# Difference between linear and parabolic velocity profiles in Stokes flows of two fluids

For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a).

When the background flow is assumed to be parabolic (i.e., pressure-driven), I have only seen boundary integral simulations, using various kinds of kernels, see e.g. Coulliette & Pozrikidis (1998). Also, there doesn't seem to be a "simple" physical interpretation of the resulting dynamics (simple in the sense that one can infer "easily" from one configuration to another).

Maybe this is very naive, but my question is what makes the parabolic flows so different from linear flows, mathematically?

To define the problem more specifically, one can consider the typical case where the equations are

$$\partial_t {\bf u}=-\nabla p + \nu \Delta {\bf u} + {\bf f}, \quad \nabla \cdot {\bf u}= 0,$$

where $$\bf f$$ is the forcing term giving the surface tension force, supported on the droplet interface. For simplicity, one can further assume the droplet is perfectly spherical, and the density and viscosity ratios are both unity.