Transcript Slide 1

Impact of Microdrops on Solids
James Sprittles & Yulii Shikhmurzaev
• Recent
experiments show that all current models of drop impact and
spreading are fundamentally flawed.
Typical microdrop simulation
(blue) compared to experiment
(black).
1 μs
• The error will be considerable for the micron scale drops
encountered in ink-jet printing.
• We are developing a universal computational platform,
implementing a new model, to describe such experiments.
Interface formation - qualitatively
Liquid
 lg
Fluid particles are advected through the contact line
from the liquid-gas to the liquid-solid interface.
Near the contact line the interface is out of
equilibrium and, notably, the surface tension takes
finite time/distance to relax to its new equilibrium
value – see figure 6.
Gas
U
Solid
 sl
6. Sketch of flow in the contact line region showing how
surface tension relaxes over a finite distance.
To model this process surface variables are introduced, the surface density

s
and the surface velocity
v s.
Governing equations
Problem formulation:
In the bulk:
θd
θd
2 μs
u
1
2
  u  0,
 u u   p  u
t

On free surfaces:
U
U, m/s
1. Sketch of a spreading droplet.
2. Contact angle-speed plot for 2 mm water droplets
impacting at different Weber number based on
impact speed (Bayer & Megaridis 06) .
4 μs
Failure of conventional models
e1
θd
Incompressible
Navier-Stokes
f
Kinematic
 v1s f  0
t
 p   n  [(u)  (u)* ]  n   1  n
f (r, t )=0
Normal and
tangential stress
n
e2
n
On liquid-solid interfaces:
 n  [u  (u)* ]  (I  nn)  12  2   (u ||  U|| )
s
s
Generalised Navier



s
2
2e
 (u  v 2 )  n 

Normal velocity
s
v 2||
 12 (u ||  U|| )   2 , v 2s   U 
 n  [(u)  (u)* ]  (I  nn)   1  0 Surface equation
of state:
s
s
s
s
1  1e
s
s
s 2  s



`Darcy type’ eqn.
s s
 (u  v1 )  n 
2
2e
2
1,2  a1,2  b( 1,2 )



(

v
)


2 2

t

s
s



1s
Surface mass
1e
At contact lines:
   ( 1s v1s )   1
continuity
t

s s
s s
Mass
balance

v

e


s
1 1
1
2 v2  e2  0
(1  4 ) 1  4 (v1||  u || )
1 cos  d   3   2
Young equation
All existing models are based on the contact angle being a function of the contact line speed and material properties:
d  f (U ,  ,  ,..)
Computational
The experimental investigations of both Bayer & Megaridis 06 and Sikalo et al 02 using millimetre sized drops have shown
this assumption incorrect – see figure 2.
10μs
Additionally, conventional models predict an infinite pressure at the contact line and/or the
incorrect kinematics there, see Shikhmurzaev 2007.
We are developing a multi-purpose finite element code extending the spine
method devised by Ruschak then improved by Scriven and co-workers.
This is capable of simulating flows where the standard approach fails, such as
pinch-off of liquid drops and coalescence of drops.
Results from a drop impact and spreading simulation are shown in figure 5.
A snapshot of the finite element mesh during a simulation can be seen in figure 7.
θd
7. Computational mesh of nodes on
triangular elements.
Extensions
θd
20μs
3. Curtain coating geometry in a frame moving
with the contact line.
Substrates of Variable Wettability
Such chemically altered solids are naturally incorporated into the IFM and can have a considerable impact on the flow field
(Sprittles & Shikhmurzaev 07).
U, cm/s
U
The development mode provides a conceptual framework for additional physical/chemical effects including thermal effects and
contact angle hysteresis.
4. Dynamic contact angle as a function of coating speed for
different flow rates (Blake & Shikhmurzaev 02).
Porous Substrates
One of the many possible generalisations of our work is the extension of the solid from impermeable to porous.
Contact angle dependence on the flow field
Experiments of Blake & Shikhmurzaev 02 and Clarke & Stattersfield 06 demonstrated that in curtain coating the contact angle is
dependent on the flow field and, in particular, the flow rate – see figure 4.
The only model to predict this effect is the Interface Formation Model (IFM), see Shikhmurzaev 2007, which we are applying to
drop impact and spreading.
5. Qualitative agreement between a simulation
using the conventional model with experiments of
Dong 06.
References
Water drops of radius 25 microns at impact speed
12.2m/s with equilibrium contact angle of 88
degrees.
Bayer & Megaridis, J. Fluid Mech., 558, 2006.
Blake & Shikhmurzaev, J. Coll. Int. Sci., 253, 2002.
Clarke & Stattersfield, Phy. Fluids, 18, 2006.
Dong, PhD, 2006.
Shikhmurzaev, Capillary Flows with Forming Interfaces, 2007.
Sikalo et al, Exper. Therm. Fluid Sci., 25, 2002.
Sprittles & Shikhmurzaev, Phy. Rev. E., 76, 2007.