Spatial steady state

Our main concern in the first six chapters was with the behavior of bound rays, which propagate without loss of power on nonabsorbing fibers. Bound rays provide a complete description of light propagation on nonabsorbing fibers sufficiently far from the source, when virtually all leaky-ray power has been lost. We call this the spatial steady state. In this chapter our attention is... 

In Sections 4-21 and 4-22, we showed that the shape of the impulse response on step and clad parabolic profiles is virtually rectangular. This conclusion is valid only in the spatial steady state. In the spatial transient, the power in tunneling rays manifests itself by adding a tail to the pulse. The power in the tail is large close to the source but becomes negligible at the onset of the spatial steady state [5]. 

We can now quantify the duration of the spatial transient, as discussed in Section 8-1. If we assume that the spatial steady state begins when the tunneling-ray power has fallen to 10% of its initial value, then Fig. 8-S(a) shows that this corresponds to G = 0.5. We can then relate G to the duration of the spatial transient through Eq. (8-17), and deduce that ... 

So far in this chapter we have assumed that the fiber is nonabsorbing, and that all power loss is due solely to radiation from leaky rays. Here we examine the combined effects of loss due to radiation and material absorption. The most significant effect is that there is no longer a spatial steady state, since every bound ray loses power as it propagates. On a sufficiently long fiber, any power that is not lost by radiation will eventually be absorbed. 

With reference to Fig. 9-1, we assume that a diffuse source excites a sufficiently long straight fiber to enable the spatial steady state to be reached at the beginning of the bend. The analysis of ray paths around the bend depends on the profile. If the profile is a step, the trajectory is a straight line between successive reflections, involving the solution of cubic and quartic polynomial equations, whereas, if the profile is a clad parabola, a paraxial approximation is used [2]. 

Snyder, A. W. and Pask, C. (1975) Optical fibre spatial transient and spatial steady state. Opt. Commun., 15, 314-16. 

The total electromagnetic fields of an optical waveguide can be expressed as the sum of two components. One component, expressible as a summation over bound modes, describes the spatial steady state, where energy is guided indefinitely along a nonabsorbing waveguide. The second component is the radiation field, which describes the spatial transient. In Chapter 24 we... 

The discrete set of bound-mode propagation constants means that the fields of the waveguide in the spatial steady state are given by the finite sum over all bound modes in Eq. (11-2). In contrast, each radiation and evanescent mode can take any of the continuum of propagation-constant values given in Table 25-1, and thus an integration over all values of is necessary. However, like bound modes, the total radiation field requires a summation over the subscript j of Eq. (25-1) to account for the transverse fields of different modes. However, rather than use / as the continuum variable, when P is real for radiation modes and imaginary for evanescent modes, we use instead the modal parameter Q, defined below, in order to simplify the notation. We take / to be the positive root of the inverse relation whence... 

Figure Bl.14.9. Imaging pulse sequence including flow and/or diflfiision encoding. Gradient pulses before and after the inversion pulse are supplemented in any of the spatial dimensions of the standard spin-echo imaging sequence. Motion weighting is achieved by switching a strong gradient pulse pair G, (see solid black line). The steady-state distribution of flow (coherent motion) as well as diffusion (spatially...

Flow which fluctuates with time, such as pulsating flow in arteries, is more difficult to experimentally quantify than steady-state motion because phase encoding of spatial coordinate(s) and/or velocity requires the acquisition of a series of transients. Then a different velocity is detected in each transient. Hence the phase-twist caused by the motion in the presence of magnetic field gradients varies from transient to transient. However if the motion is periodic, e.g., v(r,t)=VQsin (n t +( )q] with a spatially varying amplitude Vq=Vq(/-), a pulsation frequency co =co (r) and an arbitrary phase ( )q, the phase modulation of the acquired data set is described as follows ... 

Under steady-state conditions, the temperature distribution in the wall is only spatial and not time dependent. This is the case, e.g., if the boundary conditions on both sides of the wall are kept constant over a longer time period. The time to achieve such a steady-state condition is dependent on the thickness, conductivity, and specific heat of the material. If this time is much shorter than the change in time of the boundary conditions on the wall surface, then this is termed a quasi-steady-state condition. On the contrary, if this time is longer, the temperature distribution and the heat fluxes in the wall are not constant in time, and therefore the dynamic heat transfer must be analyzed (Fig. 11.32). 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

From this we can see that knowledge of k f and Rf in a conventional polymerization process readily yields a value of the ratio kp fkt. In order to obtain a value for kf wc require further information on kv. Analysis of / , data obtained under non-steady state conditions (when there is no continuous source of initiator radicals) yields the ratio kvlkx. Various non-stcady state methods have been developed including the rotating sector method, spatially intermittent polymerization and pulsed laser polymerization (PLP). The classical approach for deriving the individual values of kp and kt by combining values for kp kx. with kp/k, obtained in separate experiments can, however, be problematical because the values of kx are strongly dependent on the polymerization conditions (Section... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

Perfectly mixed stirred tank reactors have no spatial variations in composition or physical properties within the reactor or in the exit from it. Everything inside the system is uniform except at the very entrance. Molecules experience a step change in environment immediately upon entering. A perfectly mixed CSTR has only two environments one at the inlet and one inside the reactor and at the outlet. These environments are specifled by a set of compositions and operating conditions that have only two values either bi ,..., Ti or Uout, bout, , Pout, Tout- When the reactor is at a steady state, the inlet and outlet properties are related by algebraic equations. The piston flow reactors and real flow reactors show a more gradual change from inlet to outlet, and the inlet and outlet properties are related by differential equations. 

This technique is invasive however, the particle can be designed to be neutrally buoyant so that it well represents the flow of the phase of interest. An array of detectors is positioned around the reactor vessel. Calibration must be performed by positioning the particle in the vessel at a number of known locations and recording each of the detector counts. During actual measurements, the y-ray emissions from the particle are monitored over many hours as it moves freely in the system maintained at steady state. Least-squares regression methods can be applied to evaluate the temporal position of the particle and thus velocity field [13, 14]. This technique offers modest spatial resolutions of 2-5 mm and sampling frequencies up to 25 Hz. 

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

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