Rotne-Prager interaction

Evaluation of d can be carried out at the simplest possible level assuming the model of noninteracting spheres in a fluid, or one can include hydrodynamic interactions, for example, based on the Rotne-Prager (RP) approach [80, 81], which ensures a satisfactory albeit not too cumbersome treatment of sphere-sphere hydrodynamic interactions. The resulting elements of D depend upon a purely geometric tensorial component D and the translational diffusion coefficient for an isolated sphere Do, that is. 

This accounts approximately for the disturbance at bead T resulting from the motion of bead 2 a more accurate description of hydrodynamic interaction is given by the Rotne-Prager-Yamakawa tensor. When the above modification of the theory is made, equation (55) is changed to... 

If the Rotne-Prager-Yamakawa hydrodynamic interaction is included in the rigid dumbbell model, it is found that the equation for the distribution function is identical to equation (83) except that the time constant A must be replaced by... 

Rotne.J., Prager.S. Variational treatment of hydrodynamic interaction in polymers. J. Chem Phys. 50,4831-4837 (1969). 

This condition is guaranteed for the correct mobility matrix. However, the mobility matrix given by eqn (4.40) is an approximate one, and does not satisfy the inequality (4.1 ) in a certain configuration in which the beads are too close to each other. An improved formula which guarantees the inequality is proposed by Rotne and Prager. However, this correction is irrelevant for the asymptotic behaviour of N 1, which is determined by the hydrodynamic interaction between beads far apart from each other. Thus we shdl use eqn (4.40) for H, . 

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