Biological fluid mechanics at themicro and nanoscale‐

Lecture 2:Some examples of fluid flows

Anne TanguyUniversity of Lyon (France)

Some reminderI.Simple flowsII.Flow around an obstacleIII.Capillary forcesIV.Hydrodynamical instabilities

REMINDER:

The mass conservation: , for incompressible fluid:

The Navier-Stokes equation:

with

Thus:

for an incompressible and Newtonian fluid.

with veIv.)3

2(e2 S for a « Newtonian fluid ».

Claude Navier1785-1836

Georges Stokes1819-1903

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

Different regimes:

(Boger, Hur, Binnington, JNFM 1986)Re = 5.7 10-4 Re = 1.25 10-2

Born: 23 Aug 1842 in Belfast, IrelandDied: 21 Feb 1912 in Watchet, Somerset, England

Re << 1 Viscous flow (microworld)

and

Re >> 1 Ex. perfect fluids (=0)

or transient response t<<tc ,at large scales L>Lc

vt

v 2 diffusive transport of momentum

needs a time to establish

tc=10-6 s (L=10-6m) tc=106s (L=1m)

2

c Lt

T

LcL Lc=0.1mm for w=20 HzLc=10m for w=20 000 Hz

Bernouilli relation when viscosity is negligeable (ex. Re >>1):

pot

2

Upvv2

v

t

v

Along a streamline (dr // v), or everywhere for irrotational flows ( ),

For permanent flow :

For « potential flows » ( with ) :

0v

v

csteUp2

vpot

2

csteUp2

v

t pot

2

Daniel Bernouilli1700-1782

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 !)

vv0

//v0

solidfluid surface solid the at general, in

surface solid

tension surface , with

21III R

1

R

1pp

I. Simple flows

Flow along an inclined plane:

Assume: a flow along the x-direction:

Continuity equation:

Boundary conditions:

xx e)z,x(v)z,x(v

0v).v(),z(v0x

vx

then

0)hz(z

v0)hz,x('

P)hz,x(P

0)0z(v

0

)hz(cosgP)z(Pcosgz

P

)2/zh(zsing

)z(v)z(z

vsing

x

P

0

2

2

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).

3

hLsingL.dz)z(vQ

3y

2h

0z

y

)z(P

0

hsingLL

n.LLFyx

yx

Planar Couette flow:

Assume: a flow along the x-direction:

Continuity equation:

Boundary conditions:

xx e)z,x(v)z,x(v

0v).v(),z(v0x

vx

then

xeU)hz(v

0)0z(v

)z,Lx(P)z,0x(P

gradient pressure no

gzP)z(Pgz

Ph

zU)z(v)z(

z

v

x

P

0

2

2

Navier-Stokes equation:

Force applied on the upper plane: Fx=106 Pa U=1 m.s-1 h=1 nmsurface unit per h

UFx

Cylindrical Couette flow:

cst)r(rv0)rv(rr

1

r

v

r

vrr

rr

Assume: symetry around Oz + no pressure gradient along Oz:

Continuity equation:

Boundary conditions:

)r(pe)r(ve)r(v)z,,r(v rr and

0)r(rv r

e)R(v

e)R(v

22

11

0 rr

2

r22

2

2

r

v

r

v

r

1

r

v0

r

P1

r

v

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:

d/)()R(.R.R2.dz

r

1.

RR

RR2

r

v

r

v

121r

z

11Oz/1

221

22

22

2121

r

torque

Planar Poiseuille flow:

Assume: a flow along the x-direction:

Continuity equation:

Boundary conditions:

xx e)z,x(v)z,x(v

0v).v(),z(v0x

vx

then

0)hz(v

0)0z(v

)z,Lx(P)z,0x(PP

gradient Pressure

x.L

P)0(P)x(P0

z

P

)hz.(z.L2

P)z(v)z(

z

v

x

P2

2

Navier-Stokes equation:

Flow rate

small Force exerted on the upper plane: surface unit per h.L2

PFx

3

x

yxy h.

L.12

L.Pdz)z(vLQ

z

Poiseuille flow in a cylinder (Hagen-Poiseuille):

)r(v0z

vz

z

Assume: flow along Oz+ rotational invariance:

Continuity equation:

Boundary conditions:

zz e)z,r(v)z,,r(v

Gradient Pressure P)Lz(P)0z(P

0),Rr(v

gz

P

r

v

r

1v

r

1

r

v

)z(P0r

P

r

P

z2z

2

22z

2

Navier-Stokes equation:

)rR.(L4

gLPv 22

)r(z

Flow rate: 4R..L8

gLPQ

Friction force:

Total pressure force:

2z R.)gLP(F

L.R.2PFr

Jean-Louis Marie Poiseuille1797-1869

(1842)

(2010)

Rheological properties of bloodElasticity of the vesselBifurcationsThickeningNon-stationary flow…

Other example of Laminar flow with Re>>1:Lubrication hypothesis (small inclination)

0)hy(v

P)Lx(p)0x(p,U)0y(v 0

Poiseuille + Couette

cf. planar flow with x-dependence

0y

P

y

v

x

P

y

v

vv).v(

2y

2

2x

2

2

and

1

h

)x(h

)x(h

L/Q1

h

)x(h

)x(h

U

hh

L6P)x(p

2

L2t

LL00

hh

hhU.Q/L

10

L0t rateflow the with

=1.2kg.m-3 =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

Flow above an obstacle: hydraulic swell

Mass conservation: U.h=U(x).h(x)

Bernouilli along a streamline close to the surface:

then

)x(e)x(hg2

)x(UPgh

2

UP 0

2

0

2

0

(I)

(II) Case (I): dU/dx(xm)=0 then U2(x)-gh(x)<0then U(x) and h(x)

Case (II): dU/dx(x) >0 then U2(xm)-gh(xm)=0then U(x) and h(x)

U2(x)-gh(x) <0 becomes >0low velocity of surfaces wavesHydraulic swell

1gh

UFroude

wavesnalgravitatio

velocity fluid

End of Part I.