Biological fluid mechanics at the micro‐ and nanoscale Lecture 2: Some examples of fluid flows Anne Tanguy University of Lyon (France)

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

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 Georges Stokes

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

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

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

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 !)

I. Simple flows

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

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).

Planar Couette flow: Assume: a flow along the x-direction: Continuity equation: Boundary conditions: Navier-Stokes equation: Force applied on the upper plane: F x =10 6 Pa U=1 m.s -1 h=1 nm

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

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

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

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

Jean-Louis Marie Poiseuille (1842)

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

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

 =1.2kg.m -3  = Pa.s L ~ 1m, h ~ 1 cm, U ~ 0.1m/s Re ~ 6000< (L/h) 2 = x M ~ e 1.L/h ~ 10 cm Supporting pressure P M ~ Pa

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(x m )=0 then U 2 (x)-gh(x)<0 thenU(x) and h(x) Case (II): dU/dx(x) >0 then U 2 (x m )-gh(x m )=0 thenU(x)and h(x) U 2 (x)-gh(x) 0 low velocity of surfaces waves Hydraulic swell

End of Part I.