OMA rational and results

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

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;

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

Navier Stokes equations (2) For hydrostatic conditions:

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:

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

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

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

Adimensionalization – energy Energy diffusion velocity Prandtl number

Adimensionalization – species Concentration diffusion velocity Schmidt number

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

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

Summary of dimensionless numbers in terms of characteristic speeds

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

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

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.

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

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

Boundary layer – tangential stress MARANGONI velocity

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

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

Conditional dimensionless numbers Functions of the problems data

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:

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

Boundary layer regimes (buoyancy) BL1

Boundary layer regimes (Marangoni) BL2

OMA-Regime plane BL2 BL2 Marangoni boundary layer buoyancy 1 Marangoni convective-diffusive BL1 1 D2 BL1 Marangoni diffusive D1 Buoyancy convective-diffusive Buoyancy diffusive