# Newton and Stokes regime in settling velocity

Can you tell me please the differences between Newtonian and Stokes regime in settling velocities of particles in moving fluids? I know that Newtonian applies to higher Reynolds numbers. Does that mean that Newtonian regime is for turbulent flow of fluid? Or what? Or is it that Newtonian regime is when the dominant drag is the pressure drag? Or what? Can you tell me please the differences for each regime? thanks!

• In the Stokes regime everything is moving very, very, very slowly... as in mm/s or slower for air and water and objects of human scale. That's the only way to keep the viscous forces stronger than the inertial forces. May 11, 2016 at 0:36

I assume that by 'Stokes regime' you mean the drag force a object travelling through a viscous fluid experiences, in laminar flow conditions. For a perfectly spherical object and assuming flow of the fluid around the object is laminar, then acc. Stokes' law:

$$F_d=6\pi \mu Rv$$

Where $\mu$ is the dynamic viscosity of the fluid, $v$ the object's speed and $R$ its radius.

But when speed increases flow around the object becomes turbulent and Stokes' elegant expression is no longer valid.

The Reynolds Number $Re$ (a dimensionless number) is defined as:

$$Re=\frac{\rho vD}{\mu}$$

When $Re$ exceeds a critical number $Re_c$, flow ceases to be laminar and becomes turbulent and then the drag force tends to obey a different expression:

$$F_d=\rho A C_d v^2$$

Where $\rho$ is the fluid's density, $A$ the object's cross-section perpendicular to the line of motion and $C_d$ a drag coefficient depending on the shape of the object.

But I doubt if the turbulent regime was ever called 'Newton's regime', as turbulent flow really only began to become well-described and understood considerably after Newton's life time. If anything, 'Newtonian flow' often refers to laminar flow (see e.g. Newtonian viscosity).

• You can see here en.wikipedia.org/wiki/Settling the distinguish between the two regimes. I don't know if the difference is the turbulence, or the type of drag (friction/pressure) or both. Also here it gives some difference: neutrium.net/unit-operations/… I tend to think that in settling velocities, the Reynolds number determines the turbulence of the flow of the fluid ON the particle, or maybe behind it (wake formation) and NOT the flow of the fluid itself. May 11, 2016 at 1:17
• The latter url says that in settling velocity cases, the Reynolds number is not as if it is when we have fluids flowing in pipes. This is the difference I need to understand, exactly. May 11, 2016 at 1:17
• Well, I stand surprised, having never heard 'turbulent drag' referred to as 'Newtonian' (it still doesn't feel right to me and Wiki isn't always right). Yes, the $Re$ number depends on situation (flow through pipes v. flow around an object, e.g.) Nonetheless, flow is flow: flow around a stationary object or an object moving through a stationary fluid, they're the same, in terms of forces involved.
– Gert
May 11, 2016 at 1:43
• I agree that "Newtonian" is very weird. Perhaps the sedimentation sub-sub-subcommunity of fluid dynamics has adopted this terminology? I think "Rayleigh" would be a better name for the regime.
– user10851
May 11, 2016 at 8:32
• @ChrisWhite: yes, I think that too: 'Newtonian' as specifically used in sedimentation.
– Gert
May 11, 2016 at 12:59