Mass Flux arising from Velocity Gradients Rouse Chains in an Isothermal Fluid

2 The Contribution j (2) to the Mass Flux Arising from Velocity Gradients Rouse Chains in an Isothermal Fluid 

We now return to Eq. (15.1) and consider the expression for j (2) specifically for the Rouse chain model in a fluid that is at constant temperature and in which there are no external forces. For this model we can develop the velocity difference appearing in j,(2) as follows, by using Eq. (13.14) with a equal to zero AEFRXToV  

When Eq. (15.8) is substituted into the expression for i,(2), the term in Eq. (15.8) containing Vlnn does not contribute. The remaining two terms can be developed as follows AEFRXToV  

To get the first expression above we used Eq. (13.11), and to get the second expression we used Eq. 2.10 as well as Eq. 13.12. To evaluate the integrals in the second expression, Eqs. (13.26) to (13.31) were used, and to get the last line we used the fact that the flow is homogeneous and the definition of the time constants given just after Eq. (13.10). The symbol m is used for the mass of a single bead (all beads being identical), and the Ci are the eigenvalues of the Kramers matrix. Then when we use Eqs. (13.10) and (6.7), and also the first term in the series in Eq. (B.21), we finally get for the sum of the first two terms in the Taylor series expansion of the mass flux AEFRXToV  

The second term gives the influence of the flow field on the diffusivity tensor. 

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