Boundary conditions homogeneous type

Theoretical investigations of the electrooptical phenomena in nonuniform fields were performed [157], for different boundary conditions (homogeneous, homeotropic, and twisted) and different types of spatially nonuniform field. It was shown that the sensitivity and spatial resolution of a liquid... 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

Let us first discuss the properties of migration coupling in the first type of systems. Here, the solution of Laplace s equation for any homogeneous situation yields a linear dependence of on z, i.e. with boundary condition 14b and the origin of the z-axis, z = 0 at the CE,... 

Real membrane systems to be used in practice usually do not have the simple architecture assumed in the preceding quantitative treatments (single-wall, non-supported) nor do they fulfil basic boundary conditions, i.e. well mixed gas mixtures, homogeneous gas compositions and pressure (no gradient) across the membrane length (flow direction of feed/permeate). In those cases the aerodynamic conditions of the feed and permeate flow, the precise design and the type of permeate removal (sweep gas, vacuum suction) are important. 

Figures 1 is an example for (non-dimensionalized) thermal stresses (von Mises stress) under steady-state heat conduction in a rectangular body defined over (a ,j/),0 < x < 1,0 < y < 1 subject to the Dirichlet type homogeneous boundary condition (T = 0 and Ui — 0) on the boundary with a uniform internal heat generation [5]. The thermal conductivity, elastic modulus and thermal expansion coefficient are all assumed to vary in the form of ko -f kix + k2y where fc, s are constants. For the purpose of illustrations, the values of all the material properties are taken to be unity.

The experiment is carried out under diffusion control. Theoretical concentration profiles are calculated by solving Fick s second law of diffusion in the steady state with boundary conditions appropriate to the solution domain and to the substrate, taking into account its geometry and the type of reaction occurring on it. Assumption is made that the redox species are stable and not involved in a homogeneous reaction in solution. Two geometries known to produce steady-state concentration profiles have been considered (72,77) the hemisphere and the microdisk. The former only requires a radial dimension, and the diffusion equation can be solved analytically. The latter, on the other hand, necessitates cylindrical coordinates and the solution becomes much more complex. With the latter a closed form analytical ex-... 

Since and both obey the same linear homogeneous equation, equation (14), a particular (but not general) solution for Pj in terms of / is Pj- = AP + B, where A and B are constants. Equations (42)-(46) show that a solution of this type does, in fact, satisfy the appropriate boundary conditions. This solution is... 

Fig. 9.30. Spatial propagation of a sharp Cef front of the type seen in eardiae cells (type 1 wave). Shown are six successive stages of the transient pattern obtained by numerical integration of eqns,(9.11) of the model based on CICR, from which the term Vj/S related to stimulation has been removed and to which the diffusion of cytosolic Ca has been added. In these simulations, the Ca -sensitive Ca pool is assumed to be distributed homogeneously within the cell. The latter is represented as a two-dimensional mesh of 20 x 60 points and diffusion is approximated by finite differences boundary conditions are of the zero-flux type. The terms related to influx from (vq) and into kZ) the extracellular medium only appear in the points located on the borders of the mesh. The diffusion coefficient of is equal to 400 pmVs other parameter...

This boundary condition also does not identify exactfy with the type (iii) homogeneous condition given earlier. However, if we redefine the dependent variable tohe = T — T, then we have... 

Finally, we consider the type (ii) homogeneous boundary condition in physical terms. For the pipe flow problem, if we had stipulated that the tube wall was well insulated, then the heat flux at the wall is nil, so... 

We can now finish the reactor problem using the fact that the cylindrical equation was of Sturm-Liouville type with homogeneous boundary conditions. When this is the case, the solutions

orthogonal functions with respect to the weighting function r(x). Since we identified the weighting function for the reactor as r( ) = f, we can write when n is different fix)m m... 

Before we apply the Sturm-Liouville integral transform to practical problems, we should inspect the self-adjoint property more carefully. Even when the linear differential operator (Eq. 11.45) possesses self-adjointness, the self-adjoint property is not complete since it actually depends on the type of boundary conditions applied. The homogeneous boundary condition operators, defined in Eq. 11.46, are fairly general and they lead naturally to the self-adjoint property. This self-adjoint property is only correct when the boundary conditions are unmixed as defined in Eq. 11.46, that is, conditions at one end do not involve the conditions at the other end. If the boundary conditions are mixed, then the self-adjoint property may not be applicable. 

Gradient coefficients > 0 and q > 0 the expansion coefficient an>0 for the second order phase transitions. Coefficient ai(T) = ar T — Tc), E is the transition temperature of a bulk material. Note, that the coefficient flu for displacement type ferroelectrics does not depend on T, while it is temperature dependent for order-disorder type ferroelectrics (see corresponding reference in [117]). Eq is the homogeneous external field, the term Ed (P3) represents depolarization field, that increases due to the polarization inhomogeneity in confined system. Linear operator Ed P3) essentially depends on the system shape and boundary conditions. Below we consider the case when depolarization field is completely screened by the ambient free charges outside the particle, while it is nonzero inside the particle due to inhomogeneous polarization distribution (i.e., nonzero divP 0) (see Fig. 4.35b). 

It is important to note that the above models include the fixed homogeneous type boundary conditions and they both result in a Eourier sine series. 

Equation 3.43g compares the timescale for radial heat dispersion in the solid phase with the one for internal heat conduction. For catalysts with good heat conduction properties and low particle-to-bed diameter ratios, A l. In this case, the surface boimdary condition is homogeneous and of Robin type, as given by the first terms on each side of (3.42b). A similar dimensionless number related with dispersion in the axial direction also appears, but its magnitude is considered much smaller than that of the other parameters in Equation 3.43, due to the geometrical reasons explained earlier. Note that Equations 3.32 and 3.34 are obtained by integrating Equation 3.41 with respect to over the pellet domain and using Equation 3.42 as boundary conditions. 

---