Fluid Mechanics 1978 vol 84 pp

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Fluid Mechanics 1978 vol 84 pp Powered By Docstoc
					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
    adhesion, spreading and movement on a
    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
         Spreading and retraction of
                 droplets
●   Circular Drop
     –   Radial motion only
     –   Solve Poisson's
         equation for h
     –   Get an ODE for a(t), the
         drop radius
     –   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,
    get equations which         radius
    completely describe            and
    shape, position, and         length
    velocity of the droplet at
    any time. (not solved yet)
●   Two time scales:
    spreading vs. entire drop
    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
    but proportional to the gradient
    Droplet distortion due to surface
             contamination

●   Idea: droplet is drawn to regions of greater
    adherence and retracts from regions of weaker
    adherence
●   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

1) Start with lubrication equations and dynamic
  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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