Approximate horizontal motion

Whittaker [3] has given an expression for the azimuthal angle of the spherical pendulum in terms of elliptic functions of the third kind, so that, not surprisingly, there has been very little numerical discussion of its motion. Instead, we see if our approximate theory can be used to obtain a simpler picture of the motion. 

If we insert the approximate solution for w into the equation of motion for X, we obtain 

The corresponding initial conditions for F[r], which satisfies the same Mathieu s equation, are 

Numerical simulations of the spherical pendulum for arbitrary values of K and W will usually reveal a very complicated, a periodic motion of the type shown in Fig. 2, but in some cases the motion is periodic. The theory can be found in Refs. [9,11], but is summarised here. Let r be any integer or simple fraction (such as 3/2, etc.). Then solutions of Mathieu s equation of the form 

Clearly the smaller the values of the integral quantum numbers n and k, the simpler the motion. 

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In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

With the eyes closed, gently scrub the lids and eyelashes, in a horizontal motion from left to right, for approximately 20 passes across each lid. 

Fig. 7. The horizontal projection of the trajectories for the exact solution (the left curve) and the solution of the approximate equation of motion in terms of Mathieu functions (the right curve) iox n = I, k = 2.

The horizontal projection of the approximate motion for 1, A =2 is shown in Fig. 7 (the right curve), compared with the same curve from Fig. 2 for the exact motion (the left figure). 

Turning to the low temperature transition of the homopolymer of PHBA at 350 °C, it is generally accepted that the phase below this temperature is orthorhombic and converts to an approximate pseudohexagonal phase with a packing closely related to the orthorhombic phase (see Fig. 6) [27-29]. The fact that a number of the diffraction maxima retain the sharp definition at room temperature pattern combined with the streaking of the 006 line suggests both vertical and horizontal displacements of the chains [29]. As mentioned earlier, Yoon et al. has opted to describe the new phase as a smectic E whereas we prefer to interpret this new phase as a one dimensional plastic crystal where rotational freedom is permitted around the chain axis. This particular question is really a matter of semantics since both interpretations are correct. Perhaps the more important issue is which of these terminologies provides a more descriptive picture as to the nature of the molecular motions of the polymer above the 350 °C transition. As will be seen shortly in the case of the aromatic copolyesters, similar motions can be identified well below the crystal-nematic transition. 

To understand the mechanisms of solids slug flows, a two-dimensional coupled DEM/CFD numerical model was built to simulate the motion of a pre-formed slug (ca. 0.3 m long) in a 1 m long horizontal 50 mm bore pipe as shown in Fig. 1. The pipe was initially filled with a layer of particles, approximately 15 mm thick at the bottom. (The thickness of this stationary layer was determined based on experience from previous experiments and computer test runs). 

The second stage realizes a two-step procedure that re-calculates the ozone concentration over the whole space S = (tp, A, z) (, A)e l 0atmospheric boundary layer (zH 70 km), whose consideration is important in estimating the state of the regional ozonosphere. These two steps correspond to the vertical and horizontal constituents of atmospheric motion. This division is made for convenience, so that the user of the expert system can choose a synoptic scenario. According to the available estimates (Karol, 2000 Kraabol et al., 2000 Meijer and Velthoven, 1997), the processes involved in vertical mixing prevail in the dynamics of ozone concentration. It is here that, due to uncertain estimates of Dz, there are serious errors in model calculations. Therefore the units CCAB, MFDO, and MPTO (see Table 4.9) provide the user with the principal possibility to choose various approximations of the vertical profile of the eddy diffusion coefficient (Dz). 

When < = 180°, the hot wall is horizontal and at the top there is no fluid motion and the Nusselt number has its pure conduction value. When = 0°, the hot wall is horizontal and at the bottom and the flow is unstable. However, the Rayleigh number for the aspect ratio considered is too low for this to occur and the conduction value for the Nusselt number is obtained at = 0°. However, inclining the enclosure by a small amount provides this trigger leading to a value of Nusselt number that is much higher than the conduction value, e.g., Nu is calculated to be 1 when = 0° and to be approximately 2.6 when when 0 = 1°. 

Circulation models are based on the equations of motion of the geophysical fluid dynamics and on the thermodynamics of seawater. The model area is divided into finite size grid cells. The state of the ocean is described by the velocity, temperature, and salinity in each grid cell, and its time evolution can be computed from the three-dimensional model equations. To reduce the computational demands, the model ocean is usually incompressible and the vertical acceleration is neglected, the latter assumption is known as hydrostatic approximation. This removes sound waves in the ocean from the model solution. In the horizontal equations, the Boussinesq approximation is applied and small density changes are ignored except in the horizontal pressure gradient terms. This implies that such models conserve... 

Glider) Consider a glider flying at speed v at an angle 0 to the horizontal. Its motion is governed approximately by the dimensionless equations... 

We begin by considering the flow within a shallow, horizontal (a = 0) cavity as sketched in Fig. 6-7a. We assume that the ratio, d/L, is asymptotically small. We seek only the leading-order approximation within the shallow cavity. Hence the starting point for analysis is the thin-film equations, (6—1)—(6—3). In the present case of a 2D cavity, we can use a Cartesian coordinate system, and, for the present problem, we assume that the fluid is isothermal, so that the body-force term in (6-3) can be incorporated into the dynamic pressure, and hence plays no role in the fluid s motion. In this case, the governing equations become... 

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