Axisymmetric wave modes

Relations between the Plane-Wave Modes B. Total Reflection at a Vacuum Interface VII. Axisymmetric Wave Modes... 

Lehnert and Roy [10] found axial magnetic field component in the direction of propagation of photon considering the axisymmetric wave modes in Maxwell s equations with space charge in vacuo. 

One alternative that has been explored [27] is to conduct an axisymmettic computation (with its associated efficiency) and use a linear stability analysis to fractionate annular ring-shaped ligaments shed from the jet periphery in this case. A Boundary Element Method (BEM) was employed to compute the local surface dynamics. See Chap. 15 for details regarding the computational methodology. While Fig. 27.6 indicates that 3-D instabilities occur prior to pinching of axisym-metric structures, the wavelengths of the azimuthal modes appear to be comparable to those of the axisymmetric waves. Nevertheless, the axisymmetric assumption, while providing drastic simplification, still provides much room for improvement as more computational power becomes available. 

A review article gives a thorough review of creep instability of steel and other metal shells and pressure vessels at elevated temperatures. Some of the same analyses can be adopted here for polymers at lower temperatures, for example, room temperature. For example, consider the axisymmetric buckling mode of a cylindrical shell under compression. (The beam column mode can be included under column buckling. Section 4.2.) For the axisymmetric mode, where axisymmetric buckling waves occur for RJh < 33, the critical buckling stress in compression is given by... 

Poynting s theorem for the energy flow of plane waves in vacuo thus applies to the EM and EMS modes, but not to the S mode. Vector multiplication of Eqs. (52) and (53) by k, and combination with Eq. (49) and the result E C = 0, is easily shown [16,20] to result in a Poynting vector that is parallel with the group velocity C of Eq. (56). Later in Section VII.C.3 we shall return to Poynting s theorem in the case of axisymmetric photon wavepackets. 

Problem 12-3. Capillary Instability for a Thread in a Second Immiscible Fluid. In this problem, we consider the effect on capillary instability if, instead of being surrounded by air, the thread of liquid is surrounded by a second viscous immiscible fluid that is assumed to be unbounded in the radial direction. Derive a condition from which you could in principle, calculate the growth-rate parameter for an axisymmetric disturbance, a = a(k, Re, 7.) where k is the axial wave number and 7. is the ratio of the external fluid viscosity to the viscosity of the liquid thread. This condition can be simplified if either Re I or the thread is inviscid (though viscous effects still remain in the outer fluid). Evaluate a for several k values in each of these two cases. What is the qualitative effect of the second viscous fluid For example, is the range of unstable k values changed Is the fastest-growing linear mode changed relative to the case of a thread in air ... 

Bowden et al. experimentally confirmed that the first mode is an antisymmetric one, followed by an axisymmetric one (32). Figure 2 shows a flat descending front as well as axisymmetric and antisymmetric modes of ascending fronts. Figure 3 shows the stability diagram in the viscosity-front velocity plane. Most importantly, they confirmed that the stability of the fluid was a fiinction not only of the viscosity but also of the front velocity. This means that the front dynamics affects the fluids dynamics, unlike with pH fronts and chemical waves in the BZ reaction in which the front velocity does not play a role in the stability of the fluid (33-38). 

Fig. 22.11 Two threads emerging from open channel flow with different dimensionless viscosities. In (a), the thread disintegrates due to surface-driven axisymmetric or Rayleigh wave breakup. The cross-wind flow leads to a slight increase in drop size, (b) Shows a different breakup mode, as the stronger cross-flow initiates a more stochastic-wind induced breakup...

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