding of the pairwise sum and that in Table I is that the periodic boundary conditions (loop 3) are handled by Boolean and shift operations instead of branches. Branches on the 7600 causes all parallel processing to halt. 

The dynamical trajectory generated by such a process can be visualized as a movie showing the polymer motion. Figure 1 presents a frame of an = 20, N = 5 system. The duplicated chains are a consequence of the periodic boundary conditions. When part of a chain sticks out of the fundamental box, its image enters the opposite face. Rather than showing pieces of chains we have drawn complete chains for both the original and image chain. 

---