Lecture presented at the workshop "Complex Systems in Natural and Social Sciences" (CSNSS’99), 14-17 September 1999, Kazimierz Dolny, Poland

How do we respire?

 

Jaros³aw Bialecki, Janusz Holyst

Faculty of Physics

Warsaw University of Technology

fig.1. Human lung

 

 

fig.2. Lung bubble

 

 

fig.4. Experimental data transport of oxygen versus oscillation frequency of surface tension

OUR MAIN QUESTION

Is the transport of oxygen in the lung bubbles caused only by diffusion or also by micro-vortices?

 

fig.3. Experimental system for investigation phenomena in water-surfactant layer

 

 

 

 

Internal surfaces of lung bubbles are covered with a thin layer of water and a monoatomic layer of a so called surfactant (surface active agent), decreasing the surface tension of water. A simple experimental system (see above) allows „in vitro" investigations of a phenomenon in the layer of water and surfactant. One can measure the dependence of oxygen transport across the layer of water against the frequency of surface tension oscillation caused by movement of the barrier. For a certain value of frequency the transport of oxygen is maximal. For all frequencies the transport is larger in the presence of the surfactant than without the surfactant.

We have suspected, that the increase of oxygen transport could be caused by micro-vortices in the layer of water. The shape of the vortices could be similar to the Benard vortices.

A mathematical model of the system has been created: Navier-Stokes equation plus boundary conditions for surface tension. An analytical form of a stationary solution has been found. Adding small disturbance to this solution we have investigated its stability.

We have showed that for certain frequencies and certain wave numbers of disturbance the stationary solution is unstable against vortices creation. Our theoretical results are in a good agreement with experimental data. According to our calculations the critical Reynolds number should be R=18 while the experiments indicated the sudden increase of the oxygen transport for R=12.

fig.5. Parametr of stability of the stationary solution Im(w) against wave number of disturbance k and velocity of surface motion Vo. White area denotes instability of stationary solution.

 

 

 

 

fig.6. Benard vortices

 

 

fig.7. Theoretical shape of vortices for critical set of parameters. Picture (a) shows current function, picture (b) shows vector of water velocity.

 

Conclusion

Vortices in our lung bubbles allow us to live.