Vector harmonic functions

We can demonstrate this easily by substituting (8-3) into the creeping-flow equations. To satisfy the continuity equation, the harmonic function u(//) is required to satisfy the condition 

Although it is not essential, we shall see that it is convenient in the developments that follow to maintain the general solution (8-3) in dimensional form. 

XiXjXfcXi 8,jXkxi - - SikXjXi -p SuXjXk -P 8jkxix/ -p 8jixixk -p 8k[XiXj 

It is important to note that these vector harmonic functions involve only the general position vector x and its magnitude r = x and thus can be represented in any coordinate system 

let us see how the preliminary concepts in this section can be put together to achieve a general representation procedure for the solution of general classes of creeping-flow problems. We begin our discussion with the simplest case of spherical particles. 

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A. Solutions by Means of Superposition of Vector Harmonic Functions... 

A. SOLUTIONS BY MEANS OF SUPERPOSITION OF VECTOR HARMONIC FUNCTIONS... 

We begin with a powerful solution method that can be applied for general 3D flows whenever the boundaries of the domain can be expressed as a coordinate surface for some orthogonal coordinate system. In this case, we can use an invariant vector representation of the velocity and pressure fields to simultaneously represent (solve) the solutions for a complete class of related problems by using so-called vector harmonic functions, rather than solving one specific problem at a time, as is necessary when we are using standard eigenfunction expansion techniques. 

Now, the solution for p must be constructed solely from Cl and the general position vector x in the form of the vector harmonic functions. Now, there is only a single scalar function that can be constructed from Cl and the decaying harmonics, (8-5), that is linear in Cl, namely,... 

It follows, therefore, from (8-3) that u = u(ff) must be a decaying harmonic function, linear in Cl and a true vector. The only combination of Cl and the vector harmonic functions that satisfies these conditions is... 

The function u(H> is harmonic, decaying, linear in U and a true vector. Again, examining the decaying vector harmonic functions (8-5), we find that there are only two products of U and the decaying harmonics that are true vectors. Thus the most general form for u(//) is... 

There is one new factor that needs to be considered for this problem. The vector harmonic functions are irreducible tensors. Hence, for any sum of such terms to satisfy the boundary conditions, (8-28), we should also express the tensors, T and K, in terms of irreducible tensors. Now, we have seen previously that T can be expressed in terms of the rate-of-strain and vorticity tensors ... 

With this decomposition of K into irreducible form, the solution for a spherical particle in the quadratic flow part of (8-26) can be represented as the sum of the terms that involve Y, 0, and t combined with the vector harmonic functions (8-5) to produce a true scalar for p and a true vector for u This problem is posed in the problems section at the end of the chapter. 

To obtain a solution for the complete class of flows given by (8 59), we again construct a solution of the creeping-motion equations by means of the superposition of vector harmonic functions. The development of a general form for the pressure and velocity fields in the fluid exterior to the drop follows exactly the arguments of the preceding solution for a solid sphere in a linear flow, and the solution therefore takes the same general form [see Eq. (8-39) and (8 11)], that is,... 

Use the general representation of solutions for creeping flows in terms of vector harmonic functions to solve for the velocity and pressure fields in the two fluids, as well as the deformation and surfactant concentration distribution functions, at steady state. You should find... 

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with infinitely large radius. In deriving this equation we assume that the potential U p) is a harmonic function, and the Green s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an infinity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to find a Green s function in the volume V that is equal to zero at each point of the boundary surface ... 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

The solutions of the inner and outer fields can now be written as expansions in these spherical harmonic functions or vector eigenfunctions, once the incident irradiation and the boundary conditions are specified. 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

Similarly, (Now ,NewM), (Mom ,Now ) and (M N J are mutually orthogonal sets of functions. The orthogonality properties of cos m< > and sin m imply that all vector harmonics of different order m are mutually orthogonal. 

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

The explicit expansion of the spherical harmonic functions describing the orientation of the z/th internuclear vector leads to the following expression,... 

For time derivative operators, the time derivative of A is known to have a direchon different from A. However, in some cases, and as observed in spacelike derivations, we may expect that second-order derivations of A with respect to time lead to a vector having the same direction as A. This, obviously, is the case of A vectors, which are time-harmonic functions. These remarks can be extended, and therefore applied, to the case of the electromagnetic field, whose calculation is based on the following set of relations ... 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

Recall from Section 1.5 that any function in the kernel of the Laplacian (on any space of functions) is called a harmonic function. In other words, a function f is harmonic if V / = 0. The harmonic functions in the example just above are the harmonic homogeneous polynomials of degree two. We call this vector space In Exercise 2,23 we invite the reader to check that the following set is a basis of H/ ... 

In fact, every spherical harmonic function is the restriction to the sphere in L of a harmonic polynomial on Recall the vector space of restrictions of harmonic polynomials of degree f in three variables to the... 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

Associated with each operator realization of a Lie algebra we generally have a vector space on which these operators act. For the realization given by L this might be either an abstract space of angular momentum states lm), 1 = 0, 1,... m = —/, — l + 1,..., l or a concrete realization of them as spherical harmonic functions Ylm(6, (j)). We can then consider the matrix elements of the operators with respect to this vector space of states and this leads to the important concept of a matrix representation of a Lie algebra. 

Figure 3.2 Comparison of the local Cartesian and transformed coordinate systems at a general point on the unit sphere the standard spherical harmonic functions as the angular parts of the appropriate atomic orbitals are defined with reference to the local Cartesian set ex(j), ey(j) and ez(j) for each atomic position (j) with radius vector R on the unit sphere. Then the transformation of equation 3.1 is applied to construct the new local coordinate system cr(j), and JT (j).

Table 3.11 lists the real forms for the U and spherical X = 0) and vector (X = 1 ) harmonics for the 1 = 0 to 3 central harmonic functions. Group orbitals of the particular irreducible symmetries for which these central functions provide bases follow simply by making linear combinations of the u and vj at each vertex point modulated by the values of Uj and to form local resultants, which interconvert from vertex to vertex of the orbit under the actions of the symmetry operations of the point group. 

Figure 3.9 The jr-type group orbitals on the vertices of an O3 structure orbit exhibiting D3j point symmetry, displayed superimposed on the elliptical projections of the generating functions of Table 3.11. The familiar dumbbells identify tt-oriented atomic orbital components at the vertices, equation 3.21 sized to reflect the values. Table 3.12, of the corresponding vector harmonics at the orbit vertices. The motifs A, A and A" are applied as in Figure 3.8. 

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