Axisymmetry conditions

Generally, 3-D models are essential for calculating the radial distributions of spray mass, spray enthalpy, and microstructural characteristics. In some applications, axisymmetry conditions may be assumed, so that 2-D models are adequate. Similarly to normal liquid sprays, the momentum, heat and mass transfer processes between atomization gas and metal droplets may be treated using either an Eulerian or a Lagrangian approach. 

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. 

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

Up to the point of applying the boundary conditions at the surface of the drop, this problem is actually identical to the problem of a rising drop under the action of buoyancy with a clean fluid interface. The solutions, satisfying axisymmetry, and the uniform flow at infinity were given previously as (7 213) and (7 215). Now, at the drop interface, the normal velocity vanishes ... 

Assuming that all the variables in the physical problem are independent of the azimuthal coordinate (axisymmetry), the projections of Table 11.3 model equations along the three coordinate axes are given in Table 11.4 together with the boundary conditions chosen at the four peripheral boundaries of the porous medium z = 0, z = L, r=0,r= R. We will assume without proof, uniqueness of solution for the system of equations describing this ferrohydrodynamic model. 

The actual field consolidation of soil around each drain is close to an axial symmetry, which should be converted into a plane strain condition for 2D finite element program. Hird et al. (1995) proposed a geometry and permeability matching equation from axisymmetry to plane strain as follows ... 

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. 

Unless otherwise stated it is now assumed that, in addition to the condition of axisymmetry, planes normal to the axial direction will remain plane (plane strain) and all the properties are isotropic. Thus there will be only three components of stress, tr, (Tj, and cr, which will be referred to generally as a. Both the fuel and the cladding material are subjected to a number of complex interacting physical phenomena thermal gradients, elastic strains, plastic flow, creep deformation, and, most important of all, the volume changes induced by radiation. 

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