1. Biological fluid mechanics at the micro‐ and nanoscale Lectures 3: Fluid flows and capillary forces Anne Tanguy University of Lyon (France)

2. Some reminder I.Simple flows II.Flow around an obstacle III.Capillary forces IV.Hydrodynamical instabilities Lecture 3

3. II. Flow around an obstacle

4. The case of « Potential flows »

5. « Potential flow » around a fixed cylinder: Stream lines Pressure Boundary conditions: Potential  : Velocity v: Uniform flow + dipole U

6. « Potential flow » around a rotating cylinder: the Magnus force. Boundary conditions: Potential  : Velocity v: Fixed cylinder + vortex Asymetric flow: arrest points If |  |<4  R|U|sin  =  /4  RU, r=R. Else r=r P >R. Magnus Force Fz=-  U  =-∫P(R).sin  Rd  on the solid. No viscous dissipation (no drag force).  U Stream lines PressureForce

7. Air foil / birds wing Conformal mapping Joukowski’s transform: Z= g(z) = Stream linesPressureForce

8. Perfect potential flow around a sphere: Spherical coordinates Uniform flow + 3D dipole velocity decrease ~1/r 3. Viscous flow around a sphere: the Stoke’s force Boundary conditions: Low velocity decrease ~1/r. Stoke’s force.

9. II. Capillary forces Surface tension

10. Definition of capillary forces. At the interface between different phases/different chemical composition Effective force insuring the equilibrium Energy per unit surface: , « surface tension » « capillary force »

11. Water  20°C  =72.8 mN/mEthanol  20°C  =22.10 mN/m Molecular Dynamics Simulations at constant T and V (L. Joly, LPMCN) cf. lecture 7 Examples: Liquid/vapor interface

12. Comparison with gravitational forces: h area AV=h.A Total Energy: E interfaces ≈ A.(  LV +  SL -  SV ) E gravity ≈ 0.5 .g.h 2.A E gravity >> E interfaces for h>> lc « capillary length » l c =2,7 mm for water et 20°C

13. Examples: Liquid/solid interfaces (without gravity) Contact angle  0<  <90°: liquid is « partially wetting » 90°<  : liquid is « non wetting »  =0°: « complete wetting »

14. 3

15. Effect of the curvature on the pressure: Laplace’s law Δp for water drops of different radii at STPSTP Droplet radius1 mmmm0.1 mm1 μmμm10 nmnm Δp (atm)atm0.00140.01441.436143.6 11 11 (1749-1827)

16. Example: Alveoli of the lungs R ≈ 50  m  P≈2,8.10 3 Pa if water.  P smaller with a surfactant  ≈ 5 to 45.10 -3 N.m -1 Allows a common work of all the alveoli. Else: PBPB PCPC P B > P C. The small bubble will lose air

17. Example 2: droplet between 2 plates. R r  LV =70 mN.m -1  =130° V=10 -1 cm 3 h= 100  m F P = 0,95 N F C = 6,25.10 -3 N h= 1  m F P = 9500 N ! F C = 6,25.10 -2 N h= 1 nm F P = 95.10 8 N !! F C = 1,98 N E. Csapo (2007)

18. Example 3: ascent of a liquid in a thin tube (d<l c ). Jurin’s law For water at 20°C with  =0° R=1mm h=1,46 cm R=10  m h=1,46 m R=1  m h=14,6 m !

19. Sap and trees:

20. Example 4: Shape of the Meniscus in a free surface x z P ext

21. Interactions between 2 plates: I.II.III. Vertical Capillary forces:T = -2. .cos .L e z Horizontal Pressure forces:F P = ∫P(z).dz.L e x = 0.5  gL.[ h 2 2 (o)-h 1 2 (0)] e x with boundary conditions: I.h 1 (-∞)=0h 1 ’(x=0)=cotan  1 II.h2’(x=0)=-cotan  1 h2’(x=d)=cotan  2 III.h3’(x=d)=-cotan  2 h3(+∞)=0

22. I.II.III. If  1=  2 If d<<l c F P ≈ 2 .l c 2.L.(cotan  1 ) 2 /d 2 e x If d>>l c F P ≈ 2 .L.(cotan  1 ) 2.exp(-d/l c ) e x ≈ -T. cotan  1.exp(-d/l c ) e x Attractive forces (either for wetting or non-wetting surfaces) If cotan  1.cotan  2<0 If d << l c F P ≈ 2 .l c 2.L.(cotan  1 + cotan  2 ) 2 /d 2 e x Attractive Force If d=d* F P ≈ - 0.5 .L.(cotan  max ) 2 e x Max. Repulsive Force If d > d* F P <0 Repulsive Force at large distances. d*=l c.acosh(|cotan  min /cotan  max |) wetting Non wetting

24. Beetle Larva

25. III. Related instabilities

26. The Marangoni effect: Effect of boundary conditions Gradient of surface tension on the upper free surface (cf. lecture 3) Ex. Temperature gradient // surface, Chemical gradient (soap on water, Tears of wine: alcohol in water) Navier-Stokes equation: Motion in the direction of larger surface tension (flow from alcohol to water, hot places to cold places..)

28. The Bénard-Marangoni instability: Local gradient of temperature (cf. Marangoni) Flow due to coupling between T and v Fourier’s law (cf. lecture 4) Marangoni number:

29. The Taylor-Couette instability: (Couette 1921, Taylor 1923) Volumic competition between inertia and viscous forces when motion is driven by the internal cylinder. Taylor number:

30. Next lecture: From Liquid to Solid, Rheological behaviour (Lecture 6)