Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $$a$$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces applied to them. Similarly, the force $$\boldsymbol{F}_j$$ applied on one particle contributes to the velocity $$\boldsymbol{v}_i$$ of the other through a term which is linear in $$\boldsymbol{F}_j$$. I write this as follows

$$\boldsymbol{v}_1=(6\pi a)^{-1}\boldsymbol{F}_{1}+\boldsymbol{H}\left(r_{12}\right)\cdot\boldsymbol{F}_{2}$$ $$\boldsymbol{v}_2=(6\pi a)^{-1}\boldsymbol{F}_{2}+\boldsymbol{H}\left(r_{21}\right)\cdot\boldsymbol{F}_{1}$$

where $$H$$ is the hydrodynamic interaction tensor that depends on the relative positions $$\boldsymbol{r}_{ij}$$ of the two spheres ($$i=1,2$$).

Here is my question: if I wanted to look at the limit of far field, in principle I would assume that $$a\ll r_{ij}$$ and look at what happens to the equations. This can be done formally by nondimensionalising with respect to the typical distance $$\ell$$ such that $$r_{ij}\sim \ell$$, define $$\epsilon=\frac{a}{\ell}$$ and take the limit $$\epsilon\rightarrow 0$$. However, this seems to present problems, because the friction terms are proportional to $$a^{-1}$$, so would diverge in such an expansion. What am I missing? If the divergence is indeed physically relevant, what is its meaning? How can one deal with it in order to study the limit of far field?

• By '$Re=0$ flow', you mean a stationary fluid, with no bulk velocity? – Time4Tea Feb 14 at 16:01

When distance between spheres is large compared to their size, the velocity of each sphere is predominantly determined by the balance between drag force and external force acting on it, and the inter-sphere interaction force is negligible. That's why the external force term is "blowing up" in relation to interaction force when $$\epsilon\to 0$$, and if the non-dimensionalization is done correctly then the drag force term also blows up at the same rate as the external force term (so there is balance between the two as $$\epsilon\to0$$). An equivalent (and perhaps less repugnant) way to say it is that interaction force is becoming negligible in relation to the other two forces on the body. Mathematically it means that to leading order in $$\epsilon$$ the velocity of each sphere is determined by the balance between drag force and external force, and inter-sphere interaction appears only at higher order.