Time, dimensionless, various stages

Figure 4.18. Profiles of concentration of the diffusing substance developed through the bi-layer package in contact w ith the liquid food of finite volume with a = 55.3, for various values of the dimensionless time D-t/L, with R = 5. The initial concentration at time 0 is that attained at the end of the co-extrusion stage.

While there are only small thermal fluctuations in equilibrium, shear can enlarge them on small spatial scales (> the mesh size I in the following simulations) in an early stage t < yxo). In later times the fluctuations on various spatial scales appear and the system tends to a strongly fluctuating, dynamical steady state. Hence, random-source terms in the dynamic equations are indispensable at the onset. In our simulations we have therefore added the Gaussian random source terms,/y(r, t) in (3), V ffR(r, t) in (8) or (12), and 0(r, t) in (13) or (15), which are related to l/ j, rjQ, and C, respectively, to satisfy the fluctuation-dissipation relations. In our model, however, even after the dynamic equations are made dimensionless, these random source terms are still proportional to a common parameter e = which has not yet been specified, d be-... 

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