Fluid Mechanics 1978 vol 84 pp

Document Sample

On the motion of a small viscous
droplet that wets a surface

H.P.Greenspan
J. Fluid Mechanics 1978 vol 84 pp. 125-143
Overview

●   Introduction
●   Formulation
●   Procedure
●   Spreading and Retraction of Droplets
●   Creeping motion of a Droplet on a coated
surface
●   Droplet distortion due to surface
contamination
Introduction
●   Why study drops that wet surfaces?
●   Thesis statement of paper: consider
plane surface of a very small, very viscous
droplet.
Problem formulation
●   Static vs. Dynamic contact angle
●   Velocity of fluid particle at boundary (first eq. of paper)

‘
qe = À ¾ ¾ n
s

●   Differential equation for the pts. on the boundary (q)
●   Lubrication theory:
–   Depth-averaged mass conservation
–   Simplified form of Navier-Stokes for thin layer of very
viscous fluid
–   Volume conservation
–   Boundary conditions
Problem formulation (cont)

●   Equation for thickness of droplet with moving boundary
●   Non-dimensionalize:
–   PDE for height, h
–   With moving boundary at h=0
–   I.C.
–   Constant volume
Procedure
●    Express h and other dependent variables as power series
in epsilon. Keep only lowest order.
●   To kill off highest order term in expansion for boundary:
2
h0 = 0
or
2
h0 =      A t
●   Bottom line:
●   Possion's equation
●   h=0 on moving boundary
●   Conservation of volume
droplets
●   Circular Drop
–   Solve Poisson's
equation for h
–   Get an ODE for a(t), the
–   Order of magnitude:
compared to
of an earlier analysis
Creeping Motion of a Droplet on a
coated surface
●   Idea of a non-uniform coating causing motion
of the droplet
●   Let equilibrium contact angle be a function of
position on the plane, = ¾ x , y
¾                in this
case          ¾x = ¾ 1       Áx
s

●   Drop initially circular and problem is to
determine motion and shape on this surface.
●   Hypothesis: droplet is always circular and
Creeping Motion of a droplet on a
coated surface (continued)
●   New coordinate system (
,y):
●   In cylindrical coordinates,
completely describe            and
shape, position, and         length
velocity of the droplet at
any time. (not solved yet)
●   Two time scales:
moving                                time
Creeping motion (continued)
●   Get equations for the radius and the distance
drop travels
●   For small gradients of the equilibrium contact
angle the velocity of the drop will be opposite
Droplet distortion due to surface
contamination

●   Idea: droplet is drawn to regions of greater
adherence and retracts from regions of weaker
●   Express variations in surface adherence by theta
●   Solve Poisson's equation in cylindrical coordinates
●   ODE for the radius and conservation of volume
●   This leads to perturbation analysis and eventually
an expression for h
Droplet distortion due to surface
contamination (continued)
●   Specific example: unit droplet (a=1) on a
® x ,y
surface with = sin kx sin ky
●   Distortion of droplet depends on the
magnitude of the length scale of the surface
variations compared with the radius of the
droplet
Summary and major conclusions

contact angle to express the forces on the
fluid at the moving contact line.
2) Concept of dynamic contact angle
meaningful and well-founded.
3) Spreading and retraction are stable
processes; i.e. a circular contact line always
evolves in time.
Summary and major conclusions
(part two)

4) A creeping droplet expands only slightly
when creeping several diameters from its
original position.
5) Surface contamination on scale of drop
diameter is most effective in distorting
surface and contact line.

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