Time-independent boundary conditions

Equation (64) provides just one example of a phenomenon that may easily occur whenever species with different migration characteristics equilibrate sluggishly with each other and especially when they obey different boundary conditions. Namely, in uniform material subjected to time-independent boundary conditions, a steady state can be approached at long times that is not spatially uniform. We shall note a possible manifestation of such an effect in Section 5 of III. 

In the nonlinear regime and for time-independent boundary conditions we have... 

For time-independent boundary conditions, we have the general conditions for the stability of a state... 

Also, note that in Eq. [202] we have introduced a variable, I, which although inconsequential to the dynamics of the variables representing the fluid (i.e., p, q) is essential to obtain a conserved energy for SLLOD dynamics with time-independent boundary conditions. The conserved energy for the dynamics in Eqs. [202] is given by... 

A model can be constructed by maintaining the ends of the rod at fixed nonzero temperatures, that is, atx = 0, m = Tj, and at x = L, u = T2. This model is the so-called fixed nonhomogeneous, time-independent boundary conditions case. However, the technique of separation of variables requires... 

The Glansdorff-Prigogine general evolution criterion, in case of time independent boundary conditions, which are valid during the evolution of the thermodynamic steady state, can be formulated as... 

Because Eq. 4.2 has one time and two spatial derivatives, its solution requires three independent conditions an initial condition and two independent boundary conditions. Boundary conditions typically may look like... 

Class 4 Bar with fixed nonhomogeneous time-independent boundaries So far we have dealt with models that are strictly homogeneous. However, there are cases of nonhomogeneous problems that can be adjusted in such a way that the method of separation of variables can still be applied. One such case is that in which the boundary conditions are nonzero constants (fixed nonhomogeneous). This case is illustrated below. 

For the particular case that A% is time-independent, we could obtain the boundary conditions that 6 is to satisfy by inserting into the defining relation, Eq. (10-241), a complete set of states that are eigenstates of H. Thus, when x0 > x 0, we can write... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

This is a second-order ODE with independent variable z and dependent variable k C t,z), which is a function of z and of the transform parameter k. The term C(t, 0) is the initial condition and is zero for an initially relaxed system. There are two spatial boundary conditions. These are the Danckwerts conditions of Section 9.3.1. The form appropriate to the inlet of an unsteady system is a generalization of Equation (9.16) to include time dependency ... 

Solve the time-independent Schrodinger equation for this particle to obtain the energy levels and the normalized wave functions. (Note that the boundary conditions are different from those in Section 2.5.)... 

Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the reactor will be determined solely by its initial condition and its residence time, independently of what comes before or after. Practically, of course, some interaction will occur at the boundary between successive inputs of different compositions or temperatures. This is governed by diffusional behaviors which are beyond the scope of the present work. 

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