Axisymmetric coordinate system

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. 

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. 

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... 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

We first derive the kinematics of the deformation. The flow situation is shown in Fig. 14.14. Coordinate z is the vertical distance in the center of the axisymmetric bubble with the film emerging from the die at z = 0. The radius of the bubble R and its thickness 8 are a function of z. We chose a coordinate system C, embedded in the inner surface of the bubble. We discussed extensional flows in Section 3.1 where we defined the velocity field of extensional flows as... 

The analysis of the preceding section was carried out by use of a spherical coordinate system, but the majority of the results are valid for an axisymmetric body of arbitrary shape. The necessity to specify a particular particle geometry occurs only when we apply boundary conditions on the particle surface (that is, when we evaluate the coefficients Cn and Dn in the spherical coordinate form of the solution). For this purpose, an exact solution requires that the body surface be a coordinate surface in the coordinate system that is used, and this effectively restricts the application of (7-149) to streaming flow past spherical bodies, which may be solid, as subsequently considered, or spherical bubbles or drops, as considered in section H. 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

The consequences of this change can be explored first in rather general terms without the need for reference to a specific problem. To see the general situation, it is sufficient to think in terms of a local 2D Cartesian coordinate system. The resulting analysis will be apphed directly only to a 2D problem. However, as we have seen in the preceding sections, the same qualitative result will be obtained for axisymmetric or even fully 3D problems. In the simplest view, the only difference between transport across a fluid interface and previous problems is in the Taylor series approximations for the velocity components (u, v). For convenience we assume that the local coordinate system is defined so the interface corresponds to y = 0. Because the first nonzero term for the tangential component is the shp velocity, the Taylor series approximation then takes the form,... 

Problem 9-17. Heat Transfer From an Ellipsoid of Revolution at Pe S> 1. In a classic paper, Payne and Pell. J. Fluid Meek 7, 529(1960)] presented a general solution scheme for axisymmetric creeping-flow problems. Among the specific examples that they considered was the uniform, axisymmetric flow past prolate and oblate ellipsoids of revolution (spheroids). This solution was obtained with prolate and oblate ellipsoidal coordinate systems, respectively. 

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. 

FIGURE 4.3 Surface coordinate systems for two-dimensional (a), axisymmetric (6), and three-dimensional (c) natural convection flows (d) is a schematic of the temperature distribution near the boundary. 

Potential Flow around a Gas Bubble Via the Scalar Velocity Potential, An incompressible fluid with constant approach velocity (i.e., S Vapproach) flows upward past a stationary nondeformable gas bubble of radius R. This two-dimensional flow is axisymmetric about the scalar velocity potential spherical coordinates because this coordinate system provides the best match with the macroscopic boundary at r = / . The appropriate partial differential equation for is... 

A further simplification can be achieved if we make the assumption that the solution contains a completely dissociated symmetric dissolved substance (salt), so that z+ = —Z- = z. The current is axisymmetrical, and the velocity has only a single longitudinal component, therefore in the cylindrical coordinate system (x,r), Eq. (7.74) reduces to ... 

Since we model the essential part of the event by a rod with an axisymmetric strain field the velocity field in the event can be given in canonical form for a cylindrical coordinate system as... 

Here the reference pressures p Q and have been chosen for convraiiraice as those at the drop apex O of Figure 1.5. Also, a cylindrical coordinate system with its origin at O has been chosen as shown. If the drop is axisymmetric, the radius of ciurature h at O is the same for all orientations, and Equation 1.22 requires that... 

The motion is assumed to be axisymmetric and a spherical polar coordinate system r, 0 is used. All variables are independent of 0 therefore, and 9 = 0 is taken to represent the front stagnation point which is the first point the oncoming stream meets as it approaches the bubble. The assumption of axisymmetry (i.e. a spherical bubble) is a reasonable one as long as inertial and viscous forces are small relative to surface tension forces. This requires the Weber number, Wq = and the capillary number, Ca = to be small. Here a is a representative value for the surface tension coefficient. Both these conditions are usually met in the applications we are considering here because bubble sizes and rise velocities are small and surface tension is relatively large. 

Equation (39) is written in the axisymmetric spherical coordinate system of interest here and the obvious nondimensionalizations are made, unless otherwise stated. For more general conditions the reader is referred to Stone [68]. The parameter A is a measure of bulk concentration and it plays a central role in our theoretical findings described later. The second condition, i.e. the balance of the kinetic flux with the local surfactant adsorption rate takes the following dimensionless form... 

To derive the shape equations, consider the system depicted in Fig. 9. The origin of our coordinate system is chosen at the apex of the droplet with the direction of the coordinate z chosen opposite to the direction of the gravitational field. Because the droplet is axisymmetric around the z-axis, the shape of the droplet is completely determined by the function z(r), with r the radial distance to the z-axis (see Fig. 9). 

In what follows, an example of using the Finite Volume Method (FVM) to simulate the steady state motion of a spherical bubble in a surfactant solution will be discussed. Cuenot et al. [101] presented results of the numerical simulation by FVM method of a spherical bubble with a constant velocity moving through a dilute surfactant solution. The surfactant concentration in the bulk and on the surface were simultaneously calculated in a steady uniform velocity. The calculation was performed in a three-dimensional coordinate system. The problem was simplified to a two-dimensional problem because the flow around the bubble was assumed to be axisymmetric. 

In most cases a mechanical approach is more adequate. This is based on the concept that in mechanical equilibrium at each point on the interface, the curvature is adjusted such that the difference in the pressures between the two phases is balanced by the capillary pressure. This approach is particularly fruitful when applied to axisymmetric menisci, like drops and bubbles. In this ease, assuming a Cartesian coordinate system with the origin at the drop apex, O, and the vertical axes, z, in the symmetry axis and directed towards the interior of the drop, at any point, S, of the interface we have... 

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