Linear Analysis of the Basic Semi Solid Model

The linear analysis is the same as we discussed at length in Chapter 2 and is now straightforward. We linearize (5.41) and (5.42) about the nonzero steady state (u*, v*) given by

It is algebraically simpler in what follows to use general forms for the terms in the model equations (5.41) and (5.42): w is in effect another parameter here. We thus consider d u o

— = du v2u - aV ■ [ux(v)Vv] + f (u, v) (5.44) d t dv o

— = dvV2v + g(u, v), (5.45) d t which on comparison with (5.41) and (5.42) define

We now linearise the system about the steady state in the usual way by setting u u = u* + sui, v = v* + evi,

where 0 < e ^ 1. Substituting these into (5.44) and (5.45) we get the linearized equations

since f * = 0 and g* = 0. Here the superscript * denotes evaluation at the steady state. We write the linear system in the vector form

where the matrices A and D are defined by

We now look for solutions in the usual way by setting

where k is the wavevector, the c's constants and the dispersion relation X(k), giving the growth rate, is to be determined. Substituting this into the matrix equation gives

ÀI + D| k I2 - A

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