Potential flow
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Transcript Potential flow
Biological fluid mechanics at the
micro‐ and nanoscale
Lectures 3:
Fluid flows and capillary forces
Anne Tanguy
University of Lyon (France)
Lecture 3
Some reminder
I. Simple flows
II.Flow around an obstacle
III.Capillary forces
IV.Hydrodynamical instabilities
II. Flow around an obstacle
The case of « Potential flows »
incompress ible fluid.
2 v
is negligeable.
if v ( r , t 0 ) 0 then v ( r , t ) 0 always
v ( r , t ) 0 v ( r , t ) " Potential flow"
Mass conservati on : .v 0 2 0 (Laplace' s equation)
v2
Bernouilli relation :
P U pot cst
t
2
Boundary conditions ?
U
« Potential flow » around a fixed cylinder:
Boundary conditions:
Potential :
lim r v U
v r ( r R , ) 0
R2
p cos
0 ( r, ) U.r. cos
U.r. cos .1 2
2 .r
r
2
Uniform flow + dipole
Stream lines
Velocity v:
Pressure
R2
vr
U. cos .1 2
r
r
R2
v
U. sin .1 2
r
r
« Potential flow » around a rotating cylinder: the Magnus force.
U
Boundary conditions:
lim r v U
v.d r 2R.R
r R
Potential :
Velocity v:
Stream lines
vorticity
R 2
2 0 ( r, ) U.r. cos .1 2
r 2
Fixed cylinder + vortex
R2
vr
U. cos .1 2
r
r
R2
v
U. sin .1 2
r
r 2r
Asymetric flow: arrest points
If ||<4R|U|
sin=/4RU, r=R.
Else r=rP>R.
Magnus Force Fz=-U=-∫P(R).sin.Rd
on the solid.
No viscous dissipation (no drag force).
Pressure
Force
Air foil / birds wing
Conformal mapping
Joukowski’s transform:
Z= g(z) =
f (z ) known
F( Z) f (g 1 ( Z))
Stream lines
Pressure
Force
Perfect potential flow around a sphere:
Spherical coordinates
2 0 ( r , ) U.r. cos
p cos
4 .r 2
Uniform flow + 3D dipole
R3
vr
U. cos .1 3
r
r
velocity decrease ~1/r3.
R3
v
U. sin .1 3
r
2r
Viscous flow around a sphere: the Stoke’s force
Navier - Stokes equation P 2 v
with .v 0 then P ( v ) and thus 2 P 0.
Boundary conditions: lim r v 0 and v ( r R , ) U sphere
3R R 3
v r U cos
2 r 2 r 3
3
cos
P( r , ) UR 2
3R R 3
2
r
v U sin
3
4
r
4r
( r , ) P I 2 v
Low velocity decrease ~1/r.
cos
R F .ndS 6 R U
sphere
3
sin
r ( r R , ) U
2
R
3
2
rr ( r R , ) P U
Stoke’s force.
II. Capillary forces
Surface tension
Definition of capillary forces.
At the interface between different phases/different chemical composition
Effective force insuring the equilibrium
Energy per unit surface: , « surface tension »
E .A
F.dh dE .L.dh
F .L
« capillary force »
Examples: Liquid/vapor interface
Molecular Dynamics Simulations
at constant T and V
(L. Joly, LPMCN)
cf. lecture 7
Water (20°C)=72.8 mN/m
Ethanol (20°C)=22.10 mN/m
Comparison with gravitational forces:
area A
h
V=h.A
Total Energy:
Einterfaces≈ A.(LV+SL-SV)
Egravity ≈ 0.5 .g.h2.A
Egravity >> Einterfaces for h>> lc
lc
.g
« capillary length »
lc=2,7 mm for water et 20°C
Examples: Liquid/solid interfaces (without gravity)
Equilibrium of capillary forces :
SV .dl SL .dl LV . cos .dl 0
LV . cos ( SV SL )
You ng' s Law
possible if SV SL LV .
Contact angle
0<<90°: liquid is « partially wetting »
90°<: liquid is « non wetting »
=0°: « complete wetting »
3
Effect of the curvature on the pressure:
Laplace’s law
Equilibrium of forces : pressure surface tension
2 .dl 2 . sin( 1 ) 2 .dl1. sin( 2 ) Pint .dl1.dl 2 Pext .dl1.dl 2 0
f1
f1
sin( 1 ) 1
dl1
2R1
sin( 2 ) 2
dl 2
2R 2
1
1
0
Pint Pext .
R1 R 2
Laplace' s Law
(1749-1827)
Δp for water drops of different radii at STP
Droplet radius
1 mm
0.1 mm
1 μm
10 nm
Δp (atm)
0.0014
0.0144
1.436
143.6
Example: Alveoli of the lungs
R ≈ 50 mm DP≈2,8 .103 Pa if water.
