Inhomogeneous boundary conditions

Generally, the initial boundary-valne problem given by 

The solution of the transformed eqnation can then be easily obtained using DuhameTs principle with hix,t) = [a it) - b it)]x - o it). 

The methods presented here can be neatly integrated into the snbject of linear operators (Ramkrishna and Amnndson, 1985), extending considerably the variety of problems that can be solved. For the solntion nsing the method of Fonrier transforms, the reader is referred to Varma and MorbideUi (1997). 

In the previous section, we developed the finite integral transform for a general Sturm-Liouville system. Homogeneous boundary conditions were used in the analysis up to this point. Here, we would like to discuss cases where the boundary conditions are not homogeneous, and determine if complications arise which impede the inversion process. 

If the boundary conditions are not homogeneous, they can be rendered so by rearranging the dependent variable as discussed in Chapter 1. To show this, we use an example which follows the nomenclature of the previous section 

To make the boundary conditions for Y homogeneous, we could define the following auxiliary equation for u, which is simply the steady-state solution  

Having defined u as in Eq. 11.74, the governing equations for the new dependent variable Y become  

This new set of equations for Y now can be readily solved by either the method of separation of variables or the Sturm-Liouville integral transform method. We must also find u(x), but this is simply described by an elementary ODE (Lu = 0), so the Inhomogeneous boundary conditions (11.74) are not a serious impediment.  

---

Kornyshev AA, Leikin S. Huctuation theory of hydration forces the dramatic effects of inhomogeneous boundary conditions. Phys. Rev. A 1989 40 6431-6437. 

The MR theory predicts that the decay constant is independent of the nature of the surface, but the experimental data show that this is not the case and that the decay constant does depend on the nature of the surface (3). Recently, Komyshev and Leikin (KL 22) extended the Marcelja-Radic theory and demonstrated how this dependence can be explained. They replaced the homogeneous boundary conditions used in Marcelja-Radic theory by the inhomogeneous boundary conditions. According to Komyshev and Leikin, the description of the inhomogeneous character of the bound-... 

The critical parameter, although not the only one, governing the instability of a thin liquid film due to temperature induced surface tension gradients follows from the inhomogeneous boundary condition Eq. (10.6.9), the homogeneous boundary conditions introducing no parameters. With h the characteristic length scale, v a/h, and T ph the boundary condition is seen to introduce the dimensionless parameter... 

We have used elementary change of variables as a method to convert certain inhomogeneous boundary conditions to homogeneous form. This was a clear imperative in order to use, in the final steps, the properties of orthogonality. Without this property, series coefficients cannot be computed individually. 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

The quantitatives Xa, Xf in boundary conditions we call the characteristic lengths, although only Xa coincides with extrapolation length, while Xp enters inhomogeneous boundary conditions (see (3.59a)) so that it does not correspond to extrapolation length. 

Here % are components of the external normal vector to the ferroic surface. The most evident consequence of the flexocoupling is the spatially inhomogeneous boundary conditions. 

Note that in case of the essential inhomogeneous boundary conditions 2, relationships 13 remain valid, if the aforementioned boundary conditions are enforced as... 

With recent progress in micro- and nanofluidics, new interest has arisen in determining forms of hydrodynamic boundary conditions. In particular, advances in lithography to pattern substrates have raised several questions in the modeling of liquid motions over these surfaces and led to the concept of the effective tensorial slip. These effective conditions capture complicated effects of surface anisotropy and can be used to quantify flow over complex textures without the tedium of enforcing real inhomogeneous boundary conditions. 

The orientational relationships between the martensite and austenite lattice which we observe are partially in accordance with experimental results In experiments a Nishiyama-Wasserman relationship is found for those systems which we have simulated. We think that the additional rotation of the (lll)f< c planes in the simulations is an effect of boundary conditions. Experimentally bcc and fee structure coexist and the plane of contact, the habit plane, is undistorted. In our simulations we have no coexistence of these structures. But the periodic boundary conditions play a similar role like the habit plane in the real crystals. Under these considerations the fact that we find the same invariant direction as it is observed experimentally shows, that our calculations simulate the same transition process as it takes place in experiments. The same is true for the inhomogeneous shear system which we see in our simulations. 

The inhomogeneous soliton solution with boundary conditions Eq. (3.1) has the form of a hyperbolic tangent 15, 6J ... 

Both a uniform bulk fluid and an inhomogeneous fluid were simulated. The latter was in the form of a slit pore, terminated in the -direction by uniform Lennard-Jones walls. The distance between the walls for a given number of atoms was chosen so that the uniform density in the center of the cell was equal to the nominal bulk density. The effective width of the slit pore used to calculate the volume of the subsystem was taken as the region where the density was nonzero. For the bulk fluid in all directions, and for the slit pore in the lateral directions, periodic boundary conditions and the minimum image convention were used. 

Figure 7 also shows results for the thermal conductivity obtained for the slit pore, where the simulation cell was terminated by uniform Lennard-Jones walls. The results are consistent with those obtained for a bulk system using periodic boundary conditions. This indicates that the density inhomogeneity induced by the walls has little effect on the thermal conductivity. 

The intensity expression in Equation 6.5 requires all three g-values to be known. Sometimes not all g-values can be measured experimentally, and they have to be estimated on theoretical grounds. For example, the Fe(III) spectra of low-spin hemo-proteins frequently exhibit very pronounced g-anisotropy to the extent that two of the three g-values are either at helds beyond the maximum of the magnet and/or are associated with features inhomogeneously broadened beyond detection. With only the highest g-value determined the theoretical boundary condition for low-spin d5 systems with 3 < gz < 4... 

Stationary electro-convection at an electrically inhomogeneous permselective membrane.7 Once again the time-independent version of (6.4.45)-(6.4.49) with the boundary conditions (6.4.54a,b) at x — 0. 

The boundary conditions at x = 1 are replaced by those of the nonslip for the velocity and by the transport conditions at the electrically inhomogeneous surface for the electrolyte concentration and the electric potential of the form... 

This is called the partially reflecting or radiation boundary condition by analogy to the heat conduction equation analyses. It is also variously called a mixed, an inhomogeneous or the Robbins boundary condition because it mixes the value of the dependent variable p and the first... 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

In contrast to (2.2) there is an inhomogeneous term but there are no boundary conditions. (It is not true that nR(x) must be equal to unity at x = R, because when the particle arrives at R it need not exit but may jump back into the interval.) The integral equation is sufficient to determine nR(x) uniquely. In fact, suppose that (7.3) had two different solutions then their difference nR x) obeys... 

---