Moving boundaries reference frame

Traditionally the fluid mechanics of the extrusion process are summarized by the simple plate model illustrated in Fig. A7.1 and as described in Section 7.4. The motion of the screw is unchanged, but the reference frame has been moved to transform the problem to a fixed boundary problem for the observer. The flow in the rectangular channel is reduced into the x-direction flow across the channel and the z-direction flow down the channel. 

To solve the above problem, the usual method is to eliminate the moving boundary by adopting a reference frame that is fixed to the crystal-melt interface. That is, in the new reference frame and new coordinates y, the interface position is always at y=0. Hence, we let... 

When we discussed moving boundary problems, we transformed the problem into boundary-fixed reference frame and converted the moving boundary to a... 

If we attach a new reference frame (z) to the moving (stable planar) boundary, z = ,-V t. The transport equation (Fick s second law) reads in the z-system... 

This is exactly the same as the equation for a Fisher wavefront in a reference frame co-moving with the wave speed. It has solutions consistent with the boundary conditions C( —> — oc) = 0, C( —> Too) = 1 for a range of values of ao- But, as discussed in Sect. 4.2, the one with ao = 2 is dynamically preferred. Thus, the asymptotic... 

We choose a reference system, whose zero coincides with the contact point of a-f-co phases (I = 0). In the steady state, although the system moves constantly in the laboratory reference frame at certain velocity, V, the shape of the boundary y (1) and concentration profile c (I) along it remain unchanged. So, each physically small section of IPB gets an instantaneous velocity, V 1), directed along the tangent to the boundary (Figure 5.13). 

First, as mentioned in the introduction, there is little place for vacancy sinks/sources in a nanovolume. Second, if all atoms of the lattice try to move along the radial direction, this shift would immediately generate a tangential deformation and corresponding stresses. Boundaries of the shell do move but not due to lattice shift - just some atoms leave one boundary and attach to another boundary. Therefore, we can write down the continuity equations (analog of second Pick s law) in the lattice reference frame as... 

The boundary conditions in Equations 7.126 and 7.127 would be sufficient for shrinkage of a single-component shell. In our case of a binary solution, we should have additional conditions. Boundary concentrations of the main components are not fixed (we do not have an analog of Equation 7.126 for A or B), but the conservation laws are valid, of course, implying the conditions on fluxes. The sum of the three fluxes is zero in the lattice reference frame (/V +Ja +Jb = 0). Thus, two fluxes are independent. This means that one should write down the flux balance equations at both moving boundaries for one of the main components, taking into account that fluxes outside the shell are equal to zero... 

To calculate the energy dissipation close to the phase transformation front due to bulk diffusion, we search for the concentration profile close to the front [9]. In order to do this, let us put down a steady-state equation of flux balance in the element of region R (Figure 12.6a) moving at constant velocity of the boundary v, in the reference frame of the transformation boundary ... 

Additional discussion of these definitions and relationships between the different reference frames and fluxes are discussed in detail elsewhere (3,4). In principle, any reference frame for analysis may be selected however, a proper choice can reduce the mathematical difficulties. For example, in a one-dimensional diffusion process within fixed boundaries, where ideal mixing of components is a reasonable approximation, selecting the volume average frame of reference is wise because the volume average bulk velocity as defined above is zero, and hence the fluxes viewed from both a fixed reference frame and a reference frame moving... 

Substituting Eq. 7.18 into Eq. 7.3 and solving Eqs. 7.1 and 7.3 for V, 14, and Vp, the solution for the transformed boundary condition problem Is obtained, and the equations are shown by Eqs. 7.21, 7.23, and 7.26. These equations physically represent the flow due to rotation and pressure in the transformed frame of reference in Fig. 7.10. Equation 7.21 is the velocity equation for the x-direction recirculatory cross-channel flow for the observer attached to the screw, and Eq. 7.23 is the apparent velocity in the z direction for the observer attached to the moving screw. 

This boundary condition states that as the interface moves the insoluble surfactant will be redistributed by both interfacial stretching as well as by the local flow field. In a fixed frame of reference, the equation is derived by doing a mass balance over a surface element and a small time interval. Defining the surfactant concentration per unit area to be F(z, t) the equation is found to be (see Wong, Rumschitzki and Maldarelli [82]) ... 

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