About the steady state in Stokes flows

Question

Note: I have read similar questions, but since it is not totally what I want, and they are old, I prefered to write a new one.

So, if one aims to work with Navier-Stokes at microscales, the equation that is used is the Stokes equation, where we have neglected the inertia part over the viscous force. This happens when Re << 1, an even one could work with Oseen equation for a better aproximation. Now, my question is regarding to the approximation that gives the Stokes flow. Either in Stokes or Oseen, you neglect/lineralize only: $v\cdot \nabla v$, but not $\partial_t v$. The last is omitted because the flow is steady. And the question is: physically, which is the interpretation of both terms and significance of steady?

I imagine that in a fixed volume V of fluid at a position x, the speed of the fluid will evolve its speed because it is changing by itself (partial), and because it is recieving fluid from outside (convection, inertia). Is that? if you have, say an alive bacterium, that is moving inside the fluid, it can be steady because the speed of fluid around the bacterium changes only by convection? Could it be also a possible scenario where the partial is not 0 at this scale?

I'm basically interested from a physical point of view, because I want to understand the main concepts, not just the math itself.

Answer 1

The low $Re$ limit is very removed from our daily experience, so physical intuition may be tough to grasp here. As you mention, the mathematics mandate that the momentum equation transforms into $$\nabla p = \mu \nabla^2 \mathbf{u}$$ as a first order approximation. That is not to say that $\partial_t \mathbf{u}$ or $\mathbf{u} \cdot \nabla \mathbf{u}$ vanish completely, but they are much less dynamically relevant.

Physically, the low $Re$ regime is governed by dissipation. As you suggest, a bacterium can try to move around the fluid, but its momentum will constantly be dissipating into the field. (They would never think to suggest Newton's first law is being respected.) In the first order approximation that $Re \ll 1,$ any energy the bacterium tries to put into the system decays instantaneously.

Answer 2

Both terms $v\cdot\nabla v$ and $\partial_t v$ have the same order of magnitude. In fact, the former comes from the use of the Eulerian framework whereas in the Lagrangian framework only the latter exists. If the length scale of your system is $d$ and the velocity is of order $V$, then $\partial_t$ should be of order $V/d\sim V \nabla$. Now, does it mean that the flow can only be steady? Not necessarily since in many cases the momentum equation is coupled to another one which makes the forces applied to the fluid, body force or surface force, change with time. Stokes' equation means that the flow will adjust instantaneously to the change of applied force.

In the example of the bacterium (or any slow swimming body), it can change direction or speed by changing the forces applied to its surrounding fluid. In the $Re\ll 1$ limit, the Stokes equation applies and the flow adjust instantaneously. In that sense, it is steady. But if the bacterium changes velocity, the flow is seen to change by the external observer. In that case, the change for the fluid comes from varying the boundaries.

Another example is in convection at infinite Prandtl number, like that happening in the Earth mantle. Stokes equation applies with the addition of a buoyancy force proportional to the temperature. The temperature evolves with time owing to the energy balance equation which generally leads to unsteadiness, at least at high Rayleigh number. But at each instant, the momentum equation is steady, with a balance between the pressure, buoyancy and viscous forces.