1. OMA rational and results

2. Outline Introduction Equations non-dimensionalization
OMA: Order of Magnitude Analysis
Boundary layer
Regime planes diagrams

3. Introduction
OMA is a useful technique due to Luigi G. Napolitano for the QUICK ESTIMATION of the order of magnitude of relevant quantities.
An art that depends on physical insights of problem at hand;
OMA, if done properly, will provide information about the experiment a priori, i.e. in terms only of data problems;
OMA simplifies the problem and focus on important physical phenomena;
As a check to your solution when you can simplify the physics;
Example are: Marangoni- Poiseulle flow of two immiscible fluids, Protein cristal growth, g-jitter impact, Surface and Buoyancy driven free convection;

4. Navier Stokes equations (1)
Main assumptions: Newtonian liquids, incompressible flows, local equilibrium, interface parallel to gravity and Marangoni.

5. Navier Stokes equations (2)
For hydrostatic conditions:

6. Navier Stokes - adimensionalization
We proceed therefore to adimensionalize the equations choosing appropriate reference values such that each dimensionless variables varies in the range [0,1]:
One of the objective of the Order of Magnitude Analysis is to find out:

7. Adimensionalization – momentum (1)
Since diffusion is always present, we divide by its coefficient to obtain a diffusion term of order of magnitude one:

8. Adimensionalization – momentum (2)
We can define the following characteristic velocities as:
Unsteady velocity
Dissipative velocity
Buoyancy velocity

9. Adimensionalization – momentum (3)
and dimensionless numbers:
Strouhal number
Reynolds number
Grashof number

10. Adimensionalization – energy
Energy diffusion velocity
Prandtl number

11. Adimensionalization – species
Concentration diffusion velocity
Schmidt number

12. Reference equations
The non-dimensional equation can be rewritten according to:

13. Characteristic velocities and times
Time steady-state
May be used to choose the appropriate micro-gravity platform

14. Summary of dimensionless numbers in terms of characteristic speeds

15. Boundary layer Presence of a Boundary-Layer
Mass conservation is not affected by the presence of boudary layer

16. Boundary layer - momentum
The presence of boudary layer influences the ORDERS of MAGNITUDE

17. Boundary layer (1)
In presence of a Boundary-Layer reference equations are:
We have to find out what is the LEADING EQUATION (LE) wich is the one relative to the slowest process.

18. Boundary layer (2) Criteria for leading transport equation: obviously
Slowest process
Slowest diffusive velocities
Pr, Sc <1
LE: Momentum)
Pr>1, Sc <1
LE: Energy
Pr<1, Sc >1
LE: Species
Largest wins

19. Boundary layer – tangential stress
Let us to consider the tangential stresses at the interface as possible
driving force:
Curvature of surface negligible

20. Boundary layer – tangential stress
MARANGONI velocity

21. Boundary layer – tangential stress
Marangoni and buoyancy effects are driving forces.

22. Boundary layer – tangential stress Main criteria
Vr and l are not known

23. Conditional dimensionless numbers
Functions of the problems data

24. Criteria in terms of conditional numbers
Let assume N=1 (leading equation momentum, e.g. pure fluid with Pr <1)
Let assume St=0 (steady regime)
The solution is defined by the two conditional numbers:

25. Purely diffusive regimes
Let assume: D1
Similarly if Marangoni prevails: D2

26. Boundary layer regimes (buoyancy)
BL1

27. Boundary layer regimes (Marangoni)
BL2

28. OMA-Regime plane BL2 BL2 Marangoni boundary layer buoyancy1
Marangoni convective-diffusive
BL1 1 D2
BL1
Marangoni
diffusive D1
Buoyancy
convective-diffusive
Buoyancy diffusive