A nonzero constriction rate

Up to this point, the interaction of chemistry and deformation—as embodied in eqn. (15.2)—has been discussed wholly for the condition of no change in material dimensions, c (total) = 0. To make the discussion more widely applicable, and as a step toward considering interfaces, we now consider the possibility that has a nonzero value, while remaining uniform at all points in the sample. Groundwork was laid in Appendix 12B, where the idea was introduced of a material whose viscosity varies along the x-direction as Nq + n sin jx. 

In place of eqn. (14.2) where the mean value of t is equal to we propose 

Equations (15.1b) and (15.1c) are modified in a similar way then if the prescribed uniform value of is Cq, and we seek the condition Cq = + he + xe, the total left-hand side becomes Cq and the total 

It follows from eqn. (15.5) that for the strain rate to be uniform with respect to X, with magnitude Cq, we need first Bq = S0/6N0 and, second, the sum of all the terms containing the factor sin jx must be zero. They are as already shown in eqn. (15.2a) with the addition of the extra term 

A situation of particular interest is one where the supposed fluctuation n in viscosity is linked to the fluctuation c in composition from the nature of N, such a link seems unavoidable. If we express the link by writing n = kcNo, eqn. (15.2b) becomes 

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The particular complication just discussed should not obscure the main thrust of the chapter so far, which is that, to a large extent, ideas from Chapters 14 and 15 carry over to situations where an interface is present with little change. But ideas related to a nonzero constriction rate have yet to be considered to open this topic we go to part (ii) of the program on page 152 using Chapter 13 as basis or starting point. 

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