# Drag hydrodynamic force for low particle Reynolds numbers

Suppose we have a fluid particle mixture, where the particle have cylindrical shape, no gravitation force, flow is laminar (let's say $$Re_f=100$$), and we want to do numerical simulations, and we are interested in hydrodynamics forces exerted from flow on a single particle. I think the hydrodynamic forces can be categorized into three families according to the particle size.

1. If the size of the particle is relatively big, and particle Reynolds number is in the same order of flow Reynolds number. The relevant forces are frictional drag and pressure drag. So, the shape of particles play significant role here. I am not interested in modelling this.

2. If the particles are massless and their size are in the same order as the size of fluid-particles, we have $$Re_p \ll 1$$. So we have Stokes flow around the particles.

3. If the particles are small and $$Re_p \approx 1$$.

For cases 2 and 3, the forces (torques) are written as a linear function of the slip velocity, something like, $$F_H= C (v_f - v_p)$$. For case 2, it is mainly assumed that the inertial-forces (forces and torques) are negligible. The questions are:

• Why and how exactly "inertia-free" assumption is justified for case 2? Do we first assume that the particles are transported with the flow ($$v_s=0$$), and conclude that the force is zero? or particle size or mass or density play a role here?
• In general, is the drag force a function of the slip velocity, or vice versa? The conflict here is that, the slip velocity depends on the drag force physically. But in the drag formula, the force depends on the slip velocity.

Please correct me or add information to the above sentences if anything was missing or showing lack of understanding. My main purpose is to deeply understand all types of forces for case 2 and 3, and the assumptions under which the forces can be neglected.

• This question belongs in Physics Stack Exchange, not here. Jan 4 at 17:50