The Reciprocal Theorem

The reciprocal theorem is derived directly from the general integral formula, (8-111). For this purpose, we identify u and u, as well as T and T, as the solutions of two creeping-flow problems for flow past the same body but with different boundary conditions on the body surface 3 D. 

The relationship (8-213) is the famous reciprocal theorem of Lorentz.20 As we shall see shortly, it is an extremely useful result that often leads to the impression of getting something for nothing. 21 

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The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

Both Newton s equation of motion for a classical system and Schrodinger s equation for a quantum system are unchanged by time reversal, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of microscopic reversibility, sometimes also referred to as the reciprocity theorem. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of detailed balance, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility. 

As already noted, the reciprocal theorem does not include specifically conditions where sliding between particles takes place. Geometrical alterations between granular material affect reciprocity of pressure and displacement. 

There is the reciprocity theorem for the view factor For any object, sum to unity for all view factors View factor to itself is zero for plane surfaces View factor to itself is zero for convex surfaces. 

The coil sensitivity is characterized by means of the reciprocity theorem by the ratio where is the virtual field induced by a coil carrying... 

Comparison of these two expressions yields the Reciprocal Theorem ... 

On account of (1.3.10) the left-hand side vanishes, and the right-hand side may be rewritten, such that one obtains the Reciprocity Theorem... 

Formulae (10.33) and (10.34) have a clear physical interpretation. Let us introduce a unit conductance Sq = 1 S. According to the reciprocity theorem (see Chapter 8), the integral term in equation (10.33) can be treated as the complex conjugate of the electric field E (divided by the unit conductance Eo)>... 

Theorem 27 (The reciprocity theorem) The wavefield at a point r" generated by a point source located at a point r is equal to the wavefield at a point r generated by a point source located at a point r" ... 

The reciprocity theorem shows that for any wavefield we can switch between the receiver and source positions without changing the values of the observed field. This result plays an extremely important role in wavefield imaging and inversion, especially in wavefield migration, which will be discussed in Chapter 15. 

Note that we can show, by methods analogous to those used in deriving the reciprocity theorem 27 (formula (14.24)) for a scalar wavefield, that Gj satisfies the reciprocity relationship ... 

We can also use the 1-D Green s theorem, equation (15.51), to prove the reciprocity theorem in one dimension, similar to the general reciprocity theorem 27 ... 

Taking into account the reciprocity theorem (14.24), we can write the last formula in the form... 

In addition to the theoretical advantages offered by the principle of minimum potential energy, our elastic analyses will also be aided by the reciprocal theorem. This theorem is a special example of a more general class of reciprocal theorems and considers two elastic states (u ), or(i)) and where each... 

The proof of the reciprocal theorem is based upon two fundamental realizations. First, we note the relation... 

Based on this latter result, the reciprocal theorem is immediate. 

This result will serve as the cornerstone for many of our developments in the elastic theory of dislocations. We are similarly now in a position to obtain the Green function in real space which, when coupled with the reciprocal theorem, will allow for the determination of the displacement fields associated with various defects. 

The reciprocal theorem points the way to an idea that will allow for the calculation of the displacements associated with a dislocation loop of arbitrary shape. As discussed in chap. 2, the reciprocal theorem asserts that two sets of equilibrium fields (or< and f ), associated with the same... 

Within the context of the elastic Green function, the reciprocal theorem serves as a jumping off point for the construction of fundamental solutions to a number of different problems. For example, we will first show how the reciprocal theorem may be used to construct the solution for an arbitrary dislocation loop via consideration of a distribution of point forces. Later, the fundamental dislocation solution will be bootstrapped to construct solutions associated with the problem of a cracked solid. 

Hence, the only terms remaining from our implementation of the reciprocal theorem yield the rather simpler expression... 

As a first application of the reciprocal theorem, we consider its use in calculating the hydrodynamic force on a body in an undisturbed flow, u00(x). 

In particular, let us suppose that we have obtained the solution of the creeping-motion equations for uniform flow U past a stationary body 3 D. Equivalently, we may consider the case of a particle that translates with velocity —U in a fluid at rest at infinity. We denote the solution of this problem as u and the corresponding surface-force vector on 3 D as f. Then, on applying the reciprocal theorem, we find that... 

I attribute this observation about the reciprocal theorem, and other related results, to a lecture given by H. Brenner at Caltech in the 1970s. 

Problem 8-8. Reciprocal Theorem for the Navier-Stokes Equations. Derive the reciprocal theorem for the Navier-Stokes equations, not the Stokes equations. You should have a volume integral remaining. 

Problem 8-9. Torque on a Sphere in a General Stokes Flow. Use the reciprocal theorem for Stokes equations to derive the following expression for the torque exerted on a sphere of radius a that is held fixed in the Stokes flow u°°(x) ... 

Problem 8-10. Symmetry of the Grand Resistance Tensor. Use the reciprocal theorem to show that the grand resistance tensor is symmetric. The grand resistance tensor relates the hydrodynamic force/torque on a particle to its velocity/angular velocity ... 

Note To apply the reciprocal theorem, the shape of the drop for the complementary problem generally would have to be exactly the same as the shape in the problem of interest. However, because the original problem can be reduced by means of domain perturbations to an equivalent problem with the boundary conditions applied at the spherical surface, r = 1, we may also conveniently choose the drop to be spherical for this complementary problem. 

An engineering principle to bear in mind when evaluating various coil configurations for their effectiveness in NMR is the "reciprocity theorem". In practical terms it states that the maximum contribution to the signal from an element of a sample is proportional to the field at that spot if the coil were used as a transmitter coil (Hoult, 1978). 

Closely related to the preceding is the problem of calculating the pressure drop due to Stokes flow through a cylinder of arbitrary (but constant) cross section for arbitrary boundary conditions on the surfaces bounding the cylinder. A simple application of the Reciprocal theorem (B18) permits one to express this pressure drop directly in terms of the prescribed velocity field on the cylinder walls, top, and bottom. If (v, n) and denote the velocity... 

Mahnivuo, J.A. 1976. On the detection of the magnetic heart vector An appHcation of the reciprocity theorem. Acta Polytechnol. Scand. 39 112. 

Figure 4.32 illustrates macroscopic anisotropy in a simplified tissue model. In living tissue, conductivity may be 10 times larger in one direction than another. At low amplitude levels the tissue is still linear, and the principle of superposition and the reciprocity theorem are still valid. However, Ohm s law for volume conductors, J = oE, is not necessarily valid even if it still is linear the current density direction will not coincide with the E-field direction if the anisotropic structures are sufficiently small. 

A mapping of the immittance distribution in a tissue layer (tomography) is possible with an electrode system of multiple skin surface electrodes. In EIT, a current (about 1 mA) is typically injected in one electrode pair and the voltages between other electrodes are recorded (Resell et al., 1988b Bayford and Tizzard, 2012). Current injection is then successively shifted so that all electrode pairs are used. The reciprocal theorem can serve as a control of system linearity. A frequency on the order of 50 kHz is conunonly used, so a complete set of measurements with, for example, around 50 electrodes can be performed in less than 0.1 s. The images obtained have a resolution of about 1 cm at 10 cm tissue depth. 

To reduce the proximal zone contribution, and sometimes for anatomical reasons, it may be advantageous to place the PU electrodes outside the CC electrodes, instead of inside as shown in Figure 7.28. If the system is linear, transfer immittance should be the same if the CC and PU electrodes are swapped compare the reciprocity theorem. 

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