Solvent Continuum and Hydrodynamic Interaction

When a polymer chain is dispersed in a solvent, the characteristic size and time for describing the collective motion of the polymer molecule are many orders of magnirnde large in comparison with those for individual solvent molecules. As a result, the solvent background can be assumed to be a hydrodynamic continuum. Let us consider a small volume element around the spatial location r, which is big enough to consider the solvent as a continuum, at time t. The time dependence of the velocity field v(r, t) at r and t is obtained from the Newton s second law of motion, namely the rate of change of momentum of a fluid element is equal to the net force acting on it, as (Landau and Lifshitz 1959) 

As the experimental situations of our current interest generally lie in the weak velocity fields and the inertial forces become negligible for relatively long timescales relevant to polymer molecules, the first two terms on the left-hand side of the above equation may be ignored resulting in the zero-frequency linearized Navier-Stokes equation (Landau and Lifshitz 1959), 

In addition, we assume that the fluid is incompressible. This condition is given as 

The pressure field in Equation 7.2 can be eliminated by using Equation 7.3, so that the velocity field can be expressed in terms of any externally imposed force 

The fundamental property of the background fluid is how it transmits a force field F(r, t) at the location r to another location r. Combining Equations 7.2 and 7.3, the velocity field at r can be written in terms of F as 

---