1. Part VI:Viscous flows, Re<<1

2. The limit case Re<<1 - Stokes flows
The other limit case Re<<1, viscous dominated flows.
This class of flows is called the Stokes flows or viscous flows.
Mantle convection, lava flows.

3. The limit case Re<<1 - Stokes flows
The other limit case Re<<1, viscous dominated flows.
This class of flows is called the Stokes flows or viscous flows.
Mantle convection, lava flows.

4. In order for this equation to have a non trivial solution, i.e. u’=0,
the pressure gradient must be
This class of flows is called the Stokes flows or viscous flows.
Mantle convection, lava flows.

5. Viscous flows, Re<<1

6. We consider the case Re small for an incompressible fluid.
The flow in a pipe, Poiseuille flow.
V=0 R
Pin
Pout
V=0
We consider the steady flow driven by a pressure difference in a pipe of radius R and length L>>R
We consider the case Re small for an incompressible fluid.

7. R The flow in a pipe, Poiseuille flow. V=0 Pout Pin Oz r V=0 Re small
L>>R
Circular
The flow is mostly along the pipe
Viscous forces dominates over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz

8. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz
Using a cylindrical coordinate system the different operator can be expressed as:

9. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz

10. The flow in a pipe, Poiseuille flow.
Viscous forces dominates over inertia
Variations along the pipe<<variations in r
Flow in invariant by rotation around Ox 2

11. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz
Using a cylindrical coordinate system the different operator can be expressed as:

12. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz

13. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz
The velocity must remain finite at r=0
The velocity vanishes at r=R

14. The flow in a pipe, Poiseuille flow.
Viscous forces dominate over inertia
Flow is invariant along Z
Flow in invariant by rotation around Oz

15. The flow in a pipe, Poiseuille flow.

16. What is the volume flux Q at A2 for a given h and L ?

17. Assumptions:
Incompressibility
Uniform density and viscosity
Assuming a Poiseuille velocity profile in the tube between A1 and A2, the velocity writes:
with
In virtue of mass conservation and incompressibility, the volume flux at A2 can thus be written:

18. We shall now determine Uo.
To that aim, we will apply Bernoulli between A1 and a point A0 chosen such that it is along the axis of the tube and far enough from the entrance to assume U(A0)=0.
Since the velocity is invariant in the tube in the x direction, along the axis, the velocity is U1=Uo.
A2 being in the air, P2=Patm, hence
The positive root of this equation is:

19. prediction of the flow rate Q as a function of the tube length for R=3mm, h=10cm
Using the experimental device I showed you, I measured 200ml in 330s, Q~0.61ml/s
Can I use this as a validation and simply apply my results to any tubes (L,R) and any column of water (h) ?
NO: I would need to do more experiments, with different lengths of tube, radius and h.
Furthermore, my prediction assumes that I can use Poiseuille in the tube, Re<<1 or inertia is negligible compare to viscous forces. I may want to test up to which Re this prediction remains valid. In other words, I would need to establish the validity domain of this theory.

20. Let’s first estimate Re for various length of the tube I have, using my theory. Since I assume inertia to be negligible, Re has to be small.
A naive approach consist in using a single length scale L or R to calculate Re:
Defining the Reynolds number as:
At 3m, Re~350 !!!
Defining the Reynolds number as:
At 3m, Re~6e5 !!!

21. THIS SEEMS TO BE IN CONTRADICTION WITH MY ASSUMPTION !!
Let’s first estimate Re for various length of the tube I have, using my theory. Since I assume inertia to be negligible, Re has to be small.
A naive approach consist in using a single length scale L or R to calculate Re:
Defining the Reynolds number as:
THIS SEEMS TO BE IN CONTRADICTION WITH MY ASSUMPTION !!
At 3m, Re~350 !!!
Defining the Reynolds number as:
At 3m, Re~6e5 !!!

22. Let’s first estimate Re for various length of the tube I have, using my theory. Since I assume inertia to be negligible, Re has to be small.
A naive approach consist in using a single length scale L or R to calculate Re:
Defining the Reynolds number as:
If the output suggests Re>>1, I may revise my assumption.
Then I could apply Bernoulli along the pipe, but since the pressure drops, the velocity must increase.
However, in vertu of conservation of mass, in a constant radius tube, the velocity must remain constant, if the fluid is incompressible.
So this also lead to a contradiction !!!
At 3m, Re~350 !!!
Defining the Reynolds number as:
At 3m, Re~6e5 !!!

23. The apparent contradiction comes from our single length scale assumption. Indeed, inertia involves a length scale in the direction of the flow while viscosity is physically best represented using a radial length scale, the distance over which the velocity drop from U to 0.
Advection of the velocity is in the direction of the flow, thus the typical length scale of the non-linear term is the length of the tube:
Diffusion of momentum occurs in the radial direction, thus the typical length scale for the diffusion term is the radius
The Reynolds number physical interpretation is the ratio of advection to diffusion:

24. Using a length scale L for the advection and R for the diffusion,
Using our expression of Re derived from the physical meaning of Re, it is in fact small down to L~10cm.
There is no hard line defining Re small or large, it depends on the accuracy you want to reach. Generaly Re< is considered small, but you can measure very fine details in the flow structure and amplitude, you may want to reintroduce some inertia.

25. We can now estimate the pressure drop in the tube:
V0,P0
V1,P1
V2,P2 h0 h1 h2
Pressure along the tube (in cm of H2O)

26. Pressure losses derived via dimensional analysish0 h1 h2
V0,P0
V1,P1
V2,P2
According to the mass conservation, V2=V1.

27. The viscous dissipation reduces the potential energy
V1,P1
V2,P2
Vi, Pi
Vi, Patm

28. R The viscous dissipation reduces the potential energy V1,P1 V2,P2 h1
Vi, Pi
Vi, Patm
V=0 R U
V=0

29. The viscous dissipation reduces the potential energy
In the engineering literature it is presented in a form called the Darcy–Weisbach equation:
where the Reynolds number has been calculated using the D the diameter of the tube.
is the Darcy friction coefficient, for a laminar pipe (Re<2000) flow in a circular pipe it is
When the flow becomes turbulent, 4000<Re<10 000, a good approximation of the friction coefficient is given by:

30. L
We can estimate the distance over which the diffusion of the initial excess of momentum will take place:
At the entrance the flow at the boundary does not match the wall velocity U=0. The typical time it takes to diffuse from the wall to the center is govern by:

31. Summary: Viscous flows occurs at Re<~1
The N-S reduces to the Stokes equation:
In the absence of source of motion, they decay over a characteristic time scale
or, over a given small Dt, the diffusion occurs on a typical length scale:
The last two quantities are characteristic of diffusion mechanism, a random walk process