Axisymmetric systems

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Mehdizadeh and Dukovic (5) expanded the theoretical treatment and included mass transport effects in an axisymmetric system as well as a 3-D geometry. In the 3-D geometry, they assumed four peripheral low-contact-area terminals and have shown the effect of peripheral point contacts on the thickness distribution of a 200 mm wafer. Initially, the thickness near the four point contacts is very high, whereas between the contacts is very low. A time series of a growing deposit with four peripheral point contact terminals is shown in (6). Point contacts result in azimuthal nonuniformity. However, the nonuniformity in the vicinity of the contacts becomes appreciably better as the plated thickness builds up. In applications such as Damascene electroplating where the final plated thickness is usually not more than l/im, azimuthal nonuniformity can be a problem. Our solution was to implement an almost continuous peripheral contact terminal and to assume that the system is axisymmetric and that only the radial nonuniformity needs improvement. 

In Chapter Three, the boundary element method is used to solve current distributions in two-dimensional and axisymmetrical systems. More particularly, the required accurate integration of the integrals involved with the method, the use of elements specially suited for singularities and the convergence of the numerical method are treated in detail. 

The ion distribution f is a function of the two constants of motion for axisymmetric systems, H = 1/2 m v the kinetic energy and Pq = m rvQ 4 q p/c the canonical angular momentum. A third constant exists only on systems with additional symmetry, e.g., a very long cylindrical layer or a thin ring. Now we may write B = Vip x 76 y... 

The stress equilibrium in any axisymmetric system is given by a radial equation... 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

Working equations of the U-V-P scheme in axisymmetric coordinate systems... 

Using a irocedure similar to the formulation of two-dimensional forms the working equations of the U-V - P scheme in axisymmetric coordinate systems are derived on the basis of Equations (4.10) and (4.11) as... 

After the substitution of pressure via the penalty relationship the flow equations in an axisymmetric coordinate system are written as... 

Using a procedure similar to the derivation of Equation (4.53) the working equations of the continuous penalty scheme for steady-state Stokes flow in an axisymmetric coordinate system are obtained as... 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Similarly in an axisymmetric coordinate system the terms of stiffness and load matrices corresponding to the governing energy equation written as... 

Note that in polar and axisymmetric coordinate systems the stress term will include some lower-order terms that should be included in the formulations. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

Step 3 Comparing systems (7.2) and (7.3) additional terms in the members of the stiffness matrix corresponding to the axisymmetric fon-nulation are identified. Note that the measure of integration in these tenns is (r drdz). 

The two versions of the Aaberg exhaust system, namely an axisymmetri-cal version and a workbench version, both work on the same principle. In order to illustrate the principle of the Aaberg we describe the axisymmetrical version but the full theoretical, computational, and experimental basis is presented for both systems. 

The Aaberg reinforced exhaust system was first studied in the 1940s, - then more extensively in its axisymmetrical form in 1965, but it was not until... 

For premixed fuel-air systems, results are reported in various terms that can be related to a critical equivalence ratio at which the onset of some yellow flame luminosity is observed. Premixed combustion studies have been performed primarily with Bunsen-type flames [52, 53], flat flames [54], and stirred reactors [55, 56], The earliest work [57, 58] on diffusion flames dealt mainly with axisymmetric coflow (coannular) systems in which the smoke height or the volumetric or mass flow rate of the fuel at this height was used as the correlating parameter. The smoke height is considered to be a measure of the fuel s particulate formation and growth rates but is controlled by the soot particle bumup. The specific references to this early work and that mentioned in subsequent paragraphs can be found in Ref. [50],... 

Although several different system configurations have been simulated, the focus of this paper will be on the unsteady, compressible, multiphase flow in an axisymmetric ramjet combustor. After a brief discussion of the details of the geometry and the numerical model in the next section, a series of numerical simulations in which the physical complexity of the problem solved has been systematically increased are presented. For each case, the significance of the results for the combustion of high-energy fuels is elucidated. Finally, the overall accomplishments and the potential impact of the research for the simulation of other advanced chemical propulsion systems are discussed. 

For a typical case, an axisymmetric jet with a mean velocity of 100 m/s flows through the cylindrical inlet of diameter D into a cylindrical combustion chamber of twice the diameter. An annular or central exit at the end of the combustion chamber is modeled to produce choked flow. Particles are injected from the inlet-combustor junction with a streamwise velocity of 50 m/s and zero radial velocity. If the number of particles is small (that is, for low-mass loadings), the effect of the particles on the flow can be neglected. Still the flow has an effect on the particles that depends on parameters such as the size and density of the particles. Such systems are called one-way coupled systems and are discussed next. 

Chang, E. J., K. Kailasanath, and S. K. Aggarwal. 1995. Compressible flows of gas-particle systems in an axisymmetric ramjet combustor. AIAA Paper No. 95-2561. 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

Fig. 7.11 Wake configurations for drops in water (highly purified systems), reproduced from Winnikow and Chao (W8) with permission, (a) nonoscillating nitrobenzene drop = 0.280 cm, Re = 515 steady thread-like laminar wake (b) nonoscillating m-nitrotoluene drop 4 = 0.380 cm. Re = 688 steady thread accompanied by attached toroidal vortex wake (c) oscillating nitrobenzene drop 4 = 0.380 cm. Re = 686 central thread plus axisymmetric outer vortex sheet rolled inward to give inverted bottle shape of wake (d) oscillating nitrobenzene drop = 0.454 cm. Re = 775 vortex sheet in c has broken down to form vortex rings (e) oscillating nitrobenzene drop d = 0.490 cm. Re = 804 vortex rings in d now shed asymmetrically and the drop exhibits a rocking motion.

A. Zisserman, R. Saunders and J. Caldwell, Analytic solutions for axisymmetric magnetostatic systems involving iron. IEEE Trans. Magn., 1987, 23(6), 3895-3902. 

Assuming axisymmetric flow (i.e., r and z as independent variables, neglecting any 6 variations), state (not derive) the full mass-continuity and momentum equations that describe the flow in the annulus. Identify the dependent variables. Considering the characteristics and order of the system, state and discuss a set of boundary conditions that could be used to solve the system. Clearly, some approximation is required around the steam-feed entrance to retain axisymmetry. 

I. Assuming steady, incompressible, isothermal, axisymmetric flow, write out the full system of equations that describe the conservation of mass, momentum, and species... 

Transitions from steady-state to time-dependent surface-tension-driven motions are well known also and are important in meniscus-defined crystal growth systems. For example, the experiments of Preisser et al. (51) indicate the development of an azimuthal traveling wave on the axisymmetric base flow in a small-scale floating zone. 

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