Uniformly valid approximations for the conformational integrals resulting from gaussian closure in the hookean dumbbell model with internal viscosity^a)

We consider a computationally more efficient calculation of the conformational integrals resulting from applying a Gaussian closure to evaluate the higher moments of the end-to-end vector, Q, in the Hookean dumbbell model with internal viscosity. First, these integrals are expressed as functions of the eigenvalues of the chain conformation tensor (QQ). Subsequently, they are evaluated analytically in certain characteristic limiting cases. These results are then used to construct uniformly valid approximations for the integrals. The relative error incurred in these approximations, as measured by a comparison with the results obtained through a direct numerical evaluation, has been shown to be uniformly small, i.e., less than a few percent, in the entire parameter space. Further validation of the approximations to the integrals is provided from the results obtained for the shear viscosity and first normal stress difference as a function of shear rate in a steady, homogeneous shear flow which are in excellent agreement with those provided in the literature also using the Gaussian closure. The approximations are subsequently used to predict the response of the model to the startup of simple shear flow. Excellent agreement is found with the previously reported results using the Gaussian closure in this case as well. Most important, the reduction of computational workload through the use of the approximations to the conformational integrals makes it possible for this model to be used in more complicated flow simulations. As an illustrative example, we provide numerical results for the flow variables in a steady channel Poiseuille flow.