Biological fluid mechanics at the micro‐ and nanoscale Lecture 2: Some examples of fluid flows

1. Biological fluid mechanics at the micro‐ and nanoscale Lecture 2: Someexamples of fluidflows Anne Tanguy University of Lyon (France)

2. Somereminder • Simple flows • Flow around an obstacle • Capillary forces • Hydrodynamicalinstabilities

3. REMINDER: The mass conservation: , for incompressible fluid: The Navier-Stokes equation: with Thus: for an incompressible and Newtonian fluid. for a « Newtonian fluid ». Claude Navier 1785-1836 Georges Stokes 1819-1903

4. (Giesekus, Rheologica Acta, 68) Non-Newtonian liquid

5. Different regimes: Born: 23 Aug 1842 in Belfast, Ireland Died: 21 Feb 1912 in Watchet, Somerset, England Re = 5.7 10-4 Re = 1.25 10-2 (Boger, Hur, Binnington, JNFM 1986) Re << 1 Viscous flow (microworld) and Re >> 1 Ex. perfect fluids (h=0) or transient response t<<tc ,at large scales L>Lc Lc=0.1mm for w=20 Hz Lc=10mm for w=20 000 Hz diffusive transport of momentum needs a time to establish tc=10-6 s (L=10-6m) tc=106s (L=1m)

6. Bernouilli relation when viscosity is negligeable (ex. Re >>1): Along a streamline (dr // v), or everywhere for irrotational flows ( ), For permanent flow : For « potential flows » ( with ) : Daniel Bernouilli 1700-1782

7. How solve the Navier-Stokes equation ? • Non-linear equation. Many solutions. • Estimate the dominant terms of the equation (Re, permanent flow…) • Do assumptions on the geometry of the flows (laminar flow …) • Identify the boundary conditions (fluid/solid, slip/no slip, fluid/fluid..) • Ex. Fluid/Solid: rigid boundaries • (see lecture 5 !) • Ex. Fluid/Fluid: soft boundaries • (see lecture 3 !)

8. I. Simple flows

9. Flow along an inclined plane: Assume: a flow along the x-direction: Continuity equation: Boundary conditions: Navier-Stokes equation:

10. Flow along an inclined plane: Flow rate: test for rheological laws Force applied on the inclined plane: Friction and pressure compensate the weight of the fluid (stationary flow).

11. Planar Couette flow: Assume: a flow along the x-direction: Continuity equation: Boundary conditions: Navier-Stokes equation: Force applied on the upper plane: Fx=106 Pa U=1 m.s-1 h=1 nm

12. Cylindrical Couette flow: Assume: symetry around Oz + no pressure gradient along Oz: Continuity equation: Boundary conditions: radial gradient compensates radial inertia Navier-Stokes equation: no torque

13. Cylindrical Couette flow: Friction force on the cylinders: Couette Rheometer: Rotation is applied on the internal cylinder, to limit vq . Taylor-Couette instability:

14. Planar Poiseuille flow: z Assume: a flow along the x-direction: Continuity equation: Boundary conditions: Navier-Stokes equation: Flow rate small Force exerted on the upper plane:

15. Poiseuille flow in a cylinder (Hagen-Poiseuille): Assume: flow along Oz+ rotational invariance: Continuity equation: Boundary conditions: Navier-Stokes equation: Friction force: Total pressure force: Flow rate:

16. Jean-Louis Marie Poiseuille 1797-1869 (1842)

17. Rheological properties of blood Elasticity of the vessel Bifurcations Thickening Non-stationary flow… (2010)

18. Other example of Laminar flow with Re>>1: Lubrication hypothesis (small inclination) cf. planar flow with x-dependence Poiseuille + Couette

20. r=1.2kg.m-3h=2.10-5 Pa.s L ~ 1m, h ~ 1 cm, U ~ 0.1m/s Re ~ 6000< (L/h)2 = 10000 xM ~ e1.L/h ~ 10 cm Supporting pressure PM ~ 10-1Pa

22. Flow above an obstacle: hydraulic swell Mass conservation: U.h=U(x).h(x) Bernouilli along a streamline close to the surface: then (I) (II) Case (I): dU/dx(xm)=0 then U2(x)-gh(x)<0 then U(x) and h(x) Case (II): dU/dx(x) >0 then U2(xm)-gh(xm)=0 then U(x) and h(x) U2(x)-gh(x) <0 becomes >0 low velocity of surfaces waves Hydraulic swell

24. End of Part I.