DP smaller with a surfactant
≈ 5 to 45.10-3 N.m-1
Allows a common work of all the alveoli.
Else:
PB
PC
PB > PC . The small bubble will lose air
Example 2: droplet between 2 plates.
r
R
V π.a 2.h
Pressure :
1 1
Pint Pext LV .
R r
1 2 cos V. LV
Fpressure .a 2 . LV .
h
h2
R
Vertical component of the capillary force :
Fcapillary 2 .a. LV . sin LV
E. Csapo (2007)
V
h
LV=70 mN.m-1 =130° V=10-1 cm3
h= 100 mm FP= 0,95 N
FC= 6,25.10-3N
h= 1 mm FP= 9500 N ! FC= 6,25.10-2N
h= 1 nm FP= 95.108 N !! FC= 1,98 N
Example 3: ascent of a liquid in a thin tube (d<lc). Jurin’s law
Assume a spherical shape for the meniscus (r lC )
PA P0
2
(Laplace' s Law)
r
PD PB .g.h (static equilibrium of the fluid)
PB PA
PD PE P0
h
2.
2. . cos
.g.r
.g.R
Jurin' s Law
For water at 20°C with =0°
R=1mm h=1,46 cm
R=10mm h=1,46 m
R=1mm h=14,6 m !
Sap and trees:
Example 4: Shape of the Meniscus in a free surface
Pliquid( z ( x )) Pext .g.z ( x ) and Pliquid( z ( x )) Pext
z
r(x)
2
l
z(x ) c
r(x)
with lc
Pext
capillary length
g
dz 2
1
dx
and r ( x )
2
dz
dx 2
x
3/ 2
local radius of curvature of the interface
2
dz
2 d z
if
1 then z ( x ) l c . 2
dx
dx
x
z ( x ) lc . cot an . exp
lc
Interactions between 2 plates:
d 2 h( x )
h( x ) lc .
dx 2
2
I.
II.
III.
with boundary conditions:
I. h1(-∞)=0
II. h2’(x=0)=-cotan1
III. h3’(x=d)=-cotan2
h1’(x=0)=cotan1
h2’(x=d)=cotan2
h3(+∞)=0
x
h1 ( x ) lc . cot an1. exp
lc
dx
x 2lc 2
lc
cot an 2 . cosh
h2 (x)
.cot an1. cosh
cos (Jurin) if d lc
l
l c
d
d
c
sinh
lc
dx
h3 ( x ) lc . cot an 2 . exp
l
c
Vertical Capillary forces:
T = -2..cos.L ez
Horizontal Pressure forces:
FP = ∫P(z).dz.L ex = 0.5 gL.[ h22(o)-h12(0)] ex
If 1=2
I.
II.
III.
If d>>lc
If d<<lc FP ≈ 2 .lc2.L.(cotan1)2/d2 ex
FP ≈ 2 .L.(cotan1)2.exp(-d/lc) ex ≈ -T. cotan1.exp(-d/lc) ex
If cotan1.cotan2<0
Non wetting
wetting
Attractive forces (either for wetting or non-wetting surfaces)
If d << lc FP ≈ 2 .lc2.L.(cotan1+ cotan2)2/d2 ex Attractive Force
If d=d*
FP ≈ - 0.5 .L.(cotanmax)2 ex
If d > d* FP <0
Max. Repulsive Force
Repulsive Force at large distances.
d*=lc.acosh(|cotanmin/cotanmax|)
Beetle Larva
III. Related instabilities
The Marangoni effect:
Effect of boundary conditions
Gradient of surface tension on the upper free surface (cf. lecture 3)
S xz ( x, z h )
d
d
v
viscous
( x ) xz
( x, z h )
( x ) x ( x, z h ) 0
dx
dx
z
Ex. Temperature gradient // surface,
Chemical gradient (soap on water, Tears of wine: alcohol in water)
2
Navier-Stokes equation: v2x ( x, z ) 0
z
v x ( x, z )
d
.z
dx
Motion in the direction of larger surface tension
(flow from alcohol to water, hot places to cold places..)
The Bénard-Marangoni instability:
Local gradient of temperature (cf. Marangoni)
Flow due to coupling between T and v
T
Fourier’s law
(v. )T 2T
(cf. lecture 4)
t
d
.DT.h
dt
Marangoni number: Ma
motion
Ma c
dissipation
The Taylor-Couette instability:
(Couette 1921, Taylor 1923)
Volumic competition between inertia
and viscous forces
when motion is driven by the internal cylinder.
Taylor number:
12 . R .( R 2 R1 )3
inertia
Ta
Ta c
dissipation
( / )2
Next lecture:
From Liquid to Solid,
Rheological behaviour
(Lecture 6)