Axisymmetric formulations

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... 

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

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

The flow under consideration is mathematically more simple in a plane duct than in an axisymmetric round tube. However, no readily solution for smooth walls, i.e. in the case where the EPR is absent, is given in the literature. The formulations and solutions of the problem of a pulsating viscous flow to be taken for a comparison have been presented only for round tubes. The known solutions employ mainly the Laplace transformation and small-parameter methods. The present investigation follows the standard technique of mathematical physics of finding the periodic solutions described particularly by Schlichting [566] with references to some original sources. Their solution for smooth walls will serve for a comparison. 

Integral Equation Formulations in Two Dimensions, Three-dimensional and Axisymmetric Problems . 

In what follows the magnetoviscosity phenomenon is analyzed by formulating the local ferrohydrodynamic model, the upscaled volume-average model in porous media with the closure problem, and solution and discussion of a simplified zero-order steady-state isothermal incompressible axisymmetric model for non-Darcy-Forchheimer flow of a Newtonian ferrofluid in a porous medium of the... 

There has been some development in the numerical modeling of the sheet formation from swirl nozzles. A fully nonlinear model using an axisymmetric boundary element formulation has been developed for simulating the free surface shape and spray formed by simplex/pressure swirl atomizers [30, 32]. A linear instability analysis by Ponstein has been used to predict the number of droplets formed from each ring-shaped ligament shed from the parent surface. 

ABSTRACT In this study the disturbance factor in the general Hoek-Brown (HB) criterion is considered to be a gradually-attenuated variable from the excavation surface to the deep surrounding rocks. The elasto-plastic analytical solution is formulated for an axisymmetrical cavern model in which there exist a supported pressure at the wall of tunnel and a far-field pressure at infinity. The presented analytical model can well reflect the disturbance of the HB rock mass triggered by drilling and blasting excavation. 

However, the T)-value is often fixed to be a constant in the existing analysis for the underground engineering projects in the HB rock mass (Chen Tonon 2011, Fraldi Guarracino 2010, Li et al. 2009, Park Kim 2006, Shen et al. 2010, Zhong et al. 2009, Zhou Li 2011). In this study D is treated as a variable. To formulate the elasto-plastic analysis solution for an axisymmetrical cavern, a linear function is chosen to quantitatively describe D. Compared with the elastic perfectly-plastic and elastic-brittle-plastic results, the present analysis can objectively reflect the excavation disturbance of the surrounding rocks. 

These equations are formulated for the simplified axisymmetric case in which Equation 3.1 holds true. The first coordinate axis is directed along mean relative fluid velocity u, and two other axes are arbitrarily chosen in the plane normal to u. The averaged squares that appear in Equation 7.7 must be calculated in accordance with the following general rule [35] ... 

One-dimensional models for inviscid, incompressible, axisymmetric, armular liquid jets falling under gravity have been obtained by means of methods of regular perturbations for slender or long jets, integral formulations, Taylor s series expansions, weighted residuals, and variational principles [27, 47]. 

Kim, J.K., Koh, H.M., Kwon, K.J. Yi, J.S. 2000. A three-dimensional transmitting boundary formulated in Cartesian co-ordinate system for the dynamics of non-axisymmetric foundations. Earthquake Engineering and Structural Dynamics 29 1527-1546. 

The formulation of models based on this approach basically involves setting up the equations of motion in continuity, together with appropriate boundary conditions. The general forms of these equations used by various investigators [44,48] are given below using cylindrical polar coordinates, and assuming axisymmetric conditions. 

With no density differences in the fluid, there will be no source term for the radial and tangential velocities. If the inlet and boundary conditions are axisymmetric, there will be no variation in the 0 direction. Removing the tangential variations will allow us to reduce the simulation fi om 3D to 2D. The symmetric boundary condition at the center of the tube can be formulated as... 

According to the theory described above, the propagation of the overturned state of amphiphilic molecules along the surface under the droplet is described by Equation 5.197, and the propagation beyond the droplet by Equation 5.194. It is now required to formulate the boundary conditions for the probability, p(t, r), at the boundary of the spreading axisymmetric droplet, i.e., at r = r it) (Figure 5.31). 

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