Axisymmetric domains

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

In the foregoing sections the discussion of axisymmetric stagnation flows has concerned four subcases of the same general problem—semi-infinite or finite domains and rotation or no rotation. The intent of this section is to focus attention on the fact that with suitable choices of length and velocity scales these problems can be collapsed to a common representation. Generally speaking, the length scale is called L and the velocity scale is called U. Thus nondimensional variables are defined as... 

The discussion in this chapter has been dominated by axisymmetric flow. However, there is analogous behavior for planar stagnation flow in two-dimensional cartesian coordinates. In fact Hiemenz s original work was for planar stagnation flow in a semi-infinite region. The planar flow illustrated in Fig. 6.18 is for a finite domain. 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

We now transform the governing equations in cylindrical coordinates into polar coordinates. Since the motion is axisymmetric, the transformation from (r, z) to (R, 6), as shown in Fig. 3.1, is analogous to the transformation from Cartesian coordinates (x, y) to cylindrical coordinates (r, 0) in a two-dimensional domain. The stream function is related to the velocity components in polar coordinates by... 

Let us consider the flow of a highly entangled polymer in an axisymmetrical die of diameter D and length L. In the case of polymers, when the shear rates are sufficiently high or within a certain shear rate domain, variations in viscosity as a function of shear rate may be represented by a power law [30] ... 

Due to the axisymmetric nature of the problem, only a section of both domains was considered in the analysis (see Fig.2). Two different cases were investigated (i) the problem with flat rigid base (Fig. 2-left), thus simulating the drop impact of the bottle without its original base, and (11) the problem with different flexible base shapes (flat and curved, as shown in Flg.2-right) to investigate the base-shape effect. In the latter case, the bottle was allowed to bounce after the Impact. 

The local 3D computational grid in the domains in the central area of the furnace is established as shown in Fig. 4.10 [21]. The other block regions that are away from the central area of the furnace, in which the configuration and heat transfer are axisymmetric, are discretized in a 2D way. The 2D computational grid in other block regions of the furnace is the same as that used in the 2D global model as shown in Fig. 4.1. 

It is also worth noting that the volume flux of fluid across a curve joining any two arbitrary points, say, P and Q, in the flow domain is directly related to the difference in magnitude of the streamfbnction at these two points. For simplicity, let us show that this is true for 2D motions. The axisymmetric case follows in a very similar way. Thus we consider a curve, as shown in Fig. 7-4, that passes through the two points P and Q but is otherwise arbitrary. The volume flux of fluid across this curve, per unit length in the third direction, is... 

An analytic solution for such a problem can thus be sought as a superposition of separable solutions of this equation in any orthogonal curvilinear coordinate system. The most convenient coordinate system for a particular problem is dictated by the geometry of the boundaries. As a general rule, at least one of the flow boundaries should coincide with a coordinate surface. Thus, if we consider an axisymmetric coordinate system (f, rj, ), then either f = const or r] = const should correspond to one of the boundaries of the flow domain. 

Figure 7-11. A schematic representation of the domain for a uniform flow past an arbitrary, axisymmetric body. For the case of a solid sphere, this is Stokes problem.

FIG. 2 Principles of SECMID using H+ as a model adsorbate. Schematic of the transport processes in the tip/substrate domain for a reversible adsorption/desorption process at the substrate following the application of a potential step to the tip UME where the reduction of H+ is diffusion-controlled. The coordinate system and notation for the axisymmetric cylindrical geometry is also shown. Note that the diagram is not to scale as the tip/substrate separation is typically <0.01 rs. 

The radiation characteristics of axisymmetric spheroidal microorganisms, such a C. reinhardtii (Fig. lA), with major and minor chameten a and b can be predicted numerically using (i) the T-matrix method (Waterman, 1965 Mackowski, 1994 Mishchenko et al., 2002, 1995), (ii) the discrete-dipole approximation (Draine, 1988), and (iii) the finite-difference time-domain method (Liou, 2002). Most often, however, they have been approximated as homogeneous spheres with some equivalent radius r and some effective complex index of refraction nix = n +ikx (Pettier et al., 2005 Berbero u et al., 2007 Dauchet et al., 2015), as discussed in Section 3.6.1. 

When a planar domain at the centre of the channel is considered (i.e. n plane in Fig. 5.9) and given the axisymmetric assumption, Eq. (5.4.1) is stiU valid with r simply substituted by y. Alternatively a 2-D definition of the non-dimensional circulation time can be obtained by rewriting Eq. (5.4.1), valid in the observation xy plane n (Fig. 5.9) ... 

The Mie theory [1] and the T-matrix method [4] are very efficient for (multilayered) spheres and axisymmetric particles (with moderate aspect ratios), respectively. Several methods, applicable to particles of arbitrary shapes, have been used in plasmonic simulations the boundary element method (BEM) [5, 6], the DDA [7-9], the finite-difference time-domain method (FDTD) [10, 11], the finite element method (FEM] [12,13], the finite integration technique (FIT) [14] and the null-field method with discrete sources (NFM-DS) [15,16]. There is also quasi-static approximation for spheroids [12], but it is not discussed here. 

Fig. 18.5 Computational domain and initial mesh for simulating the formation and disintegration of conical swirling liquid sheets based on VOF approach, liquid injector highlighted in red (left) 3D geometry (right) 2D axisymmetric geometry...

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