Two Parallel Cylinders

We derive Deijaguin s approximation for obtaining the interaction energy between two parallel or crossed cylinders 1 and 2 of radii oj and 02 at separation H from the corresponding interaction energy between two parallel plates [16, 17]. This method is applicable when conditions [12.11] hold. 

Consider first the case of two parallel cylinders of radii and at separation H 

FIGURE 12.3 Derjaguin s approximation for the two interacting parallel cylinders 1 and 2 at separation H, having radii ai and A2 respectively. 

The interaction energy length between two parallel cylinders 1 

The interaction energy V° u H) per unit length between two parallel cylinders 1 and 2 with constant surface charge densitiesci and G2 at separation H can be 

---

Usually this would be handled in two parallel cylinders. 

A single cylinder will do this capacity however, usually it can be handled in two parallel cylinders for... 

By examining the curve for the initial compression with no unloaders, it shows that the horsepower requirement crosses the +3% overload line about one-third of the way through the suction pressure range. Figure 12-32 shows the effect of adding first one unloader and then a second one. The simplest way to handle this is a head-end unloader on each of the two parallel cylinders. 

In an attempt to clarify matters further by studying a specific example, let us evaluate the total force, acting per unit length of two parallel cylinders of radius R, symmetrically charged to a positive surface potential and... 

C.3.a. Two parallel cylinders, retardation screening neglected, solved by multiple reflection... 

Source Taken from D. Langbein, Phys. Kondens. Mat., 15, 61-86(1972) [Eqs. (41), p. 71, and Table 2, p. 79] for the energy of interaction between two parallel cylinders of length L much greater than their radii and separation. The expression given on p. 79 apparently lacked a factor tt in the denominator and should have read... 

C.3.b. Two parallel cylinders, pairwise-summation approximation, Hamaker-Lifshitz hybrid, retardation screening neglected C.3.b.l. All separations... 

C.2.b. Free energy per interaction C.2.c. Nonretarded (infinite light velocity) limit C.2.d. Light velocities taken everywhere equal to that in the medium, small Aji, Aji, q = 1 C.2.e. Hamaker-Lifshitz hybrid form C.3. Two parallel cylinders... 

Figure 10 (a) Equipotential and current lines for two parallel cylinders in the primary... 

An alternative method for calculating the asymptotic expressions for the double--layer interaction between two parallel plates at large separations is given below. This method, which was introduced by Brenner and Parsegian [6] to obtain an asymptotic expression for the interaction energy between two parallel cylinders, is based on the concept of hypothetical charge, as shown below. 

By combining Eqs. (11.77)-(11.79), we find that the double-layer interaction energy per unit length between two parallel cylinders at large separations is given by [11]... 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

Similarly, series expansion representations for the double-layer interaction between two parallel cylinders can be obtained [13, 14]. The results are given below. 

Consider first the case of two parallel soft cylinders (Fig. 15.7). With the help of Derjaguin s approximation for two parallel cylinders [8,9] (Eq. (12.38)), namely,... 

FIGURE 19.11 Two parallel cylinders of radii a 1 and 02 separated by a distance R. 

FIGURE 19.17 Interaction between two parallel cylinder of radius a and length Inb separated by a distance R. 

Repeat the preceding process, assuming that the no-slip boundary is replaced with an interface that remains flat and nondeforming and on which the shear stress vanishes. Explain any qualitative differences between the two cases. Can the results of this second calculation be applied to determine the motion of two parallel cylinders of equal radius that are acted on by equal and opposite forces along their line of centers Explain. 

If dispersion interactions between two parallel cylinder-shaped materials are considered, as shown in Figure 7.7 a, then the final equation can be written as... 

Figure 7.7 Interactions between two cylinder surfaces having the same length, L, but different radii, R, and R2. a. Between two parallel cylinders having a distance, D between them. b. Between two crossed cylinders.

The two parallel-cylinder solution (section 6.3.2) is directly applicable to a complete cylinder discussed previously (e.g., with an infinite metal plate), because the current density field is unchanged if the isoelectric plane surface is covered by a thin metal sheet. The resistance is halved and the conductance doubled (see Eq. (6.26)) ... 

---