Diapositive 1

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Biological fluid mechanics at the micro- and
nanoscale Lecture 2 Some examples of fluid
flows Anne Tanguy University of Lyon (France)
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• Some reminder
• Simple flows
• Flow around an obstacle
• Capillary forces
• Hydrodynamical instabilities

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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
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(Giesekus, Rheologica Acta, 68)
Non-Newtonian liquid
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Different regimes
Lc0.1mm for w20 Hz Lc10mm for w20 000 Hz
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Bernouilli relation when viscosity is negligeable
(ex. Re gtgt1)
Daniel Bernouilli 1700-1782
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• 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 !)

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I. Simple flows
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Flow along an inclined plane
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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).
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Planar Couette flow
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Cylindrical Couette flow
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Cylindrical Couette flow
Friction force on the cylinders
Couette Rheometer Rotation is applied on the
internal cylinder, to limit vq .
Taylor-Couette instability
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Planar Poiseuille flow
z
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Poiseuille flow in a cylinder (Hagen-Poiseuille)
Flow rate
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Jean-Louis Marie Poiseuille 1797-1869
(1842)
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Rheological properties of blood Elasticity of the
vessel Bifurcations Thickening Non-stationary
flow
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Other example of Laminar flow with
Regtgt1 Lubrication hypothesis (small inclination)
cf. planar flow with x-dependence
Poiseuille Couette
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r1.2kg.m-3 h2.10-5 Pa.s L 1m, h 1 cm, U
0.1m/s Re 6000lt (L/h)2 10000 xM e1.L/h
10 cm Supporting pressure PM 10-1Pa
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Flow above an obstacle hydraulic swell
Mass conservation U.hU(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)lt0 then U(x)
and h(x) Case (II) dU/dx(x) gt0 then
U2(xm)-gh(xm)0 then U(x) and h(x) U2(x)-gh(x)
lt0 becomes gt0 low velocity of surfaces
waves Hydraulic swell
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End of Part I.