Simulation and control of fluid flows around objects using computational fluid dynamics
Biological fluid mechanics at the micro‐ and nanoscale Lecture 2: Some examples of fluid flows
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Transcript of Biological fluid mechanics at the micro‐ and nanoscale Lecture 2: Some examples of fluid flows
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